From ecabb4077b32fa451f0c2b3fdfbc434f51b92ae2 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Sun, 8 Jan 2023 22:18:23 +0800 Subject: [PATCH 01/23] Update codes of heap.java and my_heap.java --- codes/java/chapter_heap/heap.java | 60 +++++++ codes/java/chapter_heap/my_heap.java | 170 +++++++++++++++++++ codes/java/chapter_tree/binary_tree_bfs.java | 2 +- codes/java/chapter_tree/binary_tree_dfs.java | 2 +- codes/java/include/PrintUtil.java | 14 +- codes/java/include/TreeNode.java | 21 +-- docs/chapter_heap/heap.md | 54 ++++++ 7 files changed, 307 insertions(+), 16 deletions(-) create mode 100644 codes/java/chapter_heap/heap.java create mode 100644 codes/java/chapter_heap/my_heap.java create mode 100644 docs/chapter_heap/heap.md diff --git a/codes/java/chapter_heap/heap.java b/codes/java/chapter_heap/heap.java new file mode 100644 index 000000000..16e35e603 --- /dev/null +++ b/codes/java/chapter_heap/heap.java @@ -0,0 +1,60 @@ +/** + * File: my_heap.java + * Created Time: 2023-01-07 + * Author: Krahets (krahets@163.com) + */ + +package chapter_heap; + +import include.*; +import java.util.*; + + +public class heap { + public static void testPush(Queue heap, int val) { + // 元素入堆 + heap.add(val); + + System.out.format("\n添加元素 %d 后\n", val); + PrintUtil.printHeap(heap); + } + + public static void testPoll(Queue heap) { + // 元素出堆 + int val = heap.poll(); + + System.out.format("\n出堆元素为 %d\n", val); + PrintUtil.printHeap(heap); + } + + public static void main(String[] args) { + /* 初始化堆 */ + // 初始化最小堆 + Queue minHeap = new PriorityQueue<>(); + // 初始化最大堆(使用 lambda 表达式修改 Comparator) + Queue maxHeap = new PriorityQueue<>((a, b) -> { return b - a; }); + + /* 元素入堆 */ + testPush(maxHeap, 1); + testPush(maxHeap, 3); + testPush(maxHeap, 2); + testPush(maxHeap, 5); + testPush(maxHeap, 4); + + /* 获取堆顶元素 */ + int peek = maxHeap.peek(); + System.out.format("\n堆顶元素为 %d\n", peek); + + /* 元素出堆 */ + testPoll(maxHeap); + testPoll(maxHeap); + + /* 获取堆大小 */ + int size = maxHeap.size(); + System.out.format("\n堆元素数量为 %d\n", size); + + /* 判断堆是否为空 */ + boolean isEmpty = maxHeap.isEmpty(); + System.out.format("\n堆是否为空 %b\n", isEmpty); + } +} diff --git a/codes/java/chapter_heap/my_heap.java b/codes/java/chapter_heap/my_heap.java new file mode 100644 index 000000000..39583c1d7 --- /dev/null +++ b/codes/java/chapter_heap/my_heap.java @@ -0,0 +1,170 @@ +/** + * File: my_heap.java + * Created Time: 2023-01-07 + * Author: Krahets (krahets@163.com) + */ + +package chapter_heap; + +import include.*; +import java.util.*; + +class MaxHeap { + // 使用列表而非数组,这样无需考虑扩容问题 + private List maxHeap; + + /* 构造函数,建立空堆 */ + public MaxHeap() { + maxHeap = new ArrayList<>(); + } + + /* 构造函数,堆化 nums 所有元素 */ + public MaxHeap(List nums) { + // 将元素拷贝至堆中 + maxHeap = new ArrayList<>(nums); + // 堆化除叶结点以外的其他所有结点 + for (int i = parent(size() - 1); i >= 0; i--) { + heapify(i); + } + } + + /* 获取左子结点索引 */ + private int left(int i) { + return 2 * i + 1; + } + + /* 获取右子结点索引 */ + private int right(int i) { + return 2 * i + 2; + } + + /* 获取父结点索引 */ + private int parent(int i) { + return (i - 1) / 2; + } + + /* 交换元素 */ + private void swap(int i, int j) { + int a = maxHeap.get(i), + b = maxHeap.get(j), + tmp = a; + maxHeap.set(i, b); + maxHeap.set(j, tmp); + } + + /* 获取堆大小 */ + public int size() { + return maxHeap.size(); + } + + /* 判断堆是否为空 */ + public boolean isEmpty() { + return size() == 0; + } + + /* 访问堆顶元素 */ + public int peek() { + return maxHeap.get(0); + } + + /* 元素入堆 */ + public void push(int val) { + // 添加结点 + maxHeap.add(val); + // 从底至顶堆化 + int i = size() - 1; + while (true) { + int p = parent(i); + if (p < 0 || maxHeap.get(i) <= maxHeap.get(p)) + break; + swap(i, p); + i = p; + } + } + + /* 元素出堆 */ + public int poll() { + // 判空处理 + if (isEmpty()) + throw new EmptyStackException(); + // 交换根结点与右下角(即最后一个)结点 + swap(0, size() - 1); + // 删除结点 + int val = maxHeap.remove(size() - 1); + // 从顶至底堆化 + heapify(0); + // 返回堆顶元素 + return val; + } + + /* 从结点 i 开始,从顶至底堆化 */ + private void heapify(int i) { + while (true) { + // 判断结点 i, l, r 中值最大的结点,记为 ma ; + int l = left(i), r = right(i), ma = i; + if (l < size() && maxHeap.get(l) > maxHeap.get(ma)) ma = l; + if (r < size() && maxHeap.get(r) > maxHeap.get(ma)) ma = r; + // 若结点 i 最大,则无需继续堆化,跳出 + if (ma == i) break; + // 交换结点 i 与结点 max + swap(i, ma); + // 循环向下堆化 + i = ma; + } + } + + /* 打印堆(二叉树) */ + public void print() { + Queue queue = new PriorityQueue<>((a, b) -> { return b - a; }); + queue.addAll(maxHeap); + PrintUtil.printHeap(queue); + } +} + + +public class my_heap { + public static void testPush(MaxHeap maxHeap, int val) { + // 元素入堆 + maxHeap.push(val); + + System.out.format("\n添加元素 %d 后\n", val); + maxHeap.print(); + } + + public static void testPoll(MaxHeap maxHeap) { + // 元素出堆 + int val = maxHeap.poll(); + + System.out.format("\n出堆元素为 %d\n", val); + maxHeap.print(); + } + + public static void main(String[] args) { + /* 初始化堆 */ + // 初始化最大堆 + MaxHeap maxHeap = new MaxHeap(); + + /* 元素入堆 */ + testPush(maxHeap, 1); + testPush(maxHeap, 3); + testPush(maxHeap, 2); + testPush(maxHeap, 5); + testPush(maxHeap, 4); + + /* 获取堆顶元素 */ + int peek = maxHeap.peek(); + System.out.format("\n堆顶元素为 %d\n", peek); + + /* 元素出堆 */ + testPoll(maxHeap); + testPoll(maxHeap); + + /* 获取堆大小 */ + int size = maxHeap.size(); + System.out.format("\n堆元素数量为 %d\n", size); + + /* 判断堆是否为空 */ + boolean isEmpty = maxHeap.isEmpty(); + System.out.format("\n堆是否为空 %b\n", isEmpty); + } +} diff --git a/codes/java/chapter_tree/binary_tree_bfs.java b/codes/java/chapter_tree/binary_tree_bfs.java index 450311d08..ffeb352e0 100644 --- a/codes/java/chapter_tree/binary_tree_bfs.java +++ b/codes/java/chapter_tree/binary_tree_bfs.java @@ -30,7 +30,7 @@ public class binary_tree_bfs { public static void main(String[] args) { /* 初始化二叉树 */ // 这里借助了一个从数组直接生成二叉树的函数 - TreeNode root = TreeNode.arrToTree(new Integer[] { 1, 2, 3, 4, 5, 6, 7 }); + TreeNode root = TreeNode.listToTree(Arrays.asList(1, 2, 3, 4, 5, 6, 7)); System.out.println("\n初始化二叉树\n"); PrintUtil.printTree(root); diff --git a/codes/java/chapter_tree/binary_tree_dfs.java b/codes/java/chapter_tree/binary_tree_dfs.java index ad5337f22..9f2d0507d 100644 --- a/codes/java/chapter_tree/binary_tree_dfs.java +++ b/codes/java/chapter_tree/binary_tree_dfs.java @@ -43,7 +43,7 @@ public class binary_tree_dfs { public static void main(String[] args) { /* 初始化二叉树 */ // 这里借助了一个从数组直接生成二叉树的函数 - TreeNode root = TreeNode.arrToTree(new Integer[] { 1, 2, 3, 4, 5, 6, 7 }); + TreeNode root = TreeNode.listToTree(Arrays.asList(1, 2, 3, 4, 5, 6, 7)); System.out.println("\n初始化二叉树\n"); PrintUtil.printTree(root); diff --git a/codes/java/include/PrintUtil.java b/codes/java/include/PrintUtil.java index b2593ac1f..1ebc45ba0 100755 --- a/codes/java/include/PrintUtil.java +++ b/codes/java/include/PrintUtil.java @@ -105,10 +105,16 @@ public class PrintUtil { } } - public static void printHeap(PriorityQueue queue) { - Integer[] nums = (Integer[])queue.toArray(); - TreeNode root = TreeNode.arrToTree(nums); - + /** + * Print a heap (PriorityQueue) + * @param queue + */ + public static void printHeap(Queue queue) { + List list = new ArrayList<>(queue); + System.out.print("堆的数组表示:"); + System.out.println(list); + System.out.println("堆的树状表示:"); + TreeNode root = TreeNode.listToTree(list); printTree(root); } } diff --git a/codes/java/include/TreeNode.java b/codes/java/include/TreeNode.java index 11d457a10..22a349747 100644 --- a/codes/java/include/TreeNode.java +++ b/codes/java/include/TreeNode.java @@ -23,26 +23,27 @@ public class TreeNode { /** * Generate a binary tree given an array - * @param arr + * @param list * @return */ - public static TreeNode arrToTree(Integer[] arr) { - if (arr.length == 0) + public static TreeNode listToTree(List list) { + int size = list.size(); + if (size == 0) return null; - TreeNode root = new TreeNode(arr[0]); + TreeNode root = new TreeNode(list.get(0)); Queue queue = new LinkedList<>() {{ add(root); }}; int i = 0; while(!queue.isEmpty()) { TreeNode node = queue.poll(); - if (++i >= arr.length) break; - if(arr[i] != null) { - node.left = new TreeNode(arr[i]); + if (++i >= size) break; + if (list.get(i) != null) { + node.left = new TreeNode(list.get(i)); queue.add(node.left); } - if (++i >= arr.length) break; - if(arr[i] != null) { - node.right = new TreeNode(arr[i]); + if (++i >= size) break; + if (list.get(i) != null) { + node.right = new TreeNode(list.get(i)); queue.add(node.right); } } diff --git a/docs/chapter_heap/heap.md b/docs/chapter_heap/heap.md new file mode 100644 index 000000000..b0151622f --- /dev/null +++ b/docs/chapter_heap/heap.md @@ -0,0 +1,54 @@ +# 堆 + +「堆 Heap」是一种特殊的树状数据结构,并且是一颗「完全二叉树」。