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---
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comments: true
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---
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# 13.1 回溯算法
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<u>回溯算法（backtracking algorithm）</u>是一种通过穷举来解决问题的方法，它的核心思想是从一个初始状态出发，暴力搜索所有可能的解决方案，当遇到正确的解则将其记录，直到找到解或者尝试了所有可能的选择都无法找到解为止。
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回溯算法通常采用“深度优先搜索”来遍历解空间。在“二叉树”章节中，我们提到前序、中序和后序遍历都属于深度优先搜索。接下来，我们利用前序遍历构造一个回溯问题，逐步了解回溯算法的工作原理。
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!!! question "例题一"
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给定一棵二叉树，搜索并记录所有值为 $7$ 的节点，请返回节点列表。
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对于此题，我们前序遍历这棵树，并判断当前节点的值是否为 $7$ ，若是，则将该节点的值加入结果列表 `res` 之中。相关过程实现如图 13-1 和以下代码所示：
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=== "Python"
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```python title="preorder_traversal_i_compact.py"
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def pre_order(root: TreeNode):
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"""前序遍历：例题一"""
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if root is None:
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return
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if root.val == 7:
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# 记录解
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res.append(root)
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pre_order(root.left)
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pre_order(root.right)
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```
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=== "C++"
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```cpp title="preorder_traversal_i_compact.cpp"
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/* 前序遍历：例题一 */
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void preOrder(TreeNode *root) {
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if (root == nullptr) {
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return;
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}
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if (root->val == 7) {
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// 记录解
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res.push_back(root);
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}
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preOrder(root->left);
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preOrder(root->right);
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}
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```
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=== "Java"
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```java title="preorder_traversal_i_compact.java"
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/* 前序遍历：例题一 */
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void preOrder(TreeNode root) {
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if (root == null) {
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return;
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}
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if (root.val == 7) {
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// 记录解
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res.add(root);
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}
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preOrder(root.left);
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preOrder(root.right);
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}
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```
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=== "C#"
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```csharp title="preorder_traversal_i_compact.cs"
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/* 前序遍历：例题一 */
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void PreOrder(TreeNode? root) {
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if (root == null) {
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return;
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}
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if (root.val == 7) {
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// 记录解
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res.Add(root);
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}
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PreOrder(root.left);
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PreOrder(root.right);
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}
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```
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=== "Go"
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```go title="preorder_traversal_i_compact.go"
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/* 前序遍历：例题一 */
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func preOrderI(root *TreeNode, res *[]*TreeNode) {
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if root == nil {
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return
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}
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if (root.Val).(int) == 7 {
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// 记录解
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*res = append(*res, root)
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}
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preOrderI(root.Left, res)
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preOrderI(root.Right, res)
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}
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```
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=== "Swift"
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```swift title="preorder_traversal_i_compact.swift"
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/* 前序遍历：例题一 */
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func preOrder(root: TreeNode?) {
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guard let root = root else {
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return
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}
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if root.val == 7 {
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// 记录解
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res.append(root)
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}
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preOrder(root: root.left)
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preOrder(root: root.right)
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}
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```
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=== "JS"
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```javascript title="preorder_traversal_i_compact.js"
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/* 前序遍历：例题一 */
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function preOrder(root, res) {
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if (root === null) {
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return;
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}
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if (root.val === 7) {
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// 记录解
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res.push(root);
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}
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preOrder(root.left, res);
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preOrder(root.right, res);
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}
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```
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=== "TS"
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```typescript title="preorder_traversal_i_compact.ts"
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/* 前序遍历：例题一 */
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function preOrder(root: TreeNode | null, res: TreeNode[]): void {
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if (root === null) {
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return;
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}
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if (root.val === 7) {
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// 记录解
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res.push(root);
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}
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preOrder(root.left, res);
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preOrder(root.right, res);
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}
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```
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=== "Dart"
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```dart title="preorder_traversal_i_compact.dart"
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/* 前序遍历：例题一 */
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void preOrder(TreeNode? root, List<TreeNode> res) {
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if (root == null) {
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return;
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}
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if (root.val == 7) {
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// 记录解
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res.add(root);
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}
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preOrder(root.left, res);
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preOrder(root.right, res);
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}
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```
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=== "Rust"
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```rust title="preorder_traversal_i_compact.rs"
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/* 前序遍历：例题一 */
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fn pre_order(res: &mut Vec<Rc<RefCell<TreeNode>>>, root: Option<&Rc<RefCell<TreeNode>>>) {
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if root.is_none() {
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return;
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}
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if let Some(node) = root {
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if node.borrow().val == 7 {
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// 记录解
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res.push(node.clone());
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}
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pre_order(res, node.borrow().left.as_ref());
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pre_order(res, node.borrow().right.as_ref());
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}
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}
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```
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=== "C"
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```c title="preorder_traversal_i_compact.c"
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/* 前序遍历：例题一 */
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void preOrder(TreeNode *root) {
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if (root == NULL) {
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return;
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}
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if (root->val == 7) {
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// 记录解
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res[resSize++] = root;
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}
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preOrder(root->left);
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preOrder(root->right);
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}
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```
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=== "Kotlin"
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```kotlin title="preorder_traversal_i_compact.kt"
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/* 前序遍历：例题一 */
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fun preOrder(root: TreeNode?) {
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if (root == null) {
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return
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}
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if (root._val == 7) {
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// 记录解
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res!!.add(root)
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}
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preOrder(root.left)
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preOrder(root.right)
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}
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```
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=== "Ruby"
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```ruby title="preorder_traversal_i_compact.rb"
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### 前序遍历：例题一 ###
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def pre_order(root)
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return unless root
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# 记录解
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$res << root if root.val == 7
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pre_order(root.left)
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pre_order(root.right)
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end
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```
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=== "Zig"
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```zig title="preorder_traversal_i_compact.zig"
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[class]{}-[func]{preOrder}
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```
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??? pythontutor "可视化运行"
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<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%EF%BC%9A%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E5%A6%82%E6%9E%9C%E7%B4%A2%E5%BC%95%E8%B6%85%E5%87%BA%E6%95%B0%E7%BB%84%E9%95%BF%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%AF%B9%E5%BA%94%E7%9A%84%E5%85%83%E7%B4%A0%E4%B8%BA%20None%20%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E5%BD%93%E5%89%8D%E8%8A%82%E7%82%B9%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%80%92%E5%BD%92%E6%9E%84%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A0%91%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%EF%BC%9A%E4%BE%8B%E9%A2%98%E4%B8%80%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20if%20root.val%20%3D%3D%207%3A%0A%20%20%20%20%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E8%A7%A3%0A%20%20%20%20%20%20%20%20res.append%28root%29%0A%20%20%20%20pre_order%28root.left%29%0A%20%20%20%20pre_order%28root.right%29%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%0A%20%20%20%20res%20%3D%20list%5BTreeNode%5D%28%29%0A%20%20%20%20pre_order%28root%29%0A%0A%20%20%20%20print%28%22%5Cn%E8%BE%93%E5%87%BA%E6%89%80%E6%9C%89%E5%80%BC%E4%B8%BA%207%20%E7%9A%84%E8%8A%82%E7%82%B9%22%29%0A%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20res%5D%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=126&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%EF%BC%9A%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E5%A6%82%E6%9E%9C%E7%B4%A2%E5%BC%95%E8%B6%85%E5%87%BA%E6%95%B0%E7%BB%84%E9%95%BF%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%AF%B9%E5%BA%94%E7%9A%84%E5%85%83%E7%B4%A0%E4%B8%BA%20None%20%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E5%BD%93%E5%89%8D%E8%8A%82%E7%82%B9%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%80%92%E5%BD%92%E6%9E%84%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A0%91%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%EF%BC%9A%E4%BE%8B%E9%A2%98%E4%B8%80%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20if%20root.val%20%3D%3D%207%3A%0A%20%20%20%20%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E8%A7%A3%0A%20%20%20%20%20%20%20%20res.append%28root%29%0A%20%20%20%20pre_order%28root.left%29%0A%20%20%20%20pre_order%28root.right%29%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%0A%20%20%20%20res%20%3D%20list%5BTreeNode%5D%28%29%0A%20%20%20%20pre_order%28root%29%0A%0A%20%20%20%20print%28%22%5Cn%E8%BE%93%E5%87%BA%E6%89%80%E6%9C%89%E5%80%BC%E4%B8%BA%207%20%E7%9A%84%E8%8A%82%E7%82%B9%22%29%0A%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20res%5D%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=126&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 ></a></div>
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{ class="animation-figure" }
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<p align="center"> 图 13-1 在前序遍历中搜索节点 </p>
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## 13.1.1 尝试与回退
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**之所以称之为回溯算法，是因为该算法在搜索解空间时会采用“尝试”与“回退”的策略**。当算法在搜索过程中遇到某个状态无法继续前进或无法得到满足条件的解时，它会撤销上一步的选择，退回到之前的状态，并尝试其他可能的选择。
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对于例题一，访问每个节点都代表一次“尝试”，而越过叶节点或返回父节点的 `return` 则表示“回退”。
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值得说明的是，**回退并不仅仅包括函数返回**。为解释这一点，我们对例题一稍作拓展。
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!!! question "例题二"
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在二叉树中搜索所有值为 $7$ 的节点，**请返回根节点到这些节点的路径**。
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在例题一代码的基础上，我们需要借助一个列表 `path` 记录访问过的节点路径。当访问到值为 $7$ 的节点时，则复制 `path` 并添加进结果列表 `res` 。遍历完成后，`res` 中保存的就是所有的解。代码如下所示：
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=== "Python"
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```python title="preorder_traversal_ii_compact.py"
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def pre_order(root: TreeNode):
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"""前序遍历：例题二"""
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if root is None:
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return
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# 尝试
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path.append(root)
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if root.val == 7:
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# 记录解
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res.append(list(path))
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pre_order(root.left)
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pre_order(root.right)
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# 回退
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path.pop()
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```
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=== "C++"
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```cpp title="preorder_traversal_ii_compact.cpp"
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/* 前序遍历：例题二 */
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void preOrder(TreeNode *root) {
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if (root == nullptr) {
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return;
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}
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// 尝试
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path.push_back(root);
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if (root->val == 7) {
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// 记录解
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res.push_back(path);
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}
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preOrder(root->left);
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preOrder(root->right);
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// 回退
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path.pop_back();
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}
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```
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=== "Java"
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```java title="preorder_traversal_ii_compact.java"
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/* 前序遍历：例题二 */
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void preOrder(TreeNode root) {
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if (root == null) {
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return;
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}
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// 尝试
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path.add(root);
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if (root.val == 7) {
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// 记录解
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res.add(new ArrayList<>(path));
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}
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preOrder(root.left);
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preOrder(root.right);
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// 回退
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path.remove(path.size() - 1);
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}
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```
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=== "C#"
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```csharp title="preorder_traversal_ii_compact.cs"
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/* 前序遍历：例题二 */
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void PreOrder(TreeNode? root) {
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if (root == null) {
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return;
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}
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// 尝试
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path.Add(root);
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if (root.val == 7) {
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// 记录解
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res.Add(new List<TreeNode>(path));
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}
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PreOrder(root.left);
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PreOrder(root.right);
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// 回退
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path.RemoveAt(path.Count - 1);
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}
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```
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=== "Go"
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```go title="preorder_traversal_ii_compact.go"
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/* 前序遍历：例题二 */
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func preOrderII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {
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if root == nil {
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return
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}
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// 尝试
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*path = append(*path, root)
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if root.Val.(int) == 7 {
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// 记录解
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*res = append(*res, append([]*TreeNode{}, *path...))