堆主要分为两种: + +- 「大顶堆 Max Heap」,任意结点的值 $\geq$ 其子结点的值,因此根结点的值最大; +- 「小顶堆 Min Heap」,任意结点的值 $\leq$ 其子结点的值,因此根结点的值最小; + +(图) + +!!! tip "" + + 大顶堆和小顶堆的定义、性质、操作本质上是相同的,区别只是大顶堆在求最大值,小顶堆在求最小值。 + +## 堆常用操作 + +值得说明的是,多数编程语言提供的是「优先队列 Priority Queue」,其是一种抽象数据结构,**定义为具有出队优先级的队列**。 + +而恰好,堆的定义与优先队列的操作逻辑完全吻合,大顶堆就是一个元素从大到小出队的优先队列。从使用角度看,我们可以将「优先队列」和「堆」理解为等价的数据结构,下文将统一使用 “堆” 这个名称。 + +堆的常用操作见下表(方法命名以 Java 为例)。 + +

Table. 堆的常用操作

+ +
+ +| 方法 | 描述 | +| --------- | -------------------------------------------- | +| add() | 元素入堆 | +| poll() | 堆顶元素出堆 | +| peek() | 访问堆顶元素(大 / 小顶堆分别为最大 / 小值) | +| size() | 获取堆的元素数量 | +| isEmpty() | 判断堆是否为空 | + +
+ +```java + +``` + +## 堆的实现 + +!!! tip + + 下文使用「大顶堆」来举例,「小顶堆」的用法与实现可以简单地将所有 $>$ ($<$) 替换为 $<$ ($>$) 即可。 + +我们一般使用「数组」来存储「堆」,这是因为完全二叉树非常适合用数组来表示(在二叉树章节有详细解释)。 + + + +## 堆常见应用 + +- 优先队列。 +- 堆排序。 +- 获取数据 Top K 大(小)元素。 From 80c5a0472af1c57e930ccde1238700179931067e Mon Sep 17 00:00:00 2001 From: xiongsp <37035570+xiongsp@users.noreply.github.com> Date: Sun, 8 Jan 2023 23:22:01 +0800 Subject: [PATCH 02/23] update Docker --- Dockerfile | 20 ++++++++++++++++++++ docker-compose.yml | 7 +++++++ 2 files changed, 27 insertions(+) create mode 100644 Dockerfile create mode 100644 docker-compose.yml diff --git a/Dockerfile b/Dockerfile new file mode 100644 index 000000000..5d437c846 --- /dev/null +++ b/Dockerfile @@ -0,0 +1,20 @@ +FROM python:3.9.0-buster + +RUN sed -i 's/deb.debian.org/mirrors.aliyun.com/g' /etc/apt/sources.list +RUN sed -i 's/security.debian.org/mirrors.aliyun.com/g' /etc/apt/sources.list + +RUN pip install --upgrade pip && pip install mkdocs mkdocs-material + +WORKDIR /app + +COPY codes /app/codes + +COPY docs /app/docs + +COPY mkdocs.yml /app/mkdocs.yml + +RUN mkdir ./docs/overrides && mkdocs build + +EXPOSE 8000 + +CMD ["mkdocs", "serve", "-a", "0.0.0.0:8000"] \ No newline at end of file diff --git a/docker-compose.yml b/docker-compose.yml new file mode 100644 index 000000000..6a472b173 --- /dev/null +++ b/docker-compose.yml @@ -0,0 +1,7 @@ +version: '3' +services: + hello-algo: + build: . + ports: + - "8000:8000" + \ No newline at end of file From 6b3c87399b507ef6bd43c78375c8c2cdfef4d9e5 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Mon, 9 Jan 2023 02:17:40 +0800 Subject: [PATCH 03/23] Add time complexity in stack, queue, deque. Update heap. --- .prettierrc | 4 - codes/java/chapter_heap/heap.java | 22 ++- codes/java/chapter_heap/my_heap.java | 53 +++--- docs/chapter_heap/heap.md | 230 ++++++++++++++++++++++++-- docs/chapter_stack_and_queue/deque.md | 22 +-- docs/chapter_stack_and_queue/queue.md | 20 +-- docs/chapter_stack_and_queue/stack.md | 20 +-- docs/chapter_tree/summary.md | 2 +- 8 files changed, 295 insertions(+), 78 deletions(-) delete mode 100644 .prettierrc diff --git a/.prettierrc b/.prettierrc deleted file mode 100644 index 5a938ce18..000000000 --- a/.prettierrc +++ /dev/null @@ -1,4 +0,0 @@ -{ - "tabWidth": 4, - "useTabs": false -} diff --git a/codes/java/chapter_heap/heap.java b/codes/java/chapter_heap/heap.java index 16e35e603..b964830fd 100644 --- a/codes/java/chapter_heap/heap.java +++ b/codes/java/chapter_heap/heap.java @@ -12,28 +12,26 @@ import java.util.*; public class heap { public static void testPush(Queue heap, int val) { - // 元素入堆 - heap.add(val); - + heap.add(val); // 元素入堆 System.out.format("\n添加元素 %d 后\n", val); PrintUtil.printHeap(heap); } public static void testPoll(Queue heap) { - // 元素出堆 - int val = heap.poll(); - + int val = heap.poll(); // 堆顶元素出堆 System.out.format("\n出堆元素为 %d\n", val); PrintUtil.printHeap(heap); } public static void main(String[] args) { /* 初始化堆 */ - // 初始化最小堆 + // 初始化小顶堆 Queue minHeap = new PriorityQueue<>(); - // 初始化最大堆(使用 lambda 表达式修改 Comparator) + // 初始化大顶堆(使用 lambda 表达式修改 Comparator 即可) Queue maxHeap = new PriorityQueue<>((a, b) -> { return b - a; }); + System.out.println("\n以下测试样例为大顶堆"); + /* 元素入堆 */ testPush(maxHeap, 1); testPush(maxHeap, 3); @@ -45,7 +43,7 @@ public class heap { int peek = maxHeap.peek(); System.out.format("\n堆顶元素为 %d\n", peek); - /* 元素出堆 */ + /* 堆顶元素出堆 */ testPoll(maxHeap); testPoll(maxHeap); @@ -56,5 +54,11 @@ public class heap { /* 判断堆是否为空 */ boolean isEmpty = maxHeap.isEmpty(); System.out.format("\n堆是否为空 %b\n", isEmpty); + + /* 输入列表并建堆 */ + // 时间复杂度为 O(n) ,而非 O(nlogn) + minHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4)); + System.out.println("\n输入 [1, 3, 2, 5, 4] ,建立小顶堆"); + PrintUtil.printHeap(minHeap); } } diff --git a/codes/java/chapter_heap/my_heap.java b/codes/java/chapter_heap/my_heap.java index 39583c1d7..89f5c6e23 100644 --- a/codes/java/chapter_heap/my_heap.java +++ b/codes/java/chapter_heap/my_heap.java @@ -18,13 +18,13 @@ class MaxHeap { maxHeap = new ArrayList<>(); } - /* 构造函数,堆化 nums 所有元素 */ + /* 构造函数,根据输入列表建堆 */ public MaxHeap(List nums) { - // 将元素拷贝至堆中 + // 所有元素入堆 maxHeap = new ArrayList<>(nums); // 堆化除叶结点以外的其他所有结点 for (int i = parent(size() - 1); i >= 0; i--) { - heapify(i); + siftDown(i); } } @@ -40,7 +40,7 @@ class MaxHeap { /* 获取父结点索引 */ private int parent(int i) { - return (i - 1) / 2; + return (i - 1) / 2; // 向下整除 } /* 交换元素 */ @@ -72,12 +72,20 @@ class MaxHeap { // 添加结点 maxHeap.add(val); // 从底至顶堆化 - int i = size() - 1; + siftUp(size() - 1); + } + + /* 从结点 i 开始,从底至顶堆化 */ + private void siftUp(int i) { while (true) { + // 获取结点 i 的父结点 int p = parent(i); + // 当“越过根结点”或“结点无需修复”时,结束堆化 if (p < 0 || maxHeap.get(i) <= maxHeap.get(p)) break; + // 交换两结点 swap(i, p); + // 循环向上堆化 i = p; } } @@ -87,26 +95,28 @@ class MaxHeap { // 判空处理 if (isEmpty()) throw new EmptyStackException(); - // 交换根结点与右下角(即最后一个)结点 + // 交换根结点与最右叶结点(即交换首元素与尾元素) swap(0, size() - 1); // 删除结点 int val = maxHeap.remove(size() - 1); // 从顶至底堆化 - heapify(0); + siftDown(0); // 返回堆顶元素 return val; } /* 从结点 i 开始,从顶至底堆化 */ - private void heapify(int i) { + private void siftDown(int i) { while (true) { // 判断结点 i, l, r 中值最大的结点,记为 ma ; int l = left(i), r = right(i), ma = i; - if (l < size() && maxHeap.get(l) > maxHeap.get(ma)) ma = l; - if (r < size() && maxHeap.get(r) > maxHeap.get(ma)) ma = r; - // 若结点 i 最大,则无需继续堆化,跳出 + if (l < size() && maxHeap.get(l) > maxHeap.get(ma)) + ma = l; + if (r < size() && maxHeap.get(r) > maxHeap.get(ma)) + ma = r; + // 若结点 i 最大或索引 l, r 越界,则无需继续堆化,跳出 if (ma == i) break; - // 交换结点 i 与结点 max + // 交换两结点 swap(i, ma); // 循环向下堆化 i = ma; @@ -124,26 +134,24 @@ class MaxHeap { public class my_heap { public static void testPush(MaxHeap maxHeap, int val) { - // 元素入堆 - maxHeap.push(val); - + maxHeap.push(val); // 元素入堆 System.out.format("\n添加元素 %d 后\n", val); maxHeap.print(); } public static void testPoll(MaxHeap maxHeap) { - // 元素出堆 - int val = maxHeap.poll(); - + int val = maxHeap.poll(); // 堆顶元素出堆 System.out.format("\n出堆元素为 %d\n", val); maxHeap.print(); } public static void main(String[] args) { /* 初始化堆 */ - // 初始化最大堆 + // 初始化大顶堆 MaxHeap maxHeap = new MaxHeap(); + System.out.println("\n以下测试样例为大顶堆"); + /* 元素入堆 */ testPush(maxHeap, 1); testPush(maxHeap, 3); @@ -155,7 +163,7 @@ public class my_heap { int peek = maxHeap.peek(); System.out.format("\n堆顶元素为 %d\n", peek); - /* 元素出堆 */ + /* 堆顶元素出堆 */ testPoll(maxHeap); testPoll(maxHeap); @@ -166,5 +174,10 @@ public class my_heap { /* 判断堆是否为空 */ boolean isEmpty = maxHeap.isEmpty(); System.out.format("\n堆是否为空 %b\n", isEmpty); + + /* 将输入列表堆化 */ + maxHeap = new MaxHeap(Arrays.asList(1, 3, 2, 5, 4)); + System.out.println("\n输入 [1, 3, 2, 5, 4] ,建立大顶堆"); + maxHeap.print(); } } diff --git a/docs/chapter_heap/heap.md b/docs/chapter_heap/heap.md index b0151622f..03a485f99 100644 --- a/docs/chapter_heap/heap.md +++ b/docs/chapter_heap/heap.md @@ -15,7 +15,7 @@ 值得说明的是,多数编程语言提供的是「优先队列 Priority Queue」,其是一种抽象数据结构,**定义为具有出队优先级的队列**。 -而恰好,堆的定义与优先队列的操作逻辑完全吻合,大顶堆就是一个元素从大到小出队的优先队列。从使用角度看,我们可以将「优先队列」和「堆」理解为等价的数据结构,下文将统一使用 “堆” 这个名称。 +而恰好,堆的定义与优先队列的操作逻辑完全吻合,大顶堆就是一个元素从大到小出队的优先队列。从使用角度看,我们可以将「优先队列」和「堆」理解为等价的数据结构。因此,本文与代码对两者不做特别区分,统一使用「堆」来命名。 堆的常用操作见下表(方法命名以 Java 为例)。 @@ -23,32 +23,236 @@
-| 方法 | 描述 | -| --------- | -------------------------------------------- | -| add() | 元素入堆 | -| poll() | 堆顶元素出堆 | -| peek() | 访问堆顶元素(大 / 小顶堆分别为最大 / 小值) | -| size() | 获取堆的元素数量 | -| isEmpty() | 判断堆是否为空 | +| 方法 | 描述 | 时间复杂度 | +| --------- | -------------------------------------------- | ----------- | +| add() | 元素入堆 | $O(\log n)$ | +| poll() | 堆顶元素出堆 | $O(\log n)$ | +| peek() | 访问堆顶元素(大 / 小顶堆分别为最大 / 小值) | $O(1)$ | +| size() | 获取堆的元素数量 | $O(1)$ | +| isEmpty() | 判断堆是否为空 | $O(1)$ |
+我们可以直接使用编程语言提供的堆类(或优先队列类)。 + ```java +/* 初始化堆 */ +// 初始化小顶堆 +Queue minHeap = new PriorityQueue<>(); +// 初始化大顶堆(使用 lambda 表达式修改 Comparator 即可) +Queue maxHeap = new PriorityQueue<>((a, b) -> { return b - a; }); + +/* 元素入堆 */ +maxHeap.add(1); +maxHeap.add(3); +maxHeap.add(2); +maxHeap.add(5); +maxHeap.add(4); + +/* 获取堆顶元素 */ +int peek = maxHeap.peek(); + +/* 堆顶元素出堆 */ +int val = heap.poll(); +/* 获取堆大小 */ +int size = maxHeap.size(); + +/* 判断堆是否为空 */ +boolean isEmpty = maxHeap.isEmpty(); + +/* 输入列表并建堆 */ +// 时间复杂度为 O(n) ,而非 O(nlogn) +minHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4)); ``` ## 堆的实现 !!! tip - 下文使用「大顶堆」来举例,「小顶堆」的用法与实现可以简单地将所有 $>$ ($<$) 替换为 $<$ ($>$) 即可。 + 下文使用「大顶堆」来举例,将所有 $>$ ($<$) 替换为 $<$ ($>$) 即可实现「小顶堆」。 + +### 堆的存储与表示 + +在二叉树章节我们学过,「完全二叉树」非常适合使用「数组」来表示,而堆恰好是一颗完全二叉树,因而我们一般使用「数组」来存储「堆」。 + +**二叉树指针**。使用数组表示二叉树时,数组元素都代表结点值,索引代表结点在二叉树中的位置,**结点指针通过索引映射公式来实现**。具体地,给定索引 $i$ ,那么其左子结点索引为 $2i + 1$ 、右子结点索引为 $2i + 2$ 、父结点索引为 $(i - 1) / 2$ (向下整除)。当索引越界时,代表空结点或结点不存在。我们将以上映射公式封装成函数,以便使用。 + +(图) + +```java +// 使用列表而非数组,这样无需考虑扩容问题 +List maxHeap; + +/* 构造函数,建立空堆 */ +public MaxHeap() { + maxHeap = new ArrayList<>(); +} -我们一般使用「数组」来存储「堆」,这是因为完全二叉树非常适合用数组来表示(在二叉树章节有详细解释)。 +/* 获取左子结点索引 */ +int left(int i) { + return 2 * i + 1; +} + +/* 获取右子结点索引 */ +int right(int i) { + return 2 * i + 2; +} + +/* 获取父结点索引 */ +int parent(int i) { + return (i - 1) / 2; // 向下整除 +} +``` + +### 访问堆顶元素 + +堆顶元素是二叉树的根结点,即列表首元素。 + +```java +/* 访问堆顶元素 */ +public int peek() { + return maxHeap.get(0); +} +``` + +### 元素入堆 + +给定元素 `val` ,我们先将其添加到堆的末尾。由于 `val` 可能大于其它元素,此时堆的性质可能被破坏了,我们需要修复从插入结点到根结点这条路径上的各个结点,该操作被称为「堆化 Heapify」。 + +考虑从入堆结点开始,**从底至顶执行堆化**。具体地,比较插入结点与其父结点的值,若插入结点更大则将它们交换;并循环以上操作,从底至顶地修复堆中的各个结点;直至越过根结点时结束,或当遇到无需交换的结点时提前结束。 + +设堆长度为 $n$ ,**元素入堆操作的时间复杂度为 $O(\log n)$** 。这是因为树的高度为 $O(\log n)$ ,因此堆化操作的循环轮数最多为 $O(\log n)$ 。 + +(图) +```java +/* 元素入堆 */ +void push(int val) { + // 添加结点 + maxHeap.add(val); + // 从底至顶堆化 + siftUp(size() - 1); +} + +/* 从结点 i 开始,从底至顶堆化 */ +void siftUp(int i) { + while (true) { + // 获取结点 i 的父结点 + int p = parent(i); + // 若“越过根结点”或“结点无需修复”,则结束堆化 + if (p < 0 || maxHeap.get(i) <= maxHeap.get(p)) + break; + // 交换两结点 + swap(i, p); + // 循环向上堆化 + i = p; + } +} +``` + +### 堆顶元素出堆 + +堆顶元素是二叉树根结点,即列表首元素,如果我们直接将首元素从列表中删除,则二叉树中所有结点都产生移位,这样后续使用堆化修复就很麻烦了。为了尽量减少二叉树结点变动,采取以下操作步骤: + +1. 交换列表首元素与尾元素(即交换根结点与最右叶结点); +2. 将尾元素从列表中删除(此时堆顶元素已被删除); +3. 从根结点开始,从顶至底堆化; + +顾名思义,**从顶至底堆化的操作方向与从底至顶堆化相反**,我们比较根结点的值与其两个子结点的值,将最大的子结点与根结点执行交换,并循环以上操作,直到越过叶结点时结束,或当遇到无需交换的结点时提前结束。 + +(图) + +```java +/* 元素出堆 */ +int poll() { + // 判空处理 + if (isEmpty()) + throw new EmptyStackException(); + // 交换根结点与最右叶结点(即交换首元素与尾元素) + swap(0, size() - 1); + // 删除结点 + int val = maxHeap.remove(size() - 1); + // 从顶至底堆化 + siftDown(0); + // 返回堆顶元素 + return val; +} + +/* 从结点 i 开始,从顶至底堆化 */ +void siftDown(int i) { + while (true) { + // 判断结点 i, l, r 中值最大的结点,记为 ma ; + int l = left(i), r = right(i), ma = i; + if (l < size() && maxHeap.get(l) > maxHeap.get(ma)) + ma = l; + if (r < size() && maxHeap.get(r) > maxHeap.get(ma)) + ma = r; + // 若“结点 i 最大”或“越过叶结点”,则结束堆化 + if (ma == i) break; + // 交换两结点 + swap(i, ma); + // 循环向下堆化 + i = ma; + } +} +``` + +### 输入数据并建堆 * + +给定一个列表,我们也可以将其建堆。最直接地,可以通过调用「元素入堆」方法,将列表元素依次入堆。元素入堆的时间复杂度为 $O(n)$ ,而平均长度为 $\frac{n}{2}$ ,因此该方法的总体时间复杂度为 $O(n \log n)$ 。 + +然而,存在一种更加优雅的建堆方法。设结点数量为 $n$ ,我们先将列表所有元素原封不动添加进堆,**然后迭代地对各个结点执行「从顶至底堆化」**。当然,**无需对叶结点执行堆化,**因为其没有子结点。 + +```java +/* 构造函数,根据输入列表建堆 */ +public MaxHeap(List nums) { + // 将列表元素原封不动添加进堆 + maxHeap = new ArrayList<>(nums); + // 堆化除叶结点以外的其他所有结点 + for (int i = parent(size() - 1); i >= 0; i--) { + siftDown(i); + } +} +``` + +!!! tip + + 完全二叉树的叶结点数量为 $(n + 1) / 2$ ,其中 $/$ 为向下整除。 + +那么,第二种建堆方法的时间复杂度时多少呢?我们来做一下简单推算。叶结点和需要堆化结点的数量各占约一半,即为 $O(n)$ ,二叉树高度为 $O(\log n)$ ,可得时间复杂度为 $O(n \log n)$ 。该估算结果仍不够准确,因为我们没有考虑到二叉树“底层结点远多于顶层结点”的性质。 + +设二叉树(即堆)结点数量为 $n$ ,树高度为 $h$ 。如下图所示,我们将各层的“结点数量 $\times$ 子树高度”进行求和,即可得到准确的操作数量。 + +$$ +S = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{(h-1)}\times1 +$$ + +(图) +求解上式需要借助中学的数列知识,先对 $S$ 乘以 $2$ ,可得 +$$ +\begin{aligned} +S & = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{h-1}\times1 \\ +2S & = 2^1h + 2^2(h-1) + 2^3(h-2) + \cdots + 2^{h}\times1 \\ +\end{aligned} +$$ +令下式 $2S$ 与上式 $S$ 错位相减,易得 +$$ +2S - S = S = -2^0h + 2^1 + 2^2 + \cdots + 2^{h-1} + 2^h +$$ +观察发现,$S$ 是一个等比数列,可直接借助公式求和。并且,对于高度为 $h$ 的完全二叉树,结点数量范围为 $n \in [2^h, 2^{h+1} - 1]$ ,复杂度为 $n = O(n) = O(2^h)$。 +$$ +\begin{aligned} +S & = 2 \frac{1 - 2^h}{1 - 2} - h \\ +& = 2^{h+1} - h \\ +& = O(2^h) = O(n) +\end{aligned} +$$ +以上推算表明,输入列表并建堆的时间复杂度为 $O(n)$ ,非常高效。 ## 堆常见应用 -- 优先队列。 -- 堆排序。 -- 获取数据 Top K 大(小)元素。 +- 优先队列。堆常作为实现优先队列的首选数据结构。 +- 堆排序。根据 +- 获取数据中最大的 $k$ 个元素。这即是一道经典的算法题目,也是一种实际应用 diff --git a/docs/chapter_stack_and_queue/deque.md b/docs/chapter_stack_and_queue/deque.md index 52b0540d5..4671d5c9e 100644 --- a/docs/chapter_stack_and_queue/deque.md +++ b/docs/chapter_stack_and_queue/deque.md @@ -18,16 +18,16 @@ comments: true
-| 方法 | 描述 | -| ------------ | ---------------- | -| offerFirst() | 将元素添加至队首 | -| offerLast() | 将元素添加至队尾 | -| pollFirst() | 删除队首元素 | -| pollLast() | 删除队尾元素 | -| peekFirst() | 访问队首元素 | -| peekLast() | 访问队尾元素 | -| size() | 获取队列的长度 | -| isEmpty() | 判断队列是否为空 | +| 方法 | 描述 | 时间复杂度 | +| ------------ | ---------------- | ---------- | +| offerFirst() | 将元素添加至队首 | $O(1)$ | +| offerLast() | 将元素添加至队尾 | $O(1)$ | +| pollFirst() | 删除队首元素 | $O(1)$ | +| pollLast() | 删除队尾元素 | $O(1)$ | +| peekFirst() | 访问队首元素 | $O(1)$ | +| peekLast() | 访问队尾元素 | $O(1)$ | +| size() | 获取队列的长度 | $O(1)$ | +| isEmpty() | 判断队列是否为空 | $O(1)$ |