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}
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preOrderII(root.Left, res, path)
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preOrderII(root.Right, res, path)
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// 回退
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*path = (*path)[:len(*path)-1]
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}
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```
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=== "Swift"
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```swift title="preorder_traversal_ii_compact.swift"
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/* 前序遍历：例题二 */
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func preOrder(root: TreeNode?) {
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guard let root = root else {
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return
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}
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// 尝试
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path.append(root)
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if root.val == 7 {
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// 记录解
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res.append(path)
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}
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preOrder(root: root.left)
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preOrder(root: root.right)
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// 回退
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path.removeLast()
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}
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```
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=== "JS"
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```javascript title="preorder_traversal_ii_compact.js"
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/* 前序遍历：例题二 */
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function preOrder(root, path, res) {
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if (root === null) {
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return;
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}
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// 尝试
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path.push(root);
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if (root.val === 7) {
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// 记录解
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res.push([...path]);
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}
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preOrder(root.left, path, res);
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preOrder(root.right, path, res);
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// 回退
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path.pop();
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}
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```
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=== "TS"
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```typescript title="preorder_traversal_ii_compact.ts"
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/* 前序遍历：例题二 */
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function preOrder(
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root: TreeNode | null,
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path: TreeNode[],
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res: TreeNode[][]
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): void {
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if (root === null) {
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return;
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}
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// 尝试
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path.push(root);
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if (root.val === 7) {
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// 记录解
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res.push([...path]);
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}
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preOrder(root.left, path, res);
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preOrder(root.right, path, res);
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// 回退
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path.pop();
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}
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```
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=== "Dart"
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```dart title="preorder_traversal_ii_compact.dart"
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/* 前序遍历：例题二 */
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void preOrder(
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TreeNode? root,
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List<TreeNode> path,
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List<List<TreeNode>> res,
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) {
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if (root == null) {
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return;
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}
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// 尝试
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path.add(root);
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if (root.val == 7) {
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// 记录解
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res.add(List.from(path));
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}
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preOrder(root.left, path, res);
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preOrder(root.right, path, res);
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// 回退
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path.removeLast();
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}
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```
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=== "Rust"
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```rust title="preorder_traversal_ii_compact.rs"
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/* 前序遍历：例题二 */
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fn pre_order(
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res: &mut Vec<Vec<Rc<RefCell<TreeNode>>>>,
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path: &mut Vec<Rc<RefCell<TreeNode>>>,
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|
root: Option<&Rc<RefCell<TreeNode>>>,
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) {
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|
if root.is_none() {
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|
return;
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}
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|
if let Some(node) = root {
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// 尝试
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|
path.push(node.clone());
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|
if node.borrow().val == 7 {
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// 记录解
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res.push(path.clone());
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}
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pre_order(res, path, node.borrow().left.as_ref());
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pre_order(res, path, node.borrow().right.as_ref());
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// 回退
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|
path.pop();
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}
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}
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```
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=== "C"
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|
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|
```c title="preorder_traversal_ii_compact.c"
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|
/* 前序遍历：例题二 */
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|
|
void preOrder(TreeNode *root) {
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|
if (root == NULL) {
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|
return;
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|
}
|
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|
// 尝试
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|
path[pathSize++] = root;
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|
if (root->val == 7) {
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|
// 记录解
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|
for (int i = 0; i < pathSize; ++i) {
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|
res[resSize][i] = path[i];
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|
}
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|
resSize++;
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|
}
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|
|
preOrder(root->left);
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preOrder(root->right);
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// 回退
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|
pathSize--;
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|
|
}
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|
```
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|
=== "Kotlin"
|
|
|
|
|
|
```kotlin title="preorder_traversal_ii_compact.kt"
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|
/* 前序遍历：例题二 */
|
|
|
fun preOrder(root: TreeNode?) {
|
|
|
if (root == null) {
|
|
|
return
|
|
|
}
|
|
|
// 尝试
|
|
|
path!!.add(root)
|
|
|
if (root._val == 7) {
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|
|
// 记录解
|
|
|
res!!.add(path!!.toMutableList())
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|
|
}
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|
|
preOrder(root.left)
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|
|
preOrder(root.right)
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|
// 回退
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|
|
path!!.removeAt(path!!.size - 1)
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|
|
}
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|
|
```
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|
|
=== "Ruby"
|
|
|
|
|
|
```ruby title="preorder_traversal_ii_compact.rb"
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|
|
### 前序遍历：例题二 ###
|
|
|
def pre_order(root)
|
|
|
return unless root
|
|
|
|
|
|
# 尝试
|
|
|
$path << root
|
|
|
|
|
|
# 记录解
|
|
|
$res << $path.dup if root.val == 7
|
|
|
|
|
|
pre_order(root.left)
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|
|
pre_order(root.right)
|
|
|
|
|
|
# 回退
|
|
|
$path.pop
|
|
|
end
|
|
|
```
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|
|
=== "Zig"
|
|
|
|
|
|
```zig title="preorder_traversal_ii_compact.zig"
|
|
|
[class]{}-[func]{preOrder}
|
|
|
```
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|
??? pythontutor "可视化运行"
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|
|
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%EF%BC%9A%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E5%A6%82%E6%9E%9C%E7%B4%A2%E5%BC%95%E8%B6%85%E5%87%BA%E6%95%B0%E7%BB%84%E9%95%BF%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%AF%B9%E5%BA%94%E7%9A%84%E5%85%83%E7%B4%A0%E4%B8%BA%20None%20%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E5%BD%93%E5%89%8D%E8%8A%82%E7%82%B9%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%80%92%E5%BD%92%E6%9E%84%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A0%91%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%EF%BC%9A%E4%BE%8B%E9%A2%98%E4%BA%8C%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20%E5%B0%9D%E8%AF%95%0A%20%20%20%20path.append%28root%29%0A%20%20%20%20if%20root.val%20%3D%3D%207%3A%0A%20%20%20%20%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E8%A7%A3%0A%20%20%20%20%20%20%20%20res.append%28list%28path%29%29%0A%20%20%20%20pre_order%28root.left%29%0A%20%20%20%20pre_order%28root.right%29%0A%20%20%20%20%23%20%E5%9B%9E%E9%80%80%0A%20%20%20%20path.pop%28%29%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%0A%20%20%20%20path%20%3D%20list%5BTreeNode%5D%28%29%0A%20%20%20%20res%20%3D%20list%5Blist%5BTreeNode%5D%5D%28%29%0A%20%20%20%20pre_order%28root%29%0A%0A%20%20%20%20print%28%22%5Cn%E8%BE%93%E5%87%BA%E6%89%80%E6%9C%89%E6%A0%B9%E8%8A%82%E7%82%B9%E5%88%B0%E8%8A%82%E7%82%B9%207%20%E7%9A%84%E8%B7%AF%E5%BE%84%22%29%0A%20%20%20%20for%20path%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20path%5D%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=126&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
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<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%EF%BC%9A%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E5%A6%82%E6%9E%9C%E7%B4%A2%E5%BC%95%E8%B6%85%E5%87%BA%E6%95%B0%E7%BB%84%E9%95%BF%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%AF%B9%E5%BA%94%E7%9A%84%E5%85%83%E7%B4%A0%E4%B8%BA%20None%20%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E5%BD%93%E5%89%8D%E8%8A%82%E7%82%B9%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%80%92%E5%BD%92%E6%9E%84%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A0%91%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%EF%BC%9A%E4%BE%8B%E9%A2%98%E4%BA%8C%22%22%22%0A%20%20%20%20if%20root%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20%E5%B0%9D%E8%AF%95%0A%20%20%20%20path.append%28root%29%0A%20%20%20%20if%20root.val%20%3D%3D%207%3A%0A%20%20%20%20%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E8%A7%A3%0A%20%20%20%20%20%20%20%20res.append%28list%28path%29%29%0A%20%20%20%20pre_order%28root.left%29%0A%20%20%20%20pre_order%28root.right%29%0A%20%20%20%20%23%20%E5%9B%9E%E9%80%80%0A%20%20%20%20path.pop%28%29%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%0A%20%20%20%20path%20%3D%20list%5BTreeNode%5D%28%29%0A%20%20%20%20res%20%3D%20list%5Blist%5BTreeNode%5D%5D%28%29%0A%20%20%20%20pre_order%28root%29%0A%0A%20%20%20%20print%28%22%5Cn%E8%BE%93%E5%87%BA%E6%89%80%E6%9C%89%E6%A0%B9%E8%8A%82%E7%82%B9%E5%88%B0%E8%8A%82%E7%82%B9%207%20%E7%9A%84%E8%B7%AF%E5%BE%84%22%29%0A%20%20%20%20for%20path%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20path%5D%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=126&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 ></a></div>
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在每次“尝试”中，我们通过将当前节点添加进 `path` 来记录路径；而在“回退”前，我们需要将该节点从 `path` 中弹出，**以恢复本次尝试之前的状态**。
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观察图 13-2 所示的过程，**我们可以将尝试和回退理解为“前进”与“撤销”**，两个操作互为逆向。
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=== "<1>"
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{ class="animation-figure" }
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=== "<2>"
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{ class="animation-figure" }
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=== "<3>"
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{ class="animation-figure" }
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=== "<4>"
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{ class="animation-figure" }
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=== "<5>"
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{ class="animation-figure" }
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=== "<6>"
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{ class="animation-figure" }
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=== "<7>"
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{ class="animation-figure" }
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=== "<8>"
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{ class="animation-figure" }
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=== "<9>"
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{ class="animation-figure" }
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=== "<10>"
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{ class="animation-figure" }
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=== "<11>"
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{ class="animation-figure" }
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<p align="center"> 图 13-2 尝试与回退 </p>
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## 13.1.2 剪枝
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复杂的回溯问题通常包含一个或多个约束条件，**约束条件通常可用于“剪枝”**。
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!!! question "例题三"
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在二叉树中搜索所有值为 $7$ 的节点，请返回根节点到这些节点的路径，**并要求路径中不包含值为 $3$ 的节点**。
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为了满足以上约束条件，**我们需要添加剪枝操作**：在搜索过程中，若遇到值为 $3$ 的节点，则提前返回，不再继续搜索。代码如下所示：
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=== "Python"
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```python title="preorder_traversal_iii_compact.py"
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def pre_order(root: TreeNode):
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"""前序遍历：例题三"""
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# 剪枝
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if root is None or root.val == 3:
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return
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# 尝试
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path.append(root)
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if root.val == 7:
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# 记录解
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res.append(list(path))
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pre_order(root.left)
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pre_order(root.right)
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# 回退
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path.pop()
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```
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=== "C++"
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```cpp title="preorder_traversal_iii_compact.cpp"
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/* 前序遍历：例题三 */
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void preOrder(TreeNode *root) {
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// 剪枝
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if (root == nullptr || root->val == 3) {
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return;
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}
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// 尝试
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path.push_back(root);
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if (root->val == 7) {
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// 记录解
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res.push_back(path);
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}
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preOrder(root->left);
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preOrder(root->right);
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// 回退
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path.pop_back();
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}
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```
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=== "Java"
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```java title="preorder_traversal_iii_compact.java"
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/* 前序遍历：例题三 */
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void preOrder(TreeNode root) {
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// 剪枝
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if (root == null || root.val == 3) {
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return;
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}
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// 尝试
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path.add(root);
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if (root.val == 7) {
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// 记录解
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res.add(new ArrayList<>(path));
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}
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preOrder(root.left);
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preOrder(root.right);
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// 回退
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path.remove(path.size() - 1);
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}
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```
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=== "C#"
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```csharp title="preorder_traversal_iii_compact.cs"
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/* 前序遍历：例题三 */
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void PreOrder(TreeNode? root) {
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// 剪枝
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if (root == null || root.val == 3) {
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return;
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}
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// 尝试
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path.Add(root);
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if (root.val == 7) {
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// 记录解
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res.Add(new List<TreeNode>(path));
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}
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PreOrder(root.left);
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PreOrder(root.right);
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// 回退
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path.RemoveAt(path.Count - 1);
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}
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```
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=== "Go"
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```go title="preorder_traversal_iii_compact.go"
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/* 前序遍历：例题三 */
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func preOrderIII(root *TreeNode, res *[][]*TreeNode, path *[]*TreeNode) {
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// 剪枝
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if root == nil || root.Val == 3 {
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return
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}
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// 尝试
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*path = append(*path, root)
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if root.Val.(int) == 7 {
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// 记录解
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*res = append(*res, append([]*TreeNode{}, *path...))