@@ -196,5 +196,5 @@ comments: true === "Swift" ```swift title="deque.swift" - + ``` diff --git a/docs/chapter_stack_and_queue/queue.md b/docs/chapter_stack_and_queue/queue.md index ce47edb38..65ce56f07 100644 --- a/docs/chapter_stack_and_queue/queue.md +++ b/docs/chapter_stack_and_queue/queue.md @@ -20,13 +20,13 @@ comments: true
-| 方法 | 描述 | -| --------- | ---------------------------- | -| offer() | 元素入队,即将元素添加至队尾 | -| poll() | 队首元素出队 | -| front() | 访问队首元素 | -| size() | 获取队列的长度 | -| isEmpty() | 判断队列是否为空 | +| 方法 | 描述 | 时间复杂度 | +| --------- | ---------------------------- | ---------- | +| offer() | 元素入队,即将元素添加至队尾 | $O(1)$ | +| poll() | 队首元素出队 | $O(1)$ | +| front() | 访问队首元素 | $O(1)$ | +| size() | 获取队列的长度 | $O(1)$ | +| isEmpty() | 判断队列是否为空 | $O(1)$ |
@@ -231,7 +231,7 @@ comments: true === "Swift" ```swift title="queue.swift" - + ``` ## 队列实现 @@ -621,7 +621,7 @@ comments: true === "Swift" ```swift title="linkedlist_queue.swift" - + ``` ### 基于数组的实现 @@ -1030,7 +1030,7 @@ comments: true === "Swift" ```swift title="array_queue.swift" - + ``` ## 队列典型应用 diff --git a/docs/chapter_stack_and_queue/stack.md b/docs/chapter_stack_and_queue/stack.md index 374de7424..e5f67371a 100644 --- a/docs/chapter_stack_and_queue/stack.md +++ b/docs/chapter_stack_and_queue/stack.md @@ -22,13 +22,13 @@ comments: true
-| 方法 | 描述 | -| --------- | ---------------------- | -| push() | 元素入栈(添加至栈顶) | -| pop() | 栈顶元素出栈 | -| peek() | 访问栈顶元素 | -| size() | 获取栈的长度 | -| isEmpty() | 判断栈是否为空 | +| 方法 | 描述 | 时间复杂度 | +| --------- | ---------------------- | ---------- | +| push() | 元素入栈(添加至栈顶) | $O(1)$ | +| pop() | 栈顶元素出栈 | $O(1)$ | +| peek() | 访问栈顶元素 | $O(1)$ | +| size() | 获取栈的长度 | $O(1)$ | +| isEmpty() | 判断栈是否为空 | $O(1)$ |
@@ -231,7 +231,7 @@ comments: true === "Swift" ```swift title="stack.swift" - + ``` ## 栈的实现 @@ -600,7 +600,7 @@ comments: true === "Swift" ```swift title="linkedlist_stack.swift" - + ``` ### 基于数组的实现 @@ -885,7 +885,7 @@ comments: true === "Swift" ```swift title="array_stack.swift" - + ``` !!! tip diff --git a/docs/chapter_tree/summary.md b/docs/chapter_tree/summary.md index a46a253f9..15dc2f6d3 100644 --- a/docs/chapter_tree/summary.md +++ b/docs/chapter_tree/summary.md @@ -15,4 +15,4 @@ comments: true - 前序、中序、后序遍历是深度优先搜索,体现着“走到头、再回头继续”的回溯遍历方式,通常使用递归实现。 - 二叉搜索树是一种高效的元素查找数据结构,查找、插入、删除操作的时间复杂度皆为 $O(\log n)$ 。二叉搜索树退化为链表后,各项时间复杂度劣化至 $O(n)$ ,因此如何避免退化是非常重要的课题。 - AVL 树又称平衡二叉搜索树,其通过旋转操作,使得在不断插入与删除结点后,仍然可以保持二叉树的平衡(不退化)。 -- AVL 树的旋转操作分为右旋、左旋、先右旋后左旋、先左旋后右旋。在插入或删除结点后,AVL 树会从底置顶地执行旋转操作,使树恢复平衡。 +- AVL 树的旋转操作分为右旋、左旋、先右旋后左旋、先左旋后右旋。在插入或删除结点后,AVL 树会从底至顶地执行旋转操作,使树恢复平衡。 From f1eedc369bf5345e73470d25306a27a683346683 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Tue, 10 Jan 2023 02:21:09 +0800 Subject: [PATCH 04/23] Update heap --- docs/chapter_heap/heap.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/docs/chapter_heap/heap.md b/docs/chapter_heap/heap.md index 03a485f99..dd3598a2b 100644 --- a/docs/chapter_heap/heap.md +++ b/docs/chapter_heap/heap.md @@ -253,6 +253,6 @@ $$ ## 堆常见应用 -- 优先队列。堆常作为实现优先队列的首选数据结构。 -- 堆排序。根据 -- 获取数据中最大的 $k$ 个元素。这即是一道经典的算法题目,也是一种实际应用 +- **优先队列**。堆常作为实现优先队列的首选数据结构,入队和出队操作时间复杂度为 $O(\log n)$ ,建队操作为 $O(n)$ ,皆非常高效。 +- **堆排序**。给定一组数据,我们使用其建堆,并依次全部弹出,则可以得到有序的序列。当然,堆排序一般无需弹出元素,仅需每轮将堆顶元素交换至数组尾部并减小堆的长度即可。 +- **获取最大的 $k$ 个元素**。这既是一道经典算法题目,也是一种常见应用,例如选取热度前 10 的新闻作为微博热搜,选取前 10 销量的商品等。 From bbd915d70656d0e95562fea81d827e1be7f1a171 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Tue, 10 Jan 2023 03:42:24 +0800 Subject: [PATCH 05/23] Update heap. --- mkdocs.yml | 2 ++ 1 file changed, 2 insertions(+) diff --git a/mkdocs.yml b/mkdocs.yml index 9ddae9f39..ad836ae9f 100644 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -161,6 +161,8 @@ nav: - 二叉搜索树: chapter_tree/binary_search_tree.md - AVL 树 *: chapter_tree/avl_tree.md - 小结: chapter_tree/summary.md + - 堆: + - 堆(Heap): chapter_heap/heap.md - 查找算法: - 线性查找: chapter_searching/linear_search.md - 二分查找: chapter_searching/binary_search.md From 5ab3513b5a8df6ecc95839006d1e2318fd382dd4 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Tue, 10 Jan 2023 03:42:43 +0800 Subject: [PATCH 06/23] Update heap --- codes/java/chapter_heap/my_heap.java | 2 +- docs/chapter_heap/heap.md | 317 +++++++++++++++------------ 2 files changed, 174 insertions(+), 145 deletions(-) diff --git a/codes/java/chapter_heap/my_heap.java b/codes/java/chapter_heap/my_heap.java index 89f5c6e23..8fd748217 100644 --- a/codes/java/chapter_heap/my_heap.java +++ b/codes/java/chapter_heap/my_heap.java @@ -108,7 +108,7 @@ class MaxHeap { /* 从结点 i 开始,从顶至底堆化 */ private void siftDown(int i) { while (true) { - // 判断结点 i, l, r 中值最大的结点,记为 ma ; + // 判断结点 i, l, r 中值最大的结点,记为 ma int l = left(i), r = right(i), ma = i; if (l < size() && maxHeap.get(l) > maxHeap.get(ma)) ma = l; diff --git a/docs/chapter_heap/heap.md b/docs/chapter_heap/heap.md index dd3598a2b..f41826159 100644 --- a/docs/chapter_heap/heap.md +++ b/docs/chapter_heap/heap.md @@ -1,21 +1,29 @@ +--- +comments: true +--- + # 堆 -「堆 Heap」是一种特殊的树状数据结构,并且是一颗「完全二叉树」。堆主要分为两种: +「堆 Heap」是一颗限定条件下的「完全二叉树」。根据成立条件,堆主要分为两种类型: + +- 「大顶堆 Max Heap」,任意结点的值 $\geq$ 其子结点的值; +- 「小顶堆 Min Heap」,任意结点的值 $\leq$ 其子结点的值; -- 「大顶堆 Max Heap」,任意结点的值 $\geq$ 其子结点的值,因此根结点的值最大; -- 「小顶堆 Min Heap」,任意结点的值 $\leq$ 其子结点的值,因此根结点的值最小; +由于堆是完全二叉树,因此最底层结点靠左填充,其它层结点皆被填满。 + +对于大顶堆(小顶堆),其根结点的值最大(最小)。根结点被称为「堆顶」。 (图) -!!! tip "" +!!! tip - 大顶堆和小顶堆的定义、性质、操作本质上是相同的,区别只是大顶堆在求最大值,小顶堆在求最小值。 + 大顶堆和小顶堆的定义、性质、操作本质上是相同的,区别只是大、小顶堆分别在求最大、最小值。若无特别说明,本文将使用大顶堆来举例。 ## 堆常用操作 值得说明的是,多数编程语言提供的是「优先队列 Priority Queue」,其是一种抽象数据结构,**定义为具有出队优先级的队列**。 -而恰好,堆的定义与优先队列的操作逻辑完全吻合,大顶堆就是一个元素从大到小出队的优先队列。从使用角度看,我们可以将「优先队列」和「堆」理解为等价的数据结构。因此,本文与代码对两者不做特别区分,统一使用「堆」来命名。 +而恰好,**堆的定义与优先队列的操作逻辑完全吻合**,大顶堆就是一个元素从大到小出队的优先队列。从使用角度看,我们可以将「优先队列」和「堆」理解为等价的数据结构。因此,本文与代码对两者不做特别区分,统一使用「堆」来命名。 堆的常用操作见下表(方法命名以 Java 为例)。 @@ -35,36 +43,37 @@ 我们可以直接使用编程语言提供的堆类(或优先队列类)。 -```java -/* 初始化堆 */ -// 初始化小顶堆 -Queue minHeap = new PriorityQueue<>(); -// 初始化大顶堆(使用 lambda 表达式修改 Comparator 即可) -Queue maxHeap = new PriorityQueue<>((a, b) -> { return b - a; }); - -/* 元素入堆 */ -maxHeap.add(1); -maxHeap.add(3); -maxHeap.add(2); -maxHeap.add(5); -maxHeap.add(4); - -/* 获取堆顶元素 */ -int peek = maxHeap.peek(); - -/* 堆顶元素出堆 */ -int val = heap.poll(); - -/* 获取堆大小 */ -int size = maxHeap.size(); - -/* 判断堆是否为空 */ -boolean isEmpty = maxHeap.isEmpty(); - -/* 输入列表并建堆 */ -// 时间复杂度为 O(n) ,而非 O(nlogn) -minHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4)); -``` +=== "Java" + + ```java title="heap.java" + /* 初始化堆 */ + // 初始化小顶堆 + Queue minHeap = new PriorityQueue<>(); + // 初始化大顶堆(使用 lambda 表达式修改 Comparator 即可) + Queue maxHeap = new PriorityQueue<>((a, b) -> { return b - a; }); + + /* 元素入堆 */ + maxHeap.add(1); + maxHeap.add(3); + maxHeap.add(2); + maxHeap.add(5); + maxHeap.add(4); + + /* 获取堆顶元素 */ + int peek = maxHeap.peek(); + + /* 堆顶元素出堆 */ + int val = heap.poll(); + + /* 获取堆大小 */ + int size = maxHeap.size(); + + /* 判断堆是否为空 */ + boolean isEmpty = maxHeap.isEmpty(); + + /* 输入列表并建堆 */ + minHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4)); + ``` ## 堆的实现 @@ -76,151 +85,165 @@ minHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4)); 在二叉树章节我们学过,「完全二叉树」非常适合使用「数组」来表示,而堆恰好是一颗完全二叉树,因而我们一般使用「数组」来存储「堆」。 -**二叉树指针**。使用数组表示二叉树时,数组元素都代表结点值,索引代表结点在二叉树中的位置,**结点指针通过索引映射公式来实现**。具体地,给定索引 $i$ ,那么其左子结点索引为 $2i + 1$ 、右子结点索引为 $2i + 2$ 、父结点索引为 $(i - 1) / 2$ (向下整除)。当索引越界时,代表空结点或结点不存在。我们将以上映射公式封装成函数,以便使用。 - -(图) - -```java -// 使用列表而非数组,这样无需考虑扩容问题 -List maxHeap; +**二叉树指针**。使用数组表示二叉树时,元素代表结点值,索引代表结点在二叉树中的位置,**而结点指针通过索引映射公式来实现**。 -/* 构造函数,建立空堆 */ -public MaxHeap() { - maxHeap = new ArrayList<>(); -} +具体地,给定索引 $i$ ,那么其左子结点索引为 $2i + 1$ 、右子结点索引为 $2i + 2$ 、父结点索引为 $(i - 1) / 2$ (向下整除)。