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}
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preOrderIII(root.Left, res, path)
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preOrderIII(root.Right, res, path)
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// 回退
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*path = (*path)[:len(*path)-1]
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}
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```
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=== "Swift"
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```swift title="preorder_traversal_iii_compact.swift"
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/* 前序遍历：例题三 */
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func preOrder(root: TreeNode?) {
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// 剪枝
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guard let root = root, root.val != 3 else {
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return
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}
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// 尝试
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path.append(root)
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if root.val == 7 {
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// 记录解
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res.append(path)
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}
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preOrder(root: root.left)
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preOrder(root: root.right)
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// 回退
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path.removeLast()
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}
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```
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=== "JS"
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```javascript title="preorder_traversal_iii_compact.js"
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/* 前序遍历：例题三 */
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function preOrder(root, path, res) {
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// 剪枝
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if (root === null || root.val === 3) {
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return;
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}
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// 尝试
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path.push(root);
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if (root.val === 7) {
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// 记录解
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res.push([...path]);
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}
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preOrder(root.left, path, res);
|
|
|
preOrder(root.right, path, res);
|
|
|
// 回退
|
|
|
path.pop();
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "TS"
|
|
|
|
|
|
```typescript title="preorder_traversal_iii_compact.ts"
|
|
|
/* 前序遍历：例题三 */
|
|
|
function preOrder(
|
|
|
root: TreeNode | null,
|
|
|
path: TreeNode[],
|
|
|
res: TreeNode[][]
|
|
|
): void {
|
|
|
// 剪枝
|
|
|
if (root === null || root.val === 3) {
|
|
|
return;
|
|
|
}
|
|
|
// 尝试
|
|
|
path.push(root);
|
|
|
if (root.val === 7) {
|
|
|
// 记录解
|
|
|
res.push([...path]);
|
|
|
}
|
|
|
preOrder(root.left, path, res);
|
|
|
preOrder(root.right, path, res);
|
|
|
// 回退
|
|
|
path.pop();
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Dart"
|
|
|
|
|
|
```dart title="preorder_traversal_iii_compact.dart"
|
|
|
/* 前序遍历：例题三 */
|
|
|
void preOrder(
|
|
|
TreeNode? root,
|
|
|
List<TreeNode> path,
|
|
|
List<List<TreeNode>> res,
|
|
|
) {
|
|
|
if (root == null || root.val == 3) {
|
|
|
return;
|
|
|
}
|
|
|
|
|
|
// 尝试
|
|
|
path.add(root);
|
|
|
if (root.val == 7) {
|
|
|
// 记录解
|
|
|
res.add(List.from(path));
|
|
|
}
|
|
|
preOrder(root.left, path, res);
|
|
|
preOrder(root.right, path, res);
|
|
|
// 回退
|
|
|
path.removeLast();
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Rust"
|
|
|
|
|
|
```rust title="preorder_traversal_iii_compact.rs"
|
|
|
/* 前序遍历：例题三 */
|
|
|
fn pre_order(
|
|
|
res: &mut Vec<Vec<Rc<RefCell<TreeNode>>>>,
|
|
|
path: &mut Vec<Rc<RefCell<TreeNode>>>,
|
|
|
root: Option<&Rc<RefCell<TreeNode>>>,
|
|
|
) {
|
|
|
// 剪枝
|
|
|
if root.is_none() || root.as_ref().unwrap().borrow().val == 3 {
|
|
|
return;
|
|
|
}
|
|
|
if let Some(node) = root {
|
|
|
// 尝试
|
|
|
path.push(node.clone());
|
|
|
if node.borrow().val == 7 {
|
|
|
// 记录解
|
|
|
res.push(path.clone());
|
|
|
}
|
|
|
pre_order(res, path, node.borrow().left.as_ref());
|
|
|
pre_order(res, path, node.borrow().right.as_ref());
|
|
|
// 回退
|
|
|
path.pop();
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "C"
|
|
|
|
|
|
```c title="preorder_traversal_iii_compact.c"
|
|
|
/* 前序遍历：例题三 */
|
|
|
void preOrder(TreeNode *root) {
|
|
|
// 剪枝
|
|
|
if (root == NULL || root->val == 3) {
|
|
|
return;
|
|
|
}
|
|
|
// 尝试
|
|
|
path[pathSize++] = root;
|
|
|
if (root->val == 7) {
|
|
|
// 记录解
|
|
|
for (int i = 0; i < pathSize; i++) {
|
|
|
res[resSize][i] = path[i];
|
|
|
}
|
|
|
resSize++;
|
|
|
}
|
|
|
preOrder(root->left);
|
|
|
preOrder(root->right);
|
|
|
// 回退
|
|
|
pathSize--;
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Kotlin"
|
|
|
|
|
|
```kotlin title="preorder_traversal_iii_compact.kt"
|
|
|
/* 前序遍历：例题三 */
|
|
|
fun preOrder(root: TreeNode?) {
|
|
|
// 剪枝
|
|
|
if (root == null || root._val == 3) {
|
|
|
return
|
|
|
}
|
|
|
// 尝试
|
|
|
path!!.add(root)
|
|
|
if (root._val == 7) {
|
|
|
// 记录解
|
|
|
res!!.add(path!!.toMutableList())
|
|
|
}
|
|
|
preOrder(root.left)
|
|
|
preOrder(root.right)
|
|
|
// 回退
|
|
|
path!!.removeAt(path!!.size - 1)
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Ruby"
|
|
|
|
|
|
```ruby title="preorder_traversal_iii_compact.rb"
|
|
|
### 前序遍历：例题三 ###
|
|
|
def pre_order(root)
|
|
|
# 剪枝
|
|
|
return if !root || root.val == 3
|
|
|
|
|
|
# 尝试
|
|
|
$path.append(root)
|
|
|
|
|
|
# 记录解
|
|
|
$res << $path.dup if root.val == 7
|
|
|
|
|
|
pre_order(root.left)
|
|
|
pre_order(root.right)
|
|
|
|
|
|
# 回退
|
|
|
$path.pop
|
|
|
end
|
|
|
```
|
|
|
|
|
|
=== "Zig"
|
|
|
|
|
|
```zig title="preorder_traversal_iii_compact.zig"
|
|
|
[class]{}-[func]{preOrder}
|
|
|
```
|
|
|
|
|
|
??? pythontutor "可视化运行"
|
|
|
|
|
|
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%EF%BC%9A%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E5%A6%82%E6%9E%9C%E7%B4%A2%E5%BC%95%E8%B6%85%E5%87%BA%E6%95%B0%E7%BB%84%E9%95%BF%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%AF%B9%E5%BA%94%E7%9A%84%E5%85%83%E7%B4%A0%E4%B8%BA%20None%20%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E5%BD%93%E5%89%8D%E8%8A%82%E7%82%B9%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%80%92%E5%BD%92%E6%9E%84%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A0%91%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%EF%BC%9A%E4%BE%8B%E9%A2%98%E4%B8%89%22%22%22%0A%20%20%20%20%23%20%E5%89%AA%E6%9E%9D%0A%20%20%20%20if%20root%20is%20None%20or%20root.val%20%3D%3D%203%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20%E5%B0%9D%E8%AF%95%0A%20%20%20%20path.append%28root%29%0A%20%20%20%20if%20root.val%20%3D%3D%207%3A%0A%20%20%20%20%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E8%A7%A3%0A%20%20%20%20%20%20%20%20res.append%28list%28path%29%29%0A%20%20%20%20pre_order%28root.left%29%0A%20%20%20%20pre_order%28root.right%29%0A%20%20%20%20%23%20%E5%9B%9E%E9%80%80%0A%20%20%20%20path.pop%28%29%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%0A%20%20%20%20path%20%3D%20list%5BTreeNode%5D%28%29%0A%20%20%20%20res%20%3D%20list%5Blist%5BTreeNode%5D%5D%28%29%0A%20%20%20%20pre_order%28root%29%0A%0A%20%20%20%20print%28%22%5Cn%E8%BE%93%E5%87%BA%E6%89%80%E6%9C%89%E6%A0%B9%E8%8A%82%E7%82%B9%E5%88%B0%E8%8A%82%E7%82%B9%207%20%E7%9A%84%E8%B7%AF%E5%BE%84%EF%BC%8C%E8%B7%AF%E5%BE%84%E4%B8%AD%E4%B8%8D%E5%8C%85%E5%90%AB%E5%80%BC%E4%B8%BA%203%20%E7%9A%84%E8%8A%82%E7%82%B9%22%29%0A%20%20%20%20for%20path%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20path%5D%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=126&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
|
|
|
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%EF%BC%9A%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E5%A6%82%E6%9E%9C%E7%B4%A2%E5%BC%95%E8%B6%85%E5%87%BA%E6%95%B0%E7%BB%84%E9%95%BF%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%AF%B9%E5%BA%94%E7%9A%84%E5%85%83%E7%B4%A0%E4%B8%BA%20None%20%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E5%BD%93%E5%89%8D%E8%8A%82%E7%82%B9%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%80%92%E5%BD%92%E6%9E%84%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A0%91%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20pre_order%28root%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%EF%BC%9A%E4%BE%8B%E9%A2%98%E4%B8%89%22%22%22%0A%20%20%20%20%23%20%E5%89%AA%E6%9E%9D%0A%20%20%20%20if%20root%20is%20None%20or%20root.val%20%3D%3D%203%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20%E5%B0%9D%E8%AF%95%0A%20%20%20%20path.append%28root%29%0A%20%20%20%20if%20root.val%20%3D%3D%207%3A%0A%20%20%20%20%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E8%A7%A3%0A%20%20%20%20%20%20%20%20res.append%28list%28path%29%29%0A%20%20%20%20pre_order%28root.left%29%0A%20%20%20%20pre_order%28root.right%29%0A%20%20%20%20%23%20%E5%9B%9E%E9%80%80%0A%20%20%20%20path.pop%28%29%0A%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20%E5%89%8D%E5%BA%8F%E9%81%8D%E5%8E%86%0A%20%20%20%20path%20%3D%20list%5BTreeNode%5D%28%29%0A%20%20%20%20res%20%3D%20list%5Blist%5BTreeNode%5D%5D%28%29%0A%20%20%20%20pre_order%28root%29%0A%0A%20%20%20%20print%28%22%5Cn%E8%BE%93%E5%87%BA%E6%89%80%E6%9C%89%E6%A0%B9%E8%8A%82%E7%82%B9%E5%88%B0%E8%8A%82%E7%82%B9%207%20%E7%9A%84%E8%B7%AF%E5%BE%84%EF%BC%8C%E8%B7%AF%E5%BE%84%E4%B8%AD%E4%B8%8D%E5%8C%85%E5%90%AB%E5%80%BC%E4%B8%BA%203%20%E7%9A%84%E8%8A%82%E7%82%B9%22%29%0A%20%20%20%20for%20path%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20path%5D%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=126&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">全屏观看 ></a></div>
|
|
|
|
|
|
“剪枝”是一个非常形象的名词。如图 13-3 所示，在搜索过程中，**我们“剪掉”了不满足约束条件的搜索分支**，避免许多无意义的尝试，从而提高了搜索效率。
|
|
|
|
|
|
{ class="animation-figure" }
|
|
|
|
|
|
<p align="center"> 图 13-3 根据约束条件剪枝 </p>
|
|
|
|
|
|
## 13.1.3 框架代码
|
|
|
|
|
|
接下来，我们尝试将回溯的“尝试、回退、剪枝”的主体框架提炼出来，提升代码的通用性。
|
|
|
|
|
|
在以下框架代码中，`state` 表示问题的当前状态，`choices` 表示当前状态下可以做出的选择：
|
|
|
|
|
|
=== "Python"
|
|
|
|
|
|
```python title=""
|
|
|
def backtrack(state: State, choices: list[choice], res: list[state]):
|
|
|
"""回溯算法框架"""
|
|
|
# 判断是否为解
|
|
|
if is_solution(state):
|
|
|
# 记录解
|
|
|
record_solution(state, res)
|
|
|
# 不再继续搜索
|
|
|
return
|
|
|
# 遍历所有选择
|
|
|
for choice in choices:
|
|
|
# 剪枝：判断选择是否合法
|
|
|
if is_valid(state, choice):
|
|
|
# 尝试：做出选择，更新状态
|
|
|
make_choice(state, choice)
|
|
|
backtrack(state, choices, res)
|
|
|
# 回退：撤销选择，恢复到之前的状态
|
|
|
undo_choice(state, choice)
|
|
|
```
|
|
|
|
|
|
=== "C++"
|
|
|
|
|
|
```cpp title=""
|
|
|
/* 回溯算法框架 */
|
|
|
void backtrack(State *state, vector<Choice *> &choices, vector<State *> &res) {
|
|
|
// 判断是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res);
|
|
|
// 不再继续搜索
|
|
|
return;
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (Choice choice : choices) {
|
|
|
// 剪枝：判断选择是否合法
|
|
|
if (isValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice);
|
|
|
backtrack(state, choices, res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, choice);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Java"
|
|
|
|
|
|
```java title=""
|
|
|
/* 回溯算法框架 */
|
|
|
void backtrack(State state, List<Choice> choices, List<State> res) {
|
|
|
// 判断是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res);
|
|
|
// 不再继续搜索
|
|
|
return;
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (Choice choice : choices) {
|
|
|
// 剪枝：判断选择是否合法
|
|
|
if (isValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice);
|
|
|
backtrack(state, choices, res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, choice);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "C#"
|
|
|
|
|
|
```csharp title=""
|
|
|
/* 回溯算法框架 */
|
|
|
void Backtrack(State state, List<Choice> choices, List<State> res) {
|
|
|
// 判断是否为解
|
|
|
if (IsSolution(state)) {
|
|
|
// 记录解
|
|
|
RecordSolution(state, res);
|
|
|
// 不再继续搜索
|
|
|
return;
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
foreach (Choice choice in choices) {
|
|
|
// 剪枝：判断选择是否合法
|
|
|
if (IsValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
MakeChoice(state, choice);
|
|
|
Backtrack(state, choices, res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
UndoChoice(state, choice);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Go"
|
|
|
|
|
|
```go title=""
|
|
|
/* 回溯算法框架 */
|
|
|
func backtrack(state *State, choices []Choice, res *[]State) {
|
|
|
// 判断是否为解
|
|
|
if isSolution(state) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res)
|
|
|
// 不再继续搜索
|
|
|
return
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for _, choice := range choices {
|
|
|
// 剪枝：判断选择是否合法
|
|
|
if isValid(state, choice) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice)
|
|
|
backtrack(state, choices, res)
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, choice)
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Swift"
|
|
|
|
|
|
```swift title=""
|
|
|
/* 回溯算法框架 */
|
|
|
func backtrack(state: inout State, choices: [Choice], res: inout [State]) {
|
|
|
// 判断是否为解
|
|
|
if isSolution(state: state) {
|
|
|
// 记录解
|
|
|
recordSolution(state: state, res: &res)
|
|
|
// 不再继续搜索
|
|
|
return
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for choice in choices {
|
|
|
// 剪枝：判断选择是否合法
|
|
|
if isValid(state: state, choice: choice) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state: &state, choice: choice)
|
|
|
backtrack(state: &state, choices: choices, res: &res)
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state: &state, choice: choice)
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "JS"
|
|
|
|
|
|
```javascript title=""
|
|
|
/* 回溯算法框架 */
|
|
|
function backtrack(state, choices, res) {
|
|
|
// 判断是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res);
|
|
|
// 不再继续搜索
|
|
|
return;
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (let choice of choices) {
|
|
|
// 剪枝：判断选择是否合法
|
|
|
if (isValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice);
|
|
|
backtrack(state, choices, res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, choice);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "TS"
|
|
|
|
|
|
```typescript title=""
|
|
|
/* 回溯算法框架 */
|
|
|
function backtrack(state: State, choices: Choice[], res: State[]): void {
|
|
|
// 判断是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res);
|
|
|
// 不再继续搜索
|
|
|
return;
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (let choice of choices) {
|
|
|
// 剪枝：判断选择是否合法
|
|
|
if (isValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice);
|
|
|
backtrack(state, choices, res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, choice);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Dart"
|
|
|
|
|
|
```dart title=""
|
|
|
/* 回溯算法框架 */
|
|
|
void backtrack(State state, List<Choice>, List<State> res) {
|
|
|
// 判断是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res);
|
|
|
// 不再继续搜索
|
|
|
return;
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (Choice choice in choices) {
|
|
|
// 剪枝：判断选择是否合法
|
|
|
if (isValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice);
|
|
|
backtrack(state, choices, res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, choice);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Rust"
|
|
|
|
|
|
```rust title=""
|
|
|
/* 回溯算法框架 */
|
|
|
fn backtrack(state: &mut State, choices: &Vec<Choice>, res: &mut Vec<State>) {
|
|
|
// 判断是否为解
|
|
|
if is_solution(state) {
|
|
|
// 记录解