当索引越界时,代表空结点或结点不存在。我们将以上映射公式封装成函数,以便后续使用。 -/* 获取左子结点索引 */ -int left(int i) { - return 2 * i + 1; -} +(图) -/* 获取右子结点索引 */ -int right(int i) { - return 2 * i + 2; -} +=== "Java" -/* 获取父结点索引 */ -int parent(int i) { - return (i - 1) / 2; // 向下整除 -} -``` + ```java title="my_heap.java" + // 使用列表而非数组,这样无需考虑扩容问题 + List maxHeap; + + /* 构造函数,建立空堆 */ + public MaxHeap() { + maxHeap = new ArrayList<>(); + } + + /* 获取左子结点索引 */ + int left(int i) { + return 2 * i + 1; + } + + /* 获取右子结点索引 */ + int right(int i) { + return 2 * i + 2; + } + + /* 获取父结点索引 */ + int parent(int i) { + return (i - 1) / 2; // 向下整除 + } + ``` ### 访问堆顶元素 堆顶元素是二叉树的根结点,即列表首元素。 -```java -/* 访问堆顶元素 */ -public int peek() { - return maxHeap.get(0); -} -``` +=== "Java" + + ```java title="my_heap.java" + /* 访问堆顶元素 */ + public int peek() { + return maxHeap.get(0); + } + ``` ### 元素入堆 -给定元素 `val` ,我们先将其添加到堆的末尾。由于 `val` 可能大于其它元素,此时堆的性质可能被破坏了,我们需要修复从插入结点到根结点这条路径上的各个结点,该操作被称为「堆化 Heapify」。 +给定元素 `val` ,我们先将其添加到堆的末尾。由于 `val` 可能大于堆中其它元素,此时堆的成立条件可能已经被破坏,**因此需要修复从插入结点到根结点这条路径上的各个结点**,该操作被称为「堆化 Heapify」。 考虑从入堆结点开始,**从底至顶执行堆化**。具体地,比较插入结点与其父结点的值,若插入结点更大则将它们交换;并循环以上操作,从底至顶地修复堆中的各个结点;直至越过根结点时结束,或当遇到无需交换的结点时提前结束。 -设堆长度为 $n$ ,**元素入堆操作的时间复杂度为 $O(\log n)$** 。这是因为树的高度为 $O(\log n)$ ,因此堆化操作的循环轮数最多为 $O(\log n)$ 。 +设结点总数为 $n$ ,则树的高度为 $O(\log n)$ ,易得堆化操作的循环轮数最多为 $O(\log n)$ ,**因而元素入堆操作的时间复杂度为 $O(\log n)$** 。 (图) -```java -/* 元素入堆 */ -void push(int val) { - // 添加结点 - maxHeap.add(val); - // 从底至顶堆化 - siftUp(size() - 1); -} - -/* 从结点 i 开始,从底至顶堆化 */ -void siftUp(int i) { - while (true) { - // 获取结点 i 的父结点 - int p = parent(i); - // 若“越过根结点”或“结点无需修复”,则结束堆化 - if (p < 0 || maxHeap.get(i) <= maxHeap.get(p)) - break; - // 交换两结点 - swap(i, p); - // 循环向上堆化 - i = p; +=== "Java" + + ```java title="my_heap.java" + /* 元素入堆 */ + void push(int val) { + // 添加结点 + maxHeap.add(val); + // 从底至顶堆化 + siftUp(size() - 1); + } + + /* 从结点 i 开始,从底至顶堆化 */ + void siftUp(int i) { + while (true) { + // 获取结点 i 的父结点 + int p = parent(i); + // 若“越过根结点”或“结点无需修复”,则结束堆化 + if (p < 0 || maxHeap.get(i) <= maxHeap.get(p)) + break; + // 交换两结点 + swap(i, p); + // 循环向上堆化 + i = p; + } } -} -``` + ``` ### 堆顶元素出堆 -堆顶元素是二叉树根结点,即列表首元素,如果我们直接将首元素从列表中删除,则二叉树中所有结点都产生移位,这样后续使用堆化修复就很麻烦了。为了尽量减少二叉树结点变动,采取以下操作步骤: +堆顶元素是二叉树根结点,即列表首元素,如果我们直接将首元素从列表中删除,则二叉树中所有结点都会随之发生移位(索引发生变化),这样后续使用堆化修复就很麻烦了。为了尽量减少元素索引变动,采取以下操作步骤: 1. 交换列表首元素与尾元素(即交换根结点与最右叶结点); -2. 将尾元素从列表中删除(此时堆顶元素已被删除); +2. 交换完成后,将尾元素从列表中删除(此时堆顶元素已被删除); 3. 从根结点开始,从顶至底堆化; 顾名思义,**从顶至底堆化的操作方向与从底至顶堆化相反**,我们比较根结点的值与其两个子结点的值,将最大的子结点与根结点执行交换,并循环以上操作,直到越过叶结点时结束,或当遇到无需交换的结点时提前结束。 (图) -```java -/* 元素出堆 */ -int poll() { - // 判空处理 - if (isEmpty()) - throw new EmptyStackException(); - // 交换根结点与最右叶结点(即交换首元素与尾元素) - swap(0, size() - 1); - // 删除结点 - int val = maxHeap.remove(size() - 1); - // 从顶至底堆化 - siftDown(0); - // 返回堆顶元素 - return val; -} - -/* 从结点 i 开始,从顶至底堆化 */ -void siftDown(int i) { - while (true) { - // 判断结点 i, l, r 中值最大的结点,记为 ma ; - int l = left(i), r = right(i), ma = i; - if (l < size() && maxHeap.get(l) > maxHeap.get(ma)) - ma = l; - if (r < size() && maxHeap.get(r) > maxHeap.get(ma)) - ma = r; - // 若“结点 i 最大”或“越过叶结点”,则结束堆化 - if (ma == i) break; - // 交换两结点 - swap(i, ma); - // 循环向下堆化 - i = ma; +=== "Java" + + ```java title="my_heap.java" + /* 元素出堆 */ + int poll() { + // 判空处理 + if (isEmpty()) + throw new EmptyStackException(); + // 交换根结点与最右叶结点(即交换首元素与尾元素) + swap(0, size() - 1); + // 删除结点 + int val = maxHeap.remove(size() - 1); + // 从顶至底堆化 + siftDown(0); + // 返回堆顶元素 + return val; } -} -``` + + /* 从结点 i 开始,从顶至底堆化 */ + void siftDown(int i) { + while (true) { + // 判断结点 i, l, r 中值最大的结点,记为 ma + int l = left(i), r = right(i), ma = i; + if (l < size() && maxHeap.get(l) > maxHeap.get(ma)) + ma = l; + if (r < size() && maxHeap.get(r) > maxHeap.get(ma)) + ma = r; + // 若“结点 i 最大”或“越过叶结点”,则结束堆化 + if (ma == i) break; + // 交换两结点 + swap(i, ma); + // 循环向下堆化 + i = ma; + } + } + ``` ### 输入数据并建堆 * -给定一个列表,我们也可以将其建堆。最直接地,可以通过调用「元素入堆」方法,将列表元素依次入堆。元素入堆的时间复杂度为 $O(n)$ ,而平均长度为 $\frac{n}{2}$ ,因此该方法的总体时间复杂度为 $O(n \log n)$ 。 +如果我们想要直接输入一个列表并将其建堆,那么该怎么做呢?最直接地,考虑使用「元素入堆」方法,将列表元素依次入堆。元素入堆的时间复杂度为 $O(n)$ ,而平均长度为 $\frac{n}{2}$ ,因此该方法的总体时间复杂度为 $O(n \log n)$ 。 + +然而,存在一种更加优雅的建堆方法。设结点数量为 $n$ ,我们先将列表所有元素原封不动添加进堆,**然后迭代地对各个结点执行「从顶至底堆化」**。当然,**无需对叶结点执行堆化**,因为其没有子结点。 -然而,存在一种更加优雅的建堆方法。设结点数量为 $n$ ,我们先将列表所有元素原封不动添加进堆,**然后迭代地对各个结点执行「从顶至底堆化」**。当然,**无需对叶结点执行堆化,**因为其没有子结点。 +=== "Java" -```java -/* 构造函数,根据输入列表建堆 */ -public MaxHeap(List nums) { - // 将列表元素原封不动添加进堆 - maxHeap = new ArrayList<>(nums); - // 堆化除叶结点以外的其他所有结点 - for (int i = parent(size() - 1); i >= 0; i--) { - siftDown(i); + ```java title="my_heap.java" + /* 构造函数,根据输入列表建堆 */ + public MaxHeap(List nums) { + // 将列表元素原封不动添加进堆 + maxHeap = new ArrayList<>(nums); + // 堆化除叶结点以外的其他所有结点 + for (int i = parent(size() - 1); i >= 0; i--) { + siftDown(i); + } } -} -``` + ``` !!! tip - 完全二叉树的叶结点数量为 $(n + 1) / 2$ ,其中 $/$ 为向下整除。 + 完全二叉树中,设结点总数为 $n$ ,则叶结点数量为 $(n + 1) / 2$ ,其中 $/$ 为向下整除。 -那么,第二种建堆方法的时间复杂度时多少呢?我们来做一下简单推算。叶结点和需要堆化结点的数量各占约一半,即为 $O(n)$ ,二叉树高度为 $O(\log n)$ ,可得时间复杂度为 $O(n \log n)$ 。该估算结果仍不够准确,因为我们没有考虑到二叉树“底层结点远多于顶层结点”的性质。 +那么,第二种建堆方法的时间复杂度时多少呢?我们来做一下简单推算。排除叶结点后,需要堆化结点数量为 $(n - 1)/2$ ,即为 $O(n)$ ,而二叉树高度为 $O(\log n)$ ,因此可得时间复杂度为 $O(n \log n)$ 。 + +然而,该估算结果仍不够准确,因为我们没有考虑到 **二叉树底层结点远多于顶层结点** 的特点。下面我们来换种方法推导。 设二叉树(即堆)结点数量为 $n$ ,树高度为 $h$ 。如下图所示,我们将各层的“结点数量 $\times$ 子树高度”进行求和,即可得到准确的操作数量。 @@ -231,24 +254,30 @@ $$ (图) 求解上式需要借助中学的数列知识,先对 $S$ 乘以 $2$ ,可得 + $$ \begin{aligned} -S & = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{h-1}\times1 \\ -2S & = 2^1h + 2^2(h-1) + 2^3(h-2) + \cdots + 2^{h}\times1 \\ +S & = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{h-1}\times1 \newline +2S & = 2^1h + 2^2(h-1) + 2^3(h-2) + \cdots + 2^{h}\times1 \newline \end{aligned} $$ + 令下式 $2S$ 与上式 $S$ 错位相减,易得 + $$ 2S - S = S = -2^0h + 2^1 + 2^2 + \cdots + 2^{h-1} + 2^h $$ + 观察发现,$S$ 是一个等比数列,可直接借助公式求和。并且,对于高度为 $h$ 的完全二叉树,结点数量范围为 $n \in [2^h, 2^{h+1} - 1]$ ,复杂度为 $n = O(n) = O(2^h)$。 + $$ \begin{aligned} -S & = 2 \frac{1 - 2^h}{1 - 2} - h \\ -& = 2^{h+1} - h \\ +S & = 2 \frac{1 - 2^h}{1 - 2} - h \newline +& = 2^{h+1} - h \newline & = O(2^h) = O(n) \end{aligned} $$ + 以上推算表明,输入列表并建堆的时间复杂度为 $O(n)$ ,非常高效。 ## 堆常见应用 From 777370a5f8ffb3e417f51699a1077cc4b679daad Mon Sep 17 00:00:00 2001 From: xiongsp <37035570+xiongsp@users.noreply.github.com> Date: Tue, 10 Jan 2023 12:12:10 +0800 Subject: [PATCH 07/23] Add instruction --- Dockerfile | 5 +---- README.md | 22 ++++++++++++++++++++++ 2 files changed, 23 insertions(+), 4 deletions(-) diff --git a/Dockerfile b/Dockerfile index 5d437c846..9d566fbe2 100644 --- a/Dockerfile +++ b/Dockerfile @@ -1,9 +1,6 @@ FROM python:3.9.0-buster -RUN sed -i 's/deb.debian.org/mirrors.aliyun.com/g' /etc/apt/sources.list -RUN sed -i 's/security.debian.org/mirrors.aliyun.com/g' /etc/apt/sources.list - -RUN pip install --upgrade pip && pip install mkdocs mkdocs-material +RUN pip install -i https://pypi.tuna.tsinghua.edu.cn/simple --upgrade pip && pip install -i https://pypi.tuna.tsinghua.edu.cn/simple mkdocs mkdocs-material WORKDIR /app diff --git a/README.md b/README.md index 560e73953..8bb7b345d 100644 --- a/README.md +++ b/README.md @@ -41,6 +41,28 @@ 如果感觉本书对你有所帮助,请点个 Star :star: 支持一下,谢谢! +### 本地部署 + +#### Docker + +使用本教程前,请确保Docker已经安装并启动。 + +根据如下命令离线部署。 + +```bash +git clone https://github.com/krahets/hello-algo.git +cd hello-algo +docker-compose up -d +``` + +稍等片刻,请访问`http://localhost:8000`即可访问本教程。 + +使用如下命令删除部署。 + +```bash +docker-compose down +``` + ## 推荐语 > “一本通俗易懂的数据结构与算法入门书,引导读者手脑并用地学习,强烈推荐算法初学者阅读。” From 38a3d913edf621ba3c968266d60832a5723b0daf Mon Sep 17 00:00:00 2001 From: xiongsp <37035570+xiongsp@users.noreply.github.com> Date: Tue, 10 Jan 2023 12:47:40 +0800 Subject: [PATCH 08/23] optimize the Dockerfile --- Dockerfile | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Dockerfile b/Dockerfile index 9d566fbe2..813b4a37f 100644 --- a/Dockerfile +++ b/Dockerfile @@ -1,4 +1,4 @@ -FROM python:3.9.0-buster +FROM python:3.9.0-alpine RUN pip install -i https://pypi.tuna.tsinghua.edu.cn/simple --upgrade pip && pip install -i https://pypi.tuna.tsinghua.edu.cn/simple mkdocs mkdocs-material From d1315f5cdcfcf2c0e760a61acf4cf84a33b2f74d Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Wed, 11 Jan 2023 03:37:14 +0800 Subject: [PATCH 09/23] Update Dockerfile --- Dockerfile | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/Dockerfile b/Dockerfile index 813b4a37f..f699e33ea 100644 --- a/Dockerfile +++ b/Dockerfile @@ -1,17 +1,16 @@ FROM python:3.9.0-alpine -RUN pip install -i https://pypi.tuna.tsinghua.edu.cn/simple --upgrade pip && pip install -i https://pypi.tuna.tsinghua.edu.cn/simple mkdocs mkdocs-material +RUN pip install -i https://pypi.tuna.tsinghua.edu.cn/simple --upgrade pip +RUN pip install -i https://pypi.tuna.tsinghua.edu.cn/simple mkdocs-material==9.0.2 WORKDIR /app COPY codes /app/codes - COPY docs /app/docs - COPY mkdocs.yml /app/mkdocs.yml RUN mkdir ./docs/overrides && mkdocs build EXPOSE 8000 -CMD ["mkdocs", "serve", "-a", "0.0.0.0:8000"] \ No newline at end of file +CMD ["mkdocs", "serve", "-a", "0.0.0.0:8000"] From a1579f6f7e9294f34dcb8aa8b2bbebad8206fd5b Mon Sep 17 00:00:00 2001 From: sjinzh <99076655+sjinzh@users.noreply.github.com> Date: Wed, 11 Jan 2023 17:39:28 +0800 Subject: [PATCH 10/23] add zig codes for Section 'Array', 'LinkedList' and 'List' --- codes/zig/build.zig | 55 ++++++ .../chapter_array_and_linkedlist/array.zig | 149 ++++++++++++++ .../linked_list.zig | 89 +++++++++ .../zig/chapter_array_and_linkedlist/list.zig | 85 ++++++++ .../chapter_array_and_linkedlist/my_list.zig | 182 ++++++++++++++++++ codes/zig/include/PrintUtil.zig | 32 ++- 6 files changed, 589 insertions(+), 3 deletions(-) create mode 100644 codes/zig/chapter_array_and_linkedlist/array.zig create mode 100644 codes/zig/chapter_array_and_linkedlist/linked_list.zig create mode 100644 codes/zig/chapter_array_and_linkedlist/list.zig create mode 100644 codes/zig/chapter_array_and_linkedlist/my_list.zig diff --git a/codes/zig/build.zig b/codes/zig/build.zig index 1a9fb0fba..b9bbddce0 100644 --- a/codes/zig/build.zig +++ b/codes/zig/build.zig @@ -64,4 +64,59 @@ pub fn build(b: *std.build.Builder) void { if (b.args) |args| run_cmd_leetcode_two_sum.addArgs(args); const run_step_leetcode_two_sum = b.step("run_leetcode_two_sum", "Run leetcode_two_sum"); run_step_leetcode_two_sum.dependOn(&run_cmd_leetcode_two_sum.step); + + // Section: "Array" + // Source File: "chapter_array_and_linkedlist/array.zig" + // Run Command: zig build run_array + const exe_array = b.addExecutable("array", "chapter_array_and_linkedlist/array.zig"); + exe_array.addPackagePath("include", "include/include.zig"); + exe_array.setTarget(target); + exe_array.setBuildMode(mode); + exe_array.install(); + const run_cmd_array = exe_array.run(); + run_cmd_array.step.dependOn(b.getInstallStep()); + if (b.args) |args| run_cmd_array.addArgs(args); + const run_step_array = b.step("run_array", "Run array"); + run_step_array.dependOn(&run_cmd_array.step); + + // Section: "LinkedList" + // Source File: "chapter_array_and_linkedlist/linked_list.zig" + // Run Command: zig build run_linked_list + const exe_linked_list = b.addExecutable("linked_list", "chapter_array_and_linkedlist/linked_list.zig"); + exe_linked_list.addPackagePath("include", "include/include.zig"); + exe_linked_list.setTarget(target); + exe_linked_list.setBuildMode(mode); + exe_linked_list.install(); + const run_cmd_linked_list = exe_linked_list.run(); + run_cmd_linked_list.step.dependOn(b.getInstallStep()); + if (b.args) |args| run_cmd_linked_list.addArgs(args); + const run_step_linked_list = b.step("run_linked_list", "Run linked_list"); + run_step_linked_list.dependOn(&run_cmd_linked_list.step); + + // Section: "List" + // Source File: "chapter_array_and_linkedlist/list.zig" + // Run Command: zig build run_list + const exe_list = b.addExecutable("list", "chapter_array_and_linkedlist/list.zig"); + exe_list.addPackagePath("include", "include/include.zig"); + exe_list.setTarget(target); + exe_list.setBuildMode(mode); + exe_list.install(); + const run_cmd_list = exe_list.run(); + run_cmd_list.step.dependOn(b.getInstallStep()); + if (b.args) |args| run_cmd_list.addArgs(args); + const run_step_list = b.step("run_list", "Run list"); + run_step_list.dependOn(&run_cmd_list.step); + + // Source File: "chapter_array_and_linkedlist/my_list.zig" + // Run Command: zig build run_my_list + const exe_my_list = b.addExecutable("my_list", "chapter_array_and_linkedlist/my_list.zig"); + exe_my_list.addPackagePath("include", "include/include.zig"); + exe_my_list.setTarget(target); + exe_my_list.setBuildMode(mode); + exe_my_list.install(); + const run_cmd_my_list = exe_my_list.run(); + run_cmd_my_list.step.dependOn(b.getInstallStep()); + if (b.args) |args| run_cmd_my_list.addArgs(args); + const run_step_my_list = b.step("run_my_list", "Run my_list"); + run_step_my_list.dependOn(&run_cmd_my_list.step); } diff --git a/codes/zig/chapter_array_and_linkedlist/array.zig b/codes/zig/chapter_array_and_linkedlist/array.zig new file mode 100644 index 000000000..1206168e5 --- /dev/null +++ b/codes/zig/chapter_array_and_linkedlist/array.zig @@ -0,0 +1,149 @@ +// File: array.zig +// Created Time: 2023-01-07 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); +const inc = @import("include"); + +// 随机返回一个数组元素 +pub fn randomAccess(nums: []i32) i32 { + // 在区间 [0, nums.len) 中随机抽取一个整数 + var randomIndex = std.crypto.random.intRangeLessThan(usize, 0, nums.len); + // 获取并返回随机元素 + var randomNum = nums[randomIndex]; + return randomNum; +} + +// 扩展数组长度(运行时方法) +pub fn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 { + // 初始化一个扩展长度后的数组 + var res = try mem_allocator.alloc(i32, nums.len + enlarge); + std.mem.set(i32, res, 0); + // 将原数组中的所有元素复制到新数组 + std.mem.copy(i32, res, nums); + // 返回扩展后的新数组 + return res; +} + +// 扩展数组长度(编译期方法A:初始化新数组并进行元素拷贝) +pub fn extendComptimeA(comptime nums: anytype, comptime enlarge: i32) [nums.len + enlarge]i32 { + // 初始化一个扩展长度后的数组 + var res = [_]i32{0} ** (nums.len + enlarge); + // 将原数组中的所有元素复制到新数组 + for (nums) |num, i| { + res[i] = num; + } + // 返回扩展后的新数组 + return res; +} + +// 扩展数组长度(编译期方法B: 通过数组拼接运算符“++”) +pub fn extendComptimeB(comptime nums: anytype, comptime enlarge: i32) [nums.len + enlarge]i32 { + // 数组拼接操作 + var res = nums ++ [_]i32{0} ** enlarge; + // 返回拼接后的新数组 + return res; +} + +// 在数组的索引 index 处插入元素 num +pub fn insert(nums: []i32, num: i32, index: usize) void { + // 把索引 index 以及之后的所有元素向后移动一位 + var i = nums.len - 1; + while (i > index) : (i -= 1) { + nums[i] = nums[i - 1]; + } + // 将 num 赋给 index 处元素 + nums[index] = num; +} + +// 删除索引 index 处元素 +pub fn remove(nums: []i32, index: usize) void { + // 把索引 index 之后的所有元素向前移动一位 + var i = index; + while (i < nums.len - 1) : (i += 1) { + nums[i] = nums[i + 1]; + } +} + +// 遍历数组 +pub fn traverse(nums: []i32) void { + var count: i32 = 0; + // 通过索引遍历数组 + var i: i32 = 0; + while (i < nums.len) : (i += 1) { + count += 1; + } + count = 0; + // 直接遍历数组 + for (nums) |_| { + count += 1; + } +} + +// 在数组中查找指定元素 +pub fn find(nums: []i32, target: i32) i32 { + for (nums) |num, i| { + if (num == target) return @intCast(i32, i); + } + return -1; +} + +// Driver Code +pub fn main() !void { + // 查看本地CPU架构和操作系统信息 + var native_target_info = try std.zig.system.NativeTargetInfo.detect(std.zig.CrossTarget{}); + std.debug.print("Native Info: CPU Arch = {}, OS = {}\n", .