|
|
|
record_solution(state, res);
|
|
|
// 不再继续搜索
|
|
|
return;
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for choice in choices {
|
|
|
// 剪枝：判断选择是否合法
|
|
|
if is_valid(state, choice) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
make_choice(state, choice);
|
|
|
backtrack(state, choices, res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undo_choice(state, choice);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "C"
|
|
|
|
|
|
```c title=""
|
|
|
/* 回溯算法框架 */
|
|
|
void backtrack(State *state, Choice *choices, int numChoices, State *res, int numRes) {
|
|
|
// 判断是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res, numRes);
|
|
|
// 不再继续搜索
|
|
|
return;
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (int i = 0; i < numChoices; i++) {
|
|
|
// 剪枝：判断选择是否合法
|
|
|
if (isValid(state, &choices[i])) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, &choices[i]);
|
|
|
backtrack(state, choices, numChoices, res, numRes);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, &choices[i]);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Kotlin"
|
|
|
|
|
|
```kotlin title=""
|
|
|
/* 回溯算法框架 */
|
|
|
fun backtrack(state: State?, choices: List<Choice?>, res: List<State?>?) {
|
|
|
// 判断是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res)
|
|
|
// 不再继续搜索
|
|
|
return
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (choice in choices) {
|
|
|
// 剪枝：判断选择是否合法
|
|
|
if (isValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice)
|
|
|
backtrack(state, choices, res)
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, choice)
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Ruby"
|
|
|
|
|
|
```ruby title=""
|
|
|
### 回溯算法框架 ###
|
|
|
def backtrack(state, choices, res)
|
|
|
# 判断是否为解
|
|
|
if is_solution?(state)
|
|
|
# 记录解
|
|
|
record_solution(state, res)
|
|
|
return
|
|
|
end
|
|
|
|
|
|
# 遍历所有选择
|
|
|
for choice in choices
|
|
|
# 剪枝：判断选择是否合法
|
|
|
if is_valid?(state, choice)
|
|
|
# 尝试：做出选择，更新状态
|
|
|
make_choice(state, choice)
|
|
|
backtrack(state, choices, res)
|
|
|
# 回退：撤销选择，恢复到之前的状态
|
|
|
undo_choice(state, choice)
|
|
|
end
|
|
|
end
|
|
|
end
|
|
|
```
|
|
|
|
|
|
=== "Zig"
|
|
|
|
|
|
```zig title=""
|
|
|
|
|
|
```
|
|
|
|
|
|
接下来，我们基于框架代码来解决例题三。状态 `state` 为节点遍历路径，选择 `choices` 为当前节点的左子节点和右子节点，结果 `res` 是路径列表：
|
|
|
|
|
|
=== "Python"
|
|
|
|
|
|
```python title="preorder_traversal_iii_template.py"
|
|
|
def is_solution(state: list[TreeNode]) -> bool:
|
|
|
"""判断当前状态是否为解"""
|
|
|
return state and state[-1].val == 7
|
|
|
|
|
|
def record_solution(state: list[TreeNode], res: list[list[TreeNode]]):
|
|
|
"""记录解"""
|
|
|
res.append(list(state))
|
|
|
|
|
|
def is_valid(state: list[TreeNode], choice: TreeNode) -> bool:
|
|
|
"""判断在当前状态下，该选择是否合法"""
|
|
|
return choice is not None and choice.val != 3
|
|
|
|
|
|
def make_choice(state: list[TreeNode], choice: TreeNode):
|
|
|
"""更新状态"""
|
|
|
state.append(choice)
|
|
|
|
|
|
def undo_choice(state: list[TreeNode], choice: TreeNode):
|
|
|
"""恢复状态"""
|
|
|
state.pop()
|
|
|
|
|
|
def backtrack(
|
|
|
state: list[TreeNode], choices: list[TreeNode], res: list[list[TreeNode]]
|
|
|
):
|
|
|
"""回溯算法：例题三"""
|
|
|
# 检查是否为解
|
|
|
if is_solution(state):
|
|
|
# 记录解
|
|
|
record_solution(state, res)
|
|
|
# 遍历所有选择
|
|
|
for choice in choices:
|
|
|
# 剪枝：检查选择是否合法
|
|
|
if is_valid(state, choice):
|
|
|
# 尝试：做出选择，更新状态
|
|
|
make_choice(state, choice)
|
|
|
# 进行下一轮选择
|
|
|
backtrack(state, [choice.left, choice.right], res)
|
|
|
# 回退：撤销选择，恢复到之前的状态
|
|
|
undo_choice(state, choice)
|
|
|
```
|
|
|
|
|
|
=== "C++"
|
|
|
|
|
|
```cpp title="preorder_traversal_iii_template.cpp"
|
|
|
/* 判断当前状态是否为解 */
|
|
|
bool isSolution(vector<TreeNode *> &state) {
|
|
|
return !state.empty() && state.back()->val == 7;
|
|
|
}
|
|
|
|
|
|
/* 记录解 */
|
|
|
void recordSolution(vector<TreeNode *> &state, vector<vector<TreeNode *>> &res) {
|
|
|
res.push_back(state);
|
|
|
}
|
|
|
|
|
|
/* 判断在当前状态下，该选择是否合法 */
|
|
|
bool isValid(vector<TreeNode *> &state, TreeNode *choice) {
|
|
|
return choice != nullptr && choice->val != 3;
|
|
|
}
|
|
|
|
|
|
/* 更新状态 */
|
|
|
void makeChoice(vector<TreeNode *> &state, TreeNode *choice) {
|
|
|
state.push_back(choice);
|
|
|
}
|
|
|
|
|
|
/* 恢复状态 */
|
|
|
void undoChoice(vector<TreeNode *> &state, TreeNode *choice) {
|
|
|
state.pop_back();
|
|
|
}
|
|
|
|
|
|
/* 回溯算法：例题三 */
|
|
|
void backtrack(vector<TreeNode *> &state, vector<TreeNode *> &choices, vector<vector<TreeNode *>> &res) {
|
|
|
// 检查是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res);
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (TreeNode *choice : choices) {
|
|
|
// 剪枝：检查选择是否合法
|
|
|
if (isValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice);
|
|
|
// 进行下一轮选择
|
|
|
vector<TreeNode *> nextChoices{choice->left, choice->right};
|
|
|
backtrack(state, nextChoices, res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, choice);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Java"
|
|
|
|
|
|
```java title="preorder_traversal_iii_template.java"
|
|
|
/* 判断当前状态是否为解 */
|
|
|
boolean isSolution(List<TreeNode> state) {
|
|
|
return !state.isEmpty() && state.get(state.size() - 1).val == 7;
|
|
|
}
|
|
|
|
|
|
/* 记录解 */
|
|
|
void recordSolution(List<TreeNode> state, List<List<TreeNode>> res) {
|
|
|
res.add(new ArrayList<>(state));
|
|
|
}
|
|
|
|
|
|
/* 判断在当前状态下，该选择是否合法 */
|
|
|
boolean isValid(List<TreeNode> state, TreeNode choice) {
|
|
|
return choice != null && choice.val != 3;
|
|
|
}
|
|
|
|
|
|
/* 更新状态 */
|
|
|
void makeChoice(List<TreeNode> state, TreeNode choice) {
|
|
|
state.add(choice);
|
|
|
}
|
|
|
|
|
|
/* 恢复状态 */
|
|
|
void undoChoice(List<TreeNode> state, TreeNode choice) {
|
|
|
state.remove(state.size() - 1);
|
|
|
}
|
|
|
|
|
|
/* 回溯算法：例题三 */
|
|
|
void backtrack(List<TreeNode> state, List<TreeNode> choices, List<List<TreeNode>> res) {
|
|
|
// 检查是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res);
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (TreeNode choice : choices) {
|
|
|
// 剪枝：检查选择是否合法
|
|
|
if (isValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice);
|
|
|
// 进行下一轮选择
|
|
|
backtrack(state, Arrays.asList(choice.left, choice.right), res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, choice);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "C#"
|
|
|
|
|
|
```csharp title="preorder_traversal_iii_template.cs"
|
|
|
/* 判断当前状态是否为解 */
|
|
|
bool IsSolution(List<TreeNode> state) {
|
|
|
return state.Count != 0 && state[^1].val == 7;
|
|
|
}
|
|
|
|
|
|
/* 记录解 */
|
|
|
void RecordSolution(List<TreeNode> state, List<List<TreeNode>> res) {
|
|
|
res.Add(new List<TreeNode>(state));
|
|
|
}
|
|
|
|
|
|
/* 判断在当前状态下，该选择是否合法 */
|
|
|
bool IsValid(List<TreeNode> state, TreeNode choice) {
|
|
|
return choice != null && choice.val != 3;
|
|
|
}
|
|
|
|
|
|
/* 更新状态 */
|
|
|
void MakeChoice(List<TreeNode> state, TreeNode choice) {
|
|
|
state.Add(choice);
|
|
|
}
|
|
|
|
|
|
/* 恢复状态 */
|
|
|
void UndoChoice(List<TreeNode> state, TreeNode choice) {
|
|
|
state.RemoveAt(state.Count - 1);
|
|
|
}
|
|
|
|
|
|
/* 回溯算法：例题三 */
|
|
|
void Backtrack(List<TreeNode> state, List<TreeNode> choices, List<List<TreeNode>> res) {
|
|
|
// 检查是否为解
|
|
|
if (IsSolution(state)) {
|
|
|
// 记录解
|
|
|
RecordSolution(state, res);
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
foreach (TreeNode choice in choices) {
|
|
|
// 剪枝：检查选择是否合法
|
|
|
if (IsValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
MakeChoice(state, choice);
|
|
|
// 进行下一轮选择
|
|
|
Backtrack(state, [choice.left!, choice.right!], res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
UndoChoice(state, choice);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Go"
|
|
|
|
|
|
```go title="preorder_traversal_iii_template.go"
|
|
|
/* 判断当前状态是否为解 */
|
|
|
func isSolution(state *[]*TreeNode) bool {
|
|
|
return len(*state) != 0 && (*state)[len(*state)-1].Val == 7
|
|
|
}
|
|
|
|
|
|
/* 记录解 */
|
|
|
func recordSolution(state *[]*TreeNode, res *[][]*TreeNode) {
|
|
|
*res = append(*res, append([]*TreeNode{}, *state...))
|
|
|
}
|
|
|
|
|
|
/* 判断在当前状态下，该选择是否合法 */
|
|
|
func isValid(state *[]*TreeNode, choice *TreeNode) bool {
|
|
|
return choice != nil && choice.Val != 3
|
|
|
}
|
|
|
|
|
|
/* 更新状态 */
|
|
|
func makeChoice(state *[]*TreeNode, choice *TreeNode) {
|
|
|
*state = append(*state, choice)
|
|
|
}
|
|
|
|
|
|
/* 恢复状态 */
|
|
|
func undoChoice(state *[]*TreeNode, choice *TreeNode) {
|
|
|
*state = (*state)[:len(*state)-1]
|
|
|
}
|
|
|
|
|
|
/* 回溯算法：例题三 */
|
|
|
func backtrackIII(state *[]*TreeNode, choices *[]*TreeNode, res *[][]*TreeNode) {
|
|
|
// 检查是否为解
|
|
|
if isSolution(state) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res)
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for _, choice := range *choices {
|
|
|
// 剪枝：检查选择是否合法
|
|
|
if isValid(state, choice) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice)
|
|
|
// 进行下一轮选择
|
|
|
temp := make([]*TreeNode, 0)
|
|
|
temp = append(temp, choice.Left, choice.Right)
|
|
|
backtrackIII(state, &temp, res)
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, choice)
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Swift"
|
|
|
|
|
|
```swift title="preorder_traversal_iii_template.swift"
|
|
|
/* 判断当前状态是否为解 */
|
|
|
func isSolution(state: [TreeNode]) -> Bool {
|
|
|
!state.isEmpty && state.last!.val == 7
|
|
|
}
|
|
|
|
|
|
/* 记录解 */
|
|
|
func recordSolution(state: [TreeNode], res: inout [[TreeNode]]) {
|
|
|
res.append(state)
|
|
|
}
|
|
|
|
|
|
/* 判断在当前状态下，该选择是否合法 */
|
|
|
func isValid(state: [TreeNode], choice: TreeNode?) -> Bool {
|
|
|
choice != nil && choice!.val != 3
|
|
|
}
|
|
|
|
|
|
/* 更新状态 */
|
|
|
func makeChoice(state: inout [TreeNode], choice: TreeNode) {
|
|
|
state.append(choice)
|
|
|
}
|
|
|
|
|
|
/* 恢复状态 */
|
|
|
func undoChoice(state: inout [TreeNode], choice: TreeNode) {
|
|
|
state.removeLast()
|
|
|
}
|
|
|
|
|
|
/* 回溯算法：例题三 */
|
|
|
func backtrack(state: inout [TreeNode], choices: [TreeNode], res: inout [[TreeNode]]) {
|
|
|
// 检查是否为解
|
|
|
if isSolution(state: state) {
|
|
|
recordSolution(state: state, res: &res)
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for choice in choices {
|
|
|
// 剪枝：检查选择是否合法
|
|
|
if isValid(state: state, choice: choice) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state: &state, choice: choice)
|
|
|
// 进行下一轮选择
|
|
|
backtrack(state: &state, choices: [choice.left, choice.right].compactMap { $0 }, res: &res)
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state: &state, choice: choice)
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "JS"
|
|
|
|
|
|
```javascript title="preorder_traversal_iii_template.js"
|
|
|
/* 判断当前状态是否为解 */
|
|
|
function isSolution(state) {
|
|
|
return state && state[state.length - 1]?.val === 7;
|
|
|
}
|
|
|
|
|
|
/* 记录解 */
|
|
|
function recordSolution(state, res) {
|
|
|
res.push([...state]);
|
|
|
}
|
|
|
|
|
|
/* 判断在当前状态下，该选择是否合法 */
|
|
|
function isValid(state, choice) {
|
|
|
return choice !== null && choice.val !== 3;
|
|
|
}
|
|
|
|
|
|
/* 更新状态 */
|
|
|
function makeChoice(state, choice) {
|
|
|
state.push(choice);
|
|
|
}
|
|
|
|
|
|
/* 恢复状态 */
|
|
|
function undoChoice(state) {
|
|
|
state.pop();
|
|
|
}
|
|
|
|
|
|
/* 回溯算法：例题三 */
|
|
|
function backtrack(state, choices, res) {
|
|
|
// 检查是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res);
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (const choice of choices) {
|
|
|
// 剪枝：检查选择是否合法
|
|
|
if (isValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice);
|
|
|
// 进行下一轮选择
|
|
|
backtrack(state, [choice.left, choice.right], res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "TS"
|
|
|
|
|
|
```typescript title="preorder_traversal_iii_template.ts"
|
|
|
/* 判断当前状态是否为解 */
|
|
|
function isSolution(state: TreeNode[]): boolean {
|
|
|
return state && state[state.length - 1]?.val === 7;
|
|
|
}
|
|
|
|
|
|
/* 记录解 */
|
|
|
function recordSolution(state: TreeNode[], res: TreeNode[][]): void {
|
|
|
res.push([...state]);
|
|
|
}
|
|
|
|
|
|
/* 判断在当前状态下，该选择是否合法 */
|
|
|
function isValid(state: TreeNode[], choice: TreeNode): boolean {
|
|
|
return choice !== null && choice.val !== 3;
|
|
|
}
|
|
|
|
|
|
/* 更新状态 */
|
|
|
function makeChoice(state: TreeNode[], choice: TreeNode): void {
|
|
|
state.push(choice);
|
|
|
}
|
|
|
|
|
|
/* 恢复状态 */
|
|
|
function undoChoice(state: TreeNode[]): void {
|
|
|
state.pop();
|
|
|
}
|
|
|
|
|
|
/* 回溯算法：例题三 */
|
|
|
function backtrack(
|
|
|
state: TreeNode[],
|
|
|
choices: TreeNode[],
|
|
|
res: TreeNode[][]
|
|
|
): void {
|
|
|
// 检查是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res);
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (const choice of choices) {
|
|
|
// 剪枝：检查选择是否合法
|
|
|
if (isValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice);
|
|
|
// 进行下一轮选择
|
|
|
backtrack(state, [choice.left, choice.right], res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Dart"
|
|
|
|
|
|
```dart title="preorder_traversal_iii_template.dart"
|
|
|
/* 判断当前状态是否为解 */
|
|
|
bool isSolution(List<TreeNode> state) {
|
|
|
return state.isNotEmpty && state.last.val == 7;
|
|
|
}
|
|
|
|
|
|
/* 记录解 */
|
|
|
void recordSolution(List<TreeNode> state, List<List<TreeNode>> res) {
|
|
|
res.add(List.from(state));
|
|
|
}
|
|
|
|
|
|
/* 判断在当前状态下，该选择是否合法 */
|
|
|
bool isValid(List<TreeNode> state, TreeNode? choice) {
|
|
|
return choice != null && choice.val != 3;
|
|
|
}
|
|
|
|
|
|
/* 更新状态 */
|
|
|
void makeChoice(List<TreeNode> state, TreeNode? choice) {
|
|
|
state.add(choice!);
|
|
|
}
|
|
|
|
|
|
/* 恢复状态 */
|
|
|
void undoChoice(List<TreeNode> state, TreeNode? choice) {
|
|
|
state.removeLast();
|
|
|
}
|
|
|
|
|
|
/* 回溯算法：例题三 */
|
|
|
void backtrack(
|
|
|
List<TreeNode> state,
|
|
|
List<TreeNode?> choices,
|
|
|
List<List<TreeNode>> res,
|
|
|
) {
|
|
|
// 检查是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res);
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (TreeNode? choice in choices) {
|
|
|
// 剪枝：检查选择是否合法
|
|
|
if (isValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice);
|
|
|
// 进行下一轮选择
|
|
|
backtrack(state, [choice!.left, choice.right], res);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, choice);
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Rust"
|
|
|
|
|
|
```rust title="preorder_traversal_iii_template.rs"
|
|
|
/* 判断当前状态是否为解 */
|
|
|
fn is_solution(state: &mut Vec<Rc<RefCell<TreeNode>>>) -> bool {
|
|
|
return !state.is_empty() && state.last().unwrap().borrow().val == 7;
|
|
|
}
|
|
|
|
|
|
/* 记录解 */
|
|
|
fn record_solution(
|
|
|
state: &mut Vec<Rc<RefCell<TreeNode>>>,
|
|
|
res: &mut Vec<Vec<Rc<RefCell<TreeNode>>>>,
|
|
|
) {
|
|
|
res.push(state.clone());
|
|
|
}
|
|
|
|
|
|
/* 判断在当前状态下，该选择是否合法 */
|
|
|
fn is_valid(_: &mut Vec<Rc<RefCell<TreeNode>>>, choice: Option<&Rc<RefCell<TreeNode>>>) -> bool {
|
|
|
return choice.is_some() && choice.unwrap().borrow().val != 3;
|
|
|
}
|
|
|
|
|
|
/* 更新状态 */
|
|
|
fn make_choice(state: &mut Vec<Rc<RefCell<TreeNode>>>, choice: Rc<RefCell<TreeNode>>) {
|
|
|
state.push(choice);
|
|
|
}
|
|
|
|
|
|
/* 恢复状态 */
|
|
|
fn undo_choice(state: &mut Vec<Rc<RefCell<TreeNode>>>, _: Rc<RefCell<TreeNode>>) {
|
|
|
state.pop();
|
|
|
}
|
|
|
|
|
|
/* 回溯算法：例题三 */
|
|
|
fn backtrack(
|
|
|
state: &mut Vec<Rc<RefCell<TreeNode>>>,
|
|
|
choices: &Vec<Option<&Rc<RefCell<TreeNode>>>>,
|
|
|
res: &mut Vec<Vec<Rc<RefCell<TreeNode>>>>,
|
|
|
) {
|
|
|
// 检查是否为解
|
|
|
if is_solution(state) {
|
|
|
// 记录解
|
|
|
record_solution(state, res);
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for &choice in choices.iter() {
|
|
|
// 剪枝：检查选择是否合法
|
|
|
if is_valid(state, choice) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
make_choice(state, choice.unwrap().clone());
|
|
|
// 进行下一轮选择
|
|
|
backtrack(
|
|
|
state,
|
|
|
&vec![
|
|
|
choice.unwrap().borrow().left.as_ref(),
|
|
|
choice.unwrap().borrow().right.as_ref(),
|
|
|
],
|
|
|
res,
|
|
|
);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undo_choice(state, choice.unwrap().clone());
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "C"
|
|
|
|
|
|
```c title="preorder_traversal_iii_template.c"
|
|
|
/* 判断当前状态是否为解 */
|
|
|
bool isSolution(void) {
|
|
|
return pathSize > 0 && path[pathSize - 1]->val == 7;
|
|
|
}
|
|
|
|
|
|
/* 记录解 */
|
|
|
void recordSolution(void) {
|
|
|
for (int i = 0; i < pathSize; i++) {
|
|
|
res[resSize][i] = path[i];
|
|
|
}
|
|
|
resSize++;
|
|
|
}
|
|
|
|
|
|
/* 判断在当前状态下，该选择是否合法 */
|
|
|
bool isValid(TreeNode *choice) {
|
|
|
return choice != NULL && choice->val != 3;
|
|
|
}
|
|
|
|
|
|