{native_target_info.target.cpu.arch, native_target_info.target.os.tag}); + + // 初始化数组 + const size: i32 = 5; + var arr = [_]i32{0} ** size; + std.debug.print("数组 arr = ", .{}); + inc.PrintUtil.printArray(i32, &arr); + + var array = [_]i32{ 1, 3, 2, 5, 4 }; + std.debug.print("\n数组 nums = ", .{}); + inc.PrintUtil.printArray(i32, &array); + + // 随机访问 + var randomNum = randomAccess(&array); + std.debug.print("\n在 nums 中获取随机元素 {}", .{randomNum}); + + // 长度扩展(运行时方法) + var known_at_runtime_zero: usize = 0; + var nums: []i32 = array[known_at_runtime_zero..array.len]; + var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator); + defer mem_arena.deinit(); + const mem_allocator = mem_arena.allocator(); + nums = try extend(mem_allocator, nums, 3); + std.debug.print("\n将数组长度扩展至 8 ,得到 nums = ", .{}); + inc.PrintUtil.printArray(i32, nums); + // { + // // 长度扩展(编译期方法) + // comptime var array_comptime = [_]i32{ 1, 3, 2, 5, 4 }; + // var nums_comptime = extendComptimeA(array_comptime, 3); + // // var nums_comptime = extendComptimeB(array_comptime, 3); + // std.debug.print("\n将数组长度扩展至 8 ,得到 nums_comptime = ", .{}); + // inc.PrintUtil.printArray(i32, &nums_comptime); + // } + + // 插入元素 + insert(nums, 6, 3); + std.debug.print("\n在索引 3 处插入数字 6 ,得到 nums = ", .{}); + inc.PrintUtil.printArray(i32, nums); + + // 删除元素 + remove(nums, 2); + std.debug.print("\n删除索引 2 处的元素,得到 nums = ", .{}); + inc.PrintUtil.printArray(i32, nums); + + // 遍历数组 + traverse(nums); + + // 查找元素 + var index = find(nums, 3); + std.debug.print("\n在 nums_ext 中查找元素 3 ,得到索引 = {}\n", .{index}); + + const getchar = try std.io.getStdIn().reader().readByte(); + _ = getchar; +} + diff --git a/codes/zig/chapter_array_and_linkedlist/linked_list.zig b/codes/zig/chapter_array_and_linkedlist/linked_list.zig new file mode 100644 index 000000000..495ec3c66 --- /dev/null +++ b/codes/zig/chapter_array_and_linkedlist/linked_list.zig @@ -0,0 +1,89 @@ +// File: linked_list.zig +// Created Time: 2023-01-07 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); +const inc = @import("include"); + +// 在链表的结点 n0 之后插入结点 P +pub fn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void { + var n1 = n0.?.next; + n0.?.next = P; + P.?.next = n1; +} + +// 删除链表的结点 n0 之后的首个结点 +pub fn remove(n0: ?*inc.ListNode(i32)) void { + if (n0.?.next == null) return; + // n0 -> P -> n1 + var P = n0.?.next; + var n1 = P.?.next; + n0.?.next = n1; +} + +// 访问链表中索引为 index 的结点 +pub fn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) { + var head = node; + var i: i32 = 0; + while (i < index) : (i += 1) { + head = head.?.next; + if (head == null) return null; + } + return head; +} + +// 在链表中查找值为 target 的首个结点 +pub fn find(node: ?*inc.ListNode(i32), target: i32) i32 { + var head = node; + var index: i32 = 0; + while (head != null) { + if (head.?.val == target) return index; + head = head.?.next; + index += 1; + } + return -1; +} + +// Driver Code +pub fn main() !void { + // 查看本地CPU架构和操作系统信息 + var native_target_info = try std.zig.system.NativeTargetInfo.detect(std.zig.CrossTarget{}); + std.debug.print("Native Info: CPU Arch = {}, OS = {}\n", .{native_target_info.target.cpu.arch, native_target_info.target.os.tag}); + + // 初始化链表 + // 初始化各个结点 + var n0 = inc.ListNode(i32){.val = 1}; + var n1 = inc.ListNode(i32){.val = 3}; + var n2 = inc.ListNode(i32){.val = 2}; + var n3 = inc.ListNode(i32){.val = 5}; + var n4 = inc.ListNode(i32){.val = 4}; + // 构建引用指向 + n0.next = &n1; + n1.next = &n2; + n2.next = &n3; + n3.next = &n4; + std.debug.print("初始化的链表为", .{}); + try inc.PrintUtil.printLinkedList(i32, &n0); + + // 插入结点 + var tmp = inc.ListNode(i32){.val = 0}; + insert(&n0, &tmp); + std.debug.print("插入结点后的链表为", .{}); + try inc.PrintUtil.printLinkedList(i32, &n0); + + // 删除结点 + remove(&n0); + std.debug.print("删除结点后的链表为", .{}); + try inc.PrintUtil.printLinkedList(i32, &n0); + + // 访问结点 + var node = access(&n0, 3); + std.debug.print("链表中索引 3 处的结点的值 = {}\n", .{node.?.val}); + + // 查找结点 + var index = find(&n0, 2); + std.debug.print("链表中值为 2 的结点的索引 = {}\n", .{index}); + + const getchar = try std.io.getStdIn().reader().readByte(); + _ = getchar; +} \ No newline at end of file diff --git a/codes/zig/chapter_array_and_linkedlist/list.zig b/codes/zig/chapter_array_and_linkedlist/list.zig new file mode 100644 index 000000000..c27a641bd --- /dev/null +++ b/codes/zig/chapter_array_and_linkedlist/list.zig @@ -0,0 +1,85 @@ +// File: list.zig +// Created Time: 2023-01-07 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); +const inc = @import("include"); + +// Driver Code +pub fn main() !void { + // 查看本地CPU架构和操作系统信息 + var native_target_info = try std.zig.system.NativeTargetInfo.detect(std.zig.CrossTarget{}); + std.debug.print("Native Info: CPU Arch = {}, OS = {}\n", .{native_target_info.target.cpu.arch, native_target_info.target.os.tag}); + + // 初始化列表 + var list = std.ArrayList(i32).init(std.heap.page_allocator); + // 延迟释放内存 + defer list.deinit(); + try list.appendSlice(&[_]i32{ 1, 3, 2, 5, 4 }); + std.debug.print("列表 list = ", .{}); + inc.PrintUtil.printList(i32, list); + + // 访问元素 + var num = list.items[1]; + std.debug.print("\n访问索引 1 处的元素,得到 num = {}", .{num}); + + // 更新元素 + list.items[1] = 0; + std.debug.print("\n将索引 1 处的元素更新为 0 ,得到 list = ", .{}); + inc.PrintUtil.printList(i32, list); + + // 清空列表 + list.clearRetainingCapacity(); + std.debug.print("\n清空列表后 list = ", .{}); + inc.PrintUtil.printList(i32, list); + + // 尾部添加元素 + try list.append(1); + try list.append(3); + try list.append(2); + try list.append(5); + try list.append(4); + std.debug.print("\n添加元素后 list = ", .{}); + inc.PrintUtil.printList(i32, list); + + // 中间插入元素 + try list.insert(3, 6); + std.debug.print("\n在索引 3 处插入数字 6 ,得到 list = ", .{}); + inc.PrintUtil.printList(i32, list); + + // 删除元素 + var value = list.orderedRemove(3); + _ = value; + std.debug.print("\n删除索引 3 处的元素,得到 list = ", .{}); + inc.PrintUtil.printList(i32, list); + + // 通过索引遍历列表 + var count: i32 = 0; + var i: i32 = 0; + while (i < list.items.len) : (i += 1) { + count += 1; + } + + // 直接遍历列表元素 + count = 0; + for (list.items) |_| { + count += 1; + } + + // 拼接两个列表 + var list1 = std.ArrayList(i32).init(std.heap.page_allocator); + defer list1.deinit(); + try list1.appendSlice(&[_]i32{ 6, 8, 7, 10, 9 }); + try list.insertSlice(list.items.len, list1.items); + std.debug.print("\n将列表 list1 拼接到 list 之后,得到 list = ", .{}); + inc.PrintUtil.printList(i32, list); + + // 排序列表 + std.sort.sort(i32, list.items, {}, comptime std.sort.asc(i32)); + std.debug.print("\n排序列表后 list = ", .{}); + inc.PrintUtil.printList(i32, list); + + const getchar = try std.io.getStdIn().reader().readByte(); + _ = getchar; +} + diff --git a/codes/zig/chapter_array_and_linkedlist/my_list.zig b/codes/zig/chapter_array_and_linkedlist/my_list.zig new file mode 100644 index 000000000..21aa921f3 --- /dev/null +++ b/codes/zig/chapter_array_and_linkedlist/my_list.zig @@ -0,0 +1,182 @@ +// File: my_list.zig +// Created Time: 2023-01-08 +// Author: sjinzh (sjinzh@gmail.com) + +const std = @import("std"); +const inc = @import("include"); + +// 列表类简易实现 +// 编译期泛型 +pub fn MyList(comptime T: type) type { + return struct { + const Self = @This(); + + nums: []T = undefined, // 数组(存储列表元素) + numsCapacity: usize = 10, // 列表容量 + numSize: usize = 0, // 列表长度(即当前元素数量) + extendRatio: usize = 2, // 每次列表扩容的倍数 + mem_arena: ?std.heap.ArenaAllocator = null, + mem_allocator: std.mem.Allocator = undefined, // 内存分配器 + + // 构造函数(分配内存+初始化列表) + pub fn init(self: *Self, allocator: std.mem.Allocator) !void { + if (self.mem_arena == null) { + self.mem_arena = std.heap.ArenaAllocator.init(allocator); + self.mem_allocator = self.mem_arena.?.allocator(); + } + self.nums = try self.mem_allocator.alloc(T, self.numsCapacity); + std.mem.set(T, self.nums, @as(T, 0)); + } + + // 析构函数(释放内存) + pub fn deinit(self: *Self) void { + if (self.mem_arena == null) return; + self.mem_arena.?.deinit(); + } + + // 获取列表长度(即当前元素数量) + pub fn size(self: *Self) usize { + return self.numSize; + } + + // 获取列表容量 + pub fn capacity(self: *Self) usize { + return self.numsCapacity; + } + + // 访问元素 + pub fn get(self: *Self, index: usize) T { + // 索引如果越界则抛出异常,下同 + if (index >= self.size()) @panic("索引越界"); + return self.nums[index]; + } + + // 更新元素 + pub fn set(self: *Self, index: usize, num: T) void { + // 索引如果越界则抛出异常,下同 + if (index >= self.size()) @panic("索引越界"); + self.nums[index] = num; + } + + // 尾部添加元素 + pub fn add(self: *Self, num: T) !void { + // 元素数量超出容量时,触发扩容机制 + if (self.size() == self.capacity()) try self.extendCapacity(); + self.nums[self.size()] = num; + // 更新元素数量 + self.numSize += 1; + } + + // 中间插入元素 + pub fn insert(self: *Self, index: usize, num: T) !void { + if (index >= self.size()) @panic("索引越界"); + // 元素数量超出容量时,触发扩容机制 + if (self.size() == self.capacity()) try self.extendCapacity(); + // 索引 i 以及之后的元素都向后移动一位 + var j = self.size() - 1; + while (j >= index) : (j -= 1) { + self.nums[j + 1] = self.nums[j]; + } + self.nums[index] = num; + // 更新元素数量 + self.numSize += 1; + } + + // 删除元素 + pub fn remove(self: *Self, index: usize) T { + if (index >= self.size()) @panic("索引越界"); + var num = self.nums[index]; + // 索引 i 之后的元素都向前移动一位 + var j = index; + while (j < self.size() - 1) : (j += 1) { + self.nums[j] = self.nums[j + 1]; + } + // 更新元素数量 + self.numSize -= 1; + // 返回被删除元素 + return num; + } + + // 列表扩容 + pub fn extendCapacity(self: *Self) !void { + // 新建一个长度为 size * extendRatio 的数组,并将原数组拷贝到新数组 + var newCapacity = self.capacity() * self.extendRatio; + var extend = try self.mem_allocator.alloc(T, newCapacity); + std.mem.set(T, extend, @as(T, 0)); + // 将原数组中的所有元素复制到新数组 + std.mem.copy(T, extend, self.nums); + self.nums = extend; + // 更新列表容量 + self.numsCapacity = newCapacity; + } + + // 将列表转换为数组 + pub fn toArray(self: *Self) ![]T { + // 仅转换有效长度范围内的列表元素 + var nums = try self.mem_allocator.alloc(T, self.size()); + std.mem.set(T, nums, @as(T, 0)); + for (nums) |*num, i| { + num.* = self.get(i); + } + return nums; + } + + }; +} + +// Driver Code +pub fn main() !void { + // 查看本地CPU架构和操作系统信息 + var native_target_info = try std.zig.system.NativeTargetInfo.detect(std.zig.CrossTarget{}); + std.debug.print("Native Info: CPU Arch = {}, OS = {}\n", .{native_target_info.target.cpu.arch, native_target_info.target.os.tag}); + + // 初始化列表 + var list = MyList(i32){}; + try list.init(std.heap.page_allocator); + // 延迟释放内存 + defer list.deinit(); + + // 尾部添加元素 + try list.add(1); + try list.add(3); + try list.add(2); + try list.add(5); + try list.add(4); + std.debug.print("列表 list = ", .{}); + inc.PrintUtil.printArray(i32, try list.toArray()); + std.debug.print(" ,容量 = {} ,长度 = {}", .{list.capacity(), list.size()}); + + // 中间插入元素 + try list.insert(3, 6); + std.debug.print("\n在索引 3 处插入数字 6 ,得到 list = ", .{}); + inc.PrintUtil.printArray(i32, try list.toArray()); + + // 删除元素 + _ = list.remove(3); + std.debug.print("\n删除索引 3 处的元素,得到 list = ", .{}); + inc.PrintUtil.printArray(i32, try list.toArray()); + + // 访问元素 + var num = list.get(1); + std.debug.print("\n访问索引 1 处的元素,得到 num = {}", .{num}); + + // 更新元素 + list.set(1, 0); + std.debug.print("\n将索引 1 处的元素更新为 0 ,得到 list = ", .{}); + inc.PrintUtil.printArray(i32, try list.toArray()); + + // 测试扩容机制 + list.set(1, 0); + var i: i32 = 0; + while (i < 10) : (i += 1) { + // 在 i = 5 时,列表长度将超出列表容量,此时触发扩容机制 + try list.add(i); + } + std.debug.print("\n扩容后的列表 list = ", .{}); + inc.PrintUtil.printArray(i32, try list.toArray()); + std.debug.print(" ,容量 = {} ,长度 = {}\n", .{list.capacity(), list.size()}); + + const getchar = try std.io.getStdIn().reader().readByte(); + _ = getchar; +} + diff --git a/codes/zig/include/PrintUtil.zig b/codes/zig/include/PrintUtil.zig index 9e6b18b05..4d2d20893 100644 --- a/codes/zig/include/PrintUtil.zig +++ b/codes/zig/include/PrintUtil.zig @@ -7,16 +7,42 @@ const ListNode = @import("ListNode.zig").ListNode; const TreeNode = @import("TreeNode.zig").TreeNode; // Print an array -// 编译期泛型 pub fn printArray(comptime T: type, nums: []T) void { std.debug.print("[", .{}); if (nums.len > 0) { for (nums) |num, j| { - std.debug.print("{}{s}", .{num, if (j == nums.len-1) "]\n" else ", " }); + std.debug.print("{}{s}", .{num, if (j == nums.len-1) "]" else ", " }); } } else { std.debug.print("]", .{}); - std.debug.print("\n", .{}); + } +} + +// Print a list +pub fn printList(comptime T: type, list: std.ArrayList(T)) void { + std.debug.print("[", .{}); + if (list.items.len > 0) { + for (list.items) |value, i| { + std.debug.print("{}{s}", .{value, if (i == list.items.len-1) "]" else ", " }); + } + } else { + std.debug.print("]", .{}); + } + +} + +// Print a linked list +pub fn printLinkedList(comptime T: type, node: ?*ListNode(T)) !void { + if (node == null) return; + var list = std.ArrayList(i32).init(std.heap.page_allocator); + defer list.deinit(); + var head = node; + while (head != null) { + try list.append(head.?.val); + head = head.?.next; + } + for (list.items) |value, i| { + std.debug.print("{}{s}", .{value, if (i == list.items.len-1) "\n" else "->" }); } } From 82f9542abe8cee81289c5adc1aac5dfaf82329b0 Mon Sep 17 00:00:00 2001 From: sjinzh <99076655+sjinzh@users.noreply.github.com> Date: Wed, 11 Jan 2023 17:43:52 +0800 Subject: [PATCH 11/23] add zig codes for Section 'Array', 'LinkedList' and 'List' --- codes/zig/chapter_array_and_linkedlist/array.zig | 4 ---- codes/zig/chapter_array_and_linkedlist/linked_list.zig | 4 ---- codes/zig/chapter_array_and_linkedlist/list.zig | 4 ---- codes/zig/chapter_array_and_linkedlist/my_list.zig | 4 ---- 4 files changed, 16 deletions(-) diff --git a/codes/zig/chapter_array_and_linkedlist/array.zig b/codes/zig/chapter_array_and_linkedlist/array.zig index 1206168e5..55000bf9b 100644 --- a/codes/zig/chapter_array_and_linkedlist/array.zig +++ b/codes/zig/chapter_array_and_linkedlist/array.zig @@ -90,10 +90,6 @@ pub fn find(nums: []i32, target: i32) i32 { // Driver Code pub fn main() !void { - // 查看本地CPU架构和操作系统信息 - var native_target_info = try std.zig.system.NativeTargetInfo.detect(std.zig.CrossTarget{}); - std.debug.print("Native Info: CPU Arch = {}, OS = {}\n", .{native_target_info.target.cpu.arch, native_target_info.target.os.tag}); - // 初始化数组 const size: i32 = 5; var arr = [_]i32{0} ** size; diff --git a/codes/zig/chapter_array_and_linkedlist/linked_list.zig b/codes/zig/chapter_array_and_linkedlist/linked_list.zig index 495ec3c66..fbaa3f8d7 100644 --- a/codes/zig/chapter_array_and_linkedlist/linked_list.zig +++ b/codes/zig/chapter_array_and_linkedlist/linked_list.zig @@ -46,10 +46,6 @@ pub fn find(node: ?*inc.ListNode(i32), target: i32) i32 { // Driver Code pub fn main() !void { - // 查看本地CPU架构和操作系统信息 - var native_target_info = try std.zig.system.NativeTargetInfo.detect(std.zig.CrossTarget{}); - std.debug.print("Native Info: CPU Arch = {}, OS = {}\n", .{native_target_info.target.cpu.arch, native_target_info.target.os.tag}); - // 初始化链表 // 初始化各个结点 var n0 = inc.