/* 更新状态 */
|
|
|
void makeChoice(TreeNode *choice) {
|
|
|
path[pathSize++] = choice;
|
|
|
}
|
|
|
|
|
|
/* 恢复状态 */
|
|
|
void undoChoice(void) {
|
|
|
pathSize--;
|
|
|
}
|
|
|
|
|
|
/* 回溯算法：例题三 */
|
|
|
void backtrack(TreeNode *choices[2]) {
|
|
|
// 检查是否为解
|
|
|
if (isSolution()) {
|
|
|
// 记录解
|
|
|
recordSolution();
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (int i = 0; i < 2; i++) {
|
|
|
TreeNode *choice = choices[i];
|
|
|
// 剪枝：检查选择是否合法
|
|
|
if (isValid(choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(choice);
|
|
|
// 进行下一轮选择
|
|
|
TreeNode *nextChoices[2] = {choice->left, choice->right};
|
|
|
backtrack(nextChoices);
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice();
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Kotlin"
|
|
|
|
|
|
```kotlin title="preorder_traversal_iii_template.kt"
|
|
|
/* 判断当前状态是否为解 */
|
|
|
fun isSolution(state: MutableList<TreeNode?>): Boolean {
|
|
|
return state.isNotEmpty() && state[state.size - 1]?._val == 7
|
|
|
}
|
|
|
|
|
|
/* 记录解 */
|
|
|
fun recordSolution(state: MutableList<TreeNode?>?, res: MutableList<MutableList<TreeNode?>?>) {
|
|
|
res.add(state!!.toMutableList())
|
|
|
}
|
|
|
|
|
|
/* 判断在当前状态下，该选择是否合法 */
|
|
|
fun isValid(state: MutableList<TreeNode?>?, choice: TreeNode?): Boolean {
|
|
|
return choice != null && choice._val != 3
|
|
|
}
|
|
|
|
|
|
/* 更新状态 */
|
|
|
fun makeChoice(state: MutableList<TreeNode?>, choice: TreeNode?) {
|
|
|
state.add(choice)
|
|
|
}
|
|
|
|
|
|
/* 恢复状态 */
|
|
|
fun undoChoice(state: MutableList<TreeNode?>, choice: TreeNode?) {
|
|
|
state.removeLast()
|
|
|
}
|
|
|
|
|
|
/* 回溯算法：例题三 */
|
|
|
fun backtrack(
|
|
|
state: MutableList<TreeNode?>,
|
|
|
choices: MutableList<TreeNode?>,
|
|
|
res: MutableList<MutableList<TreeNode?>?>
|
|
|
) {
|
|
|
// 检查是否为解
|
|
|
if (isSolution(state)) {
|
|
|
// 记录解
|
|
|
recordSolution(state, res)
|
|
|
}
|
|
|
// 遍历所有选择
|
|
|
for (choice in choices) {
|
|
|
// 剪枝：检查选择是否合法
|
|
|
if (isValid(state, choice)) {
|
|
|
// 尝试：做出选择，更新状态
|
|
|
makeChoice(state, choice)
|
|
|
// 进行下一轮选择
|
|
|
backtrack(state, mutableListOf(choice!!.left, choice.right), res)
|
|
|
// 回退：撤销选择，恢复到之前的状态
|
|
|
undoChoice(state, choice)
|
|
|
}
|
|
|
}
|
|
|
}
|
|
|
```
|
|
|
|
|
|
=== "Ruby"
|
|
|
|
|
|
```ruby title="preorder_traversal_iii_template.rb"
|
|
|
### 判断当前状态是否为解 ###
|
|
|
def is_solution?(state)
|
|
|
!state.empty? && state.last.val == 7
|
|
|
end
|
|
|
|
|
|
### 记录解 ###
|
|
|
def record_solution(state, res)
|
|
|
res << state.dup
|
|
|
end
|
|
|
|
|
|
### 判断在当前状态下，该选择是否合法 ###
|
|
|
def is_valid?(state, choice)
|
|
|
choice && choice.val != 3
|
|
|
end
|
|
|
|
|
|
### 更新状态 ###
|
|
|
def make_choice(state, choice)
|
|
|
state << choice
|
|
|
end
|
|
|
|
|
|
### 恢复状态 ###
|
|
|
def undo_choice(state, choice)
|
|
|
state.pop
|
|
|
end
|
|
|
|
|
|
### 回溯算法：例题三 ###
|
|
|
def backtrack(state, choices, res)
|
|
|
# 检查是否为解
|
|
|
record_solution(state, res) if is_solution?(state)
|
|
|
|
|
|
# 遍历所有选择
|
|
|
for choice in choices
|
|
|
# 剪枝：检查选择是否合法
|
|
|
if is_valid?(state, choice)
|
|
|
# 尝试：做出选择，更新状态
|
|
|
make_choice(state, choice)
|
|
|
# 进行下一轮选择
|
|
|
backtrack(state, [choice.left, choice.right], res)
|
|
|
# 回退：撤销选择，恢复到之前的状态
|
|
|
undo_choice(state, choice)
|
|
|
end
|
|
|
end
|
|
|
end
|
|
|
```
|
|
|
|
|
|
=== "Zig"
|
|
|
|
|
|
```zig title="preorder_traversal_iii_template.zig"
|
|
|
[class]{}-[func]{isSolution}
|
|
|
|
|
|
[class]{}-[func]{recordSolution}
|
|
|
|
|
|
[class]{}-[func]{isValid}
|
|
|
|
|
|
[class]{}-[func]{makeChoice}
|
|
|
|
|
|
[class]{}-[func]{undoChoice}
|
|
|
|
|
|
[class]{}-[func]{backtrack}
|
|
|
```
|
|
|
|
|
|
??? pythontutor "可视化运行"
|
|
|
|
|
|
<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%EF%BC%9A%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E5%A6%82%E6%9E%9C%E7%B4%A2%E5%BC%95%E8%B6%85%E5%87%BA%E6%95%B0%E7%BB%84%E9%95%BF%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%AF%B9%E5%BA%94%E7%9A%84%E5%85%83%E7%B4%A0%E4%B8%BA%20None%20%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E5%BD%93%E5%89%8D%E8%8A%82%E7%82%B9%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%80%92%E5%BD%92%E6%9E%84%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A0%91%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20is_solution%28state%3A%20list%5BTreeNode%5D%29%20-%3E%20bool%3A%0A%20%20%20%20%22%22%22%E5%88%A4%E6%96%AD%E5%BD%93%E5%89%8D%E7%8A%B6%E6%80%81%E6%98%AF%E5%90%A6%E4%B8%BA%E8%A7%A3%22%22%22%0A%20%20%20%20return%20state%20and%20state%5B-1%5D.val%20%3D%3D%207%0A%0Adef%20record_solution%28state%3A%20list%5BTreeNode%5D,%20res%3A%20list%5Blist%5BTreeNode%5D%5D%29%3A%0A%20%20%20%20%22%22%22%E8%AE%B0%E5%BD%95%E8%A7%A3%22%22%22%0A%20%20%20%20res.append%28list%28state%29%29%0A%0Adef%20is_valid%28state%3A%20list%5BTreeNode%5D,%20choice%3A%20TreeNode%29%20-%3E%20bool%3A%0A%20%20%20%20%22%22%22%E5%88%A4%E6%96%AD%E5%9C%A8%E5%BD%93%E5%89%8D%E7%8A%B6%E6%80%81%E4%B8%8B%EF%BC%8C%E8%AF%A5%E9%80%89%E6%8B%A9%E6%98%AF%E5%90%A6%E5%90%88%E6%B3%95%22%22%22%0A%20%20%20%20return%20choice%20is%20not%20None%20and%20choice.val%20!%3D%203%0A%0Adef%20make_choice%28state%3A%20list%5BTreeNode%5D,%20choice%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22%E6%9B%B4%E6%96%B0%E7%8A%B6%E6%80%81%22%22%22%0A%20%20%20%20state.append%28choice%29%0A%0Adef%20undo_choice%28state%3A%20list%5BTreeNode%5D,%20choice%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22%E6%81%A2%E5%A4%8D%E7%8A%B6%E6%80%81%22%22%22%0A%20%20%20%20state.pop%28%29%0A%0Adef%20backtrack%28%0A%20%20%20%20state%3A%20list%5BTreeNode%5D,%20choices%3A%20list%5BTreeNode%5D,%20res%3A%20list%5Blist%5BTreeNode%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22%E5%9B%9E%E6%BA%AF%E7%AE%97%E6%B3%95%EF%BC%9A%E4%BE%8B%E9%A2%98%E4%B8%89%22%22%22%0A%20%20%20%20%23%20%E6%A3%80%E6%9F%A5%E6%98%AF%E5%90%A6%E4%B8%BA%E8%A7%A3%0A%20%20%20%20if%20is_solution%28state%29%3A%0A%20%20%20%20%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E8%A7%A3%0A%20%20%20%20%20%20%20%20record_solution%28state,%20res%29%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E6%89%80%E6%9C%89%E9%80%89%E6%8B%A9%0A%20%20%20%20for%20choice%20in%20choices%3A%0A%20%20%20%20%20%20%20%20%23%20%E5%89%AA%E6%9E%9D%EF%BC%9A%E6%A3%80%E6%9F%A5%E9%80%89%E6%8B%A9%E6%98%AF%E5%90%A6%E5%90%88%E6%B3%95%0A%20%20%20%20%20%20%20%20if%20is_valid%28state,%20choice%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E5%B0%9D%E8%AF%95%EF%BC%9A%E5%81%9A%E5%87%BA%E9%80%89%E6%8B%A9%EF%BC%8C%E6%9B%B4%E6%96%B0%E7%8A%B6%E6%80%81%0A%20%20%20%20%20%20%20%20%20%20%20%20make_choice%28state,%20choice%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%BF%9B%E8%A1%8C%E4%B8%8B%E4%B8%80%E8%BD%AE%E9%80%89%E6%8B%A9%0A%20%20%20%20%20%20%20%20%20%20%20%20backtrack%28state,%20%5Bchoice.left,%20choice.right%5D,%20res%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E5%9B%9E%E9%80%80%EF%BC%9A%E6%92%A4%E9%94%80%E9%80%89%E6%8B%A9%EF%BC%8C%E6%81%A2%E5%A4%8D%E5%88%B0%E4%B9%8B%E5%89%8D%E7%9A%84%E7%8A%B6%E6%80%81%0A%20%20%20%20%20%20%20%20%20%20%20%20undo_choice%28state,%20choice%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20%E5%9B%9E%E6%BA%AF%E7%AE%97%E6%B3%95%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20backtrack%28state%3D%5B%5D,%20choices%3D%5Broot%5D,%20res%3Dres%29%0A%20%20%20%20print%28%22%5Cn%E8%BE%93%E5%87%BA%E6%89%80%E6%9C%89%E8%B7%AF%E5%BE%84%22%29%0A%20%20%20%20for%20path%20in%20res%3A%0A%20%20%20%20%20%20%20%20print%28%5Bnode.val%20for%20node%20in%20path%5D%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=138&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
|
|
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<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%20%3D%200%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20list_to_tree_dfs%28arr%3A%20list%5Bint%5D,%20i%3A%20int%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%EF%BC%9A%E9%80%92%E5%BD%92%22%22%22%0A%20%20%20%20%23%20%E5%A6%82%E6%9E%9C%E7%B4%A2%E5%BC%95%E8%B6%85%E5%87%BA%E6%95%B0%E7%BB%84%E9%95%BF%E5%BA%A6%EF%BC%8C%E6%88%96%E8%80%85%E5%AF%B9%E5%BA%94%E7%9A%84%E5%85%83%E7%B4%A0%E4%B8%BA%20None%20%EF%BC%8C%E5%88%99%E8%BF%94%E5%9B%9E%20None%0A%20%20%20%20if%20i%20%3C%200%20or%20i%20%3E%3D%20len%28arr%29%20or%20arr%5Bi%5D%20is%20None%3A%0A%20%20%20%20%20%20%20%20return%20None%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E5%BD%93%E5%89%8D%E8%8A%82%E7%82%B9%0A%20%20%20%20root%20%3D%20TreeNode%28arr%5Bi%5D%29%0A%20%20%20%20%23%20%E9%80%92%E5%BD%92%E6%9E%84%E5%BB%BA%E5%B7%A6%E5%8F%B3%E5%AD%90%E6%A0%91%0A%20%20%20%20root.left%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%201%29%0A%20%20%20%20root.right%20%3D%20list_to_tree_dfs%28arr,%202%20*%20i%20%2B%202%29%0A%20%20%20%20return%20root%0A%0Adef%20list_to_tree%28arr%3A%20list%5Bint%5D%29%20-%3E%20TreeNode%20%7C%20None%3A%0A%20%20%20%20%22%22%22%E5%B0%86%E5%88%97%E8%A1%A8%E5%8F%8D%E5%BA%8F%E5%88%97%E5%8C%96%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%A0%91%22%22%22%0A%20%20%20%20return%20list_to_tree_dfs%28arr,%200%29%0A%0A%0Adef%20is_solution%28state%3A%20list%5BTreeNode%5D%29%20-%3E%20bool%3A%0A%20%20%20%20%22%22%22%E5%88%A4%E6%96%AD%E5%BD%93%E5%89%8D%E7%8A%B6%E6%80%81%E6%98%AF%E5%90%A6%E4%B8%BA%E8%A7%A3%22%22%22%0A%20%20%20%20return%20state%20and%20state%5B-1%5D.val%20%3D%3D%207%0A%0Adef%20record_solution%28state%3A%20list%5BTreeNode%5D,%20res%3A%20list%5Blist%5BTreeNode%5D%5D%29%3A%0A%20%20%20%20%22%22%22%E8%AE%B0%E5%BD%95%E8%A7%A3%22%22%22%0A%20%20%20%20res.append%28list%28state%29%29%0A%0Adef%20is_valid%28state%3A%20list%5BTreeNode%5D,%20choice%3A%20TreeNode%29%20-%3E%20bool%3A%0A%20%20%20%20%22%22%22%E5%88%A4%E6%96%AD%E5%9C%A8%E5%BD%93%E5%89%8D%E7%8A%B6%E6%80%81%E4%B8%8B%EF%BC%8C%E8%AF%A5%E9%80%89%E6%8B%A9%E6%98%AF%E5%90%A6%E5%90%88%E6%B3%95%22%22%22%0A%20%20%20%20return%20choice%20is%20not%20None%20and%20choice.val%20!%3D%203%0A%0Adef%20make_choice%28state%3A%20list%5BTreeNode%5D,%20choice%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22%E6%9B%B4%E6%96%B0%E7%8A%B6%E6%80%81%22%22%22%0A%20%20%20%20state.append%28choice%29%0A%0Adef%20undo_choice%28state%3A%20list%5BTreeNode%5D,%20choice%3A%20TreeNode%29%3A%0A%20%20%20%20%22%22%22%E6%81%A2%E5%A4%8D%E7%8A%B6%E6%80%81%22%22%22%0A%20%20%20%20state.pop%28%29%0A%0Adef%20backtrack%28%0A%20%20%20%20state%3A%20list%5BTreeNode%5D,%20choices%3A%20list%5BTreeNode%5D,%20res%3A%20list%5Blist%5BTreeNode%5D%5D%0A%29%3A%0A%20%20%20%20%22%22%22%E5%9B%9E%E6%BA%AF%E7%AE%97%E6%B3%95%EF%BC%9A%E4%BE%8B%E9%A2%98%E4%B8%89%22%22%22%0A%20%20%20%20%23%20%E6%A3%80%E6%9F%A5%E6%98%AF%E5%90%A6%E4%B8%BA%E8%A7%A3%0A%20%20%20%20if%20is_solution%28state%29%3A%0A%20%20%20%20%20%20%20%20%23%20%E8%AE%B0%E5%BD%95%E8%A7%A3%0A%20%20%20%20%20%20%20%20record_solution%28state,%20res%29%0A%20%20%20%20%23%20%E9%81%8D%E5%8E%86%E6%89%80%E6%9C%89%E9%80%89%E6%8B%A9%0A%20%20%20%20for%20choice%20in%20choices%3A%0A%20%20%20%20%20%20%20%20%23%20%E5%89%AA%E6%9E%9D%EF%BC%9A%E6%A3%80%E6%9F%A5%E9%80%89%E6%8B%A9%E6%98%AF%E5%90%A6%E5%90%88%E6%B3%95%0A%20%20%20%20%20%20%20%20if%20is_valid%28state,%20choice%29%3A%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E5%B0%9D%E8%AF%95%EF%BC%9A%E5%81%9A%E5%87%BA%E9%80%89%E6%8B%A9%EF%BC%8C%E6%9B%B4%E6%96%B0%E7%8A%B6%E6%80%81%0A%20%20%20%20%20%20%20%20%20%20%20%20make_choice%28state,%20choice%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%BF%9B%E8%A1%8C%E4%B8%8B%E4%B8%80%E8%BD%AE%E9%80%89%E6%8B%A9%0A%20%20%20%20%20%20%20%20%20%20%20%20backtrack%28state,%20%5Bchoice.left,%20choice.right%5D,%20res%29%0A%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E5%9B%9E%E9%80%80%EF%BC%9A%E6%92%A4%E9%94%80%E9%80%89%E6%8B%A9%EF%BC%8C%E6%81%A2%E5%A4%8D%E5%88%B0%E4%B9%8B%E5%89%8D%E7%9A%84%E7%8A%B6%E6%80%81%0A%20%20%20%20%20%20%20%20%20%20%20%20undo_choice%28state,%20choice%29%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20root%20%3D%20list_to_tree%28%5B1,%207,%203,%204,%205,%206,%207%5D%29%0A%0A%20%20%20%20%23%20%E5%9B%9E%E6%BA%AF%E7%AE%97%E6%B3%95%0A%20%20%20%20res%20%3D%20%5B%5D%0A%20%20%20%20backtrack%28state%3D%5B%5D,%20choices%3D%5Broot%5D,%20res%3Dres%29%0A%2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target="_blank" rel="noopener noreferrer">全屏观看 ></a></div>
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根据题意，我们在找到值为 $7$ 的节点后应该继续搜索，**因此需要将记录解之后的 `return` 语句删除**。图 13-4 对比了保留或删除 `return` 语句的搜索过程。
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{ class="animation-figure" }
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<p align="center"> 图 13-4 保留与删除 return 的搜索过程对比 </p>
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相比基于前序遍历的代码实现，基于回溯算法框架的代码实现虽然显得啰唆，但通用性更好。实际上，**许多回溯问题可以在该框架下解决**。我们只需根据具体问题来定义 `state` 和 `choices` ，并实现框架中的各个方法即可。
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## 13.1.4 常用术语
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为了更清晰地分析算法问题，我们总结一下回溯算法中常用术语的含义，并对照例题三给出对应示例，如表 13-1 所示。
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<p align="center"> 表 13-1 常见的回溯算法术语 </p>
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<div class="center-table" markdown>
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| 名词 | 定义 | 例题三 |
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| ---------------------- | -------------------------------------------------------------------------- | -------------------------------------------------------------------- |
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| 解（solution） | 解是满足问题特定条件的答案，可能有一个或多个 | 根节点到节点 $7$ 的满足约束条件的所有路径 |
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| 约束条件（constraint） | 约束条件是问题中限制解的可行性的条件，通常用于剪枝 | 路径中不包含节点 $3$ |
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| 状态（state） | 状态表示问题在某一时刻的情况，包括已经做出的选择 | 当前已访问的节点路径，即 `path` 节点列表 |
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| 尝试（attempt） | 尝试是根据可用选择来探索解空间的过程，包括做出选择，更新状态，检查是否为解 | 递归访问左（右）子节点，将节点添加进 `path` ，判断节点的值是否为 $7$ |
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| 回退（backtracking） | 回退指遇到不满足约束条件的状态时，撤销前面做出的选择，回到上一个状态 | 当越过叶节点、结束节点访问、遇到值为 $3$ 的节点时终止搜索，函数返回 |
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| 剪枝（pruning） | 剪枝是根据问题特性和约束条件避免无意义的搜索路径的方法，可提高搜索效率 | 当遇到值为 $3$ 的节点时，则不再继续搜索 |
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</div>
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!!! tip
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问题、解、状态等概念是通用的，在分治、回溯、动态规划、贪心等算法中都有涉及。
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## 13.1.5 优点与局限性
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回溯算法本质上是一种深度优先搜索算法，它尝试所有可能的解决方案直到找到满足条件的解。这种方法的优点在于能够找到所有可能的解决方案，而且在合理的剪枝操作下，具有很高的效率。
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然而，在处理大规模或者复杂问题时，**回溯算法的运行效率可能难以接受**。
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- **时间**：回溯算法通常需要遍历状态空间的所有可能，时间复杂度可以达到指数阶或阶乘阶。
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- **空间**：在递归调用中需要保存当前的状态（例如路径、用于剪枝的辅助变量等），当深度很大时，空间需求可能会变得很大。
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即便如此，**回溯算法仍然是某些搜索问题和约束满足问题的最佳解决方案**。对于这些问题，由于无法预测哪些选择可生成有效的解，因此我们必须对所有可能的选择进行遍历。在这种情况下，**关键是如何优化效率**，常见的效率优化方法有两种。
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- **剪枝**：避免搜索那些肯定不会产生解的路径，从而节省时间和空间。
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- **启发式搜索**：在搜索过程中引入一些策略或者估计值，从而优先搜索最有可能产生有效解的路径。
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## 13.1.6 回溯典型例题
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回溯算法可用于解决许多搜索问题、约束满足问题和组合优化问题。
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**搜索问题**：这类问题的目标是找到满足特定条件的解决方案。
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- 全排列问题：给定一个集合，求出其所有可能的排列组合。
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- 子集和问题：给定一个集合和一个目标和，找到集合中所有和为目标和的子集。
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- 汉诺塔问题：给定三根柱子和一系列大小不同的圆盘，要求将所有圆盘从一根柱子移动到另一根柱子，每次只能移动一个圆盘，且不能将大圆盘放在小圆盘上。
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**约束满足问题**：这类问题的目标是找到满足所有约束条件的解。
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- $n$ 皇后：在 $n \times n$ 的棋盘上放置 $n$ 个皇后，使得它们互不攻击。
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- 数独：在 $9 \times 9$ 的网格中填入数字 $1$ ~ $9$ ，使得每行、每列和每个 $3 \times 3$ 子网格中的数字不重复。
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- 图着色问题：给定一个无向图，用最少的颜色给图的每个顶点着色，使得相邻顶点颜色不同。
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**组合优化问题**：这类问题的目标是在一个组合空间中找到满足某些条件的最优解。
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- 0-1 背包问题：给定一组物品和一个背包，每个物品有一定的价值和重量，要求在背包容量限制内，选择物品使得总价值最大。
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- 旅行商问题：在一个图中，从一个点出发，访问所有其他点恰好一次后返回起点，求最短路径。
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- 最大团问题：给定一个无向图，找到最大的完全子图，即子图中的任意两个顶点之间都有边相连。
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请注意，对于许多组合优化问题，回溯不是最优解决方案。
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- 0-1 背包问题通常使用动态规划解决，以达到更高的时间效率。
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- 旅行商是一个著名的 NP-Hard 问题，常用解法有遗传算法和蚁群算法等。
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- 最大团问题是图论中的一个经典问题，可用贪心算法等启发式算法来解决。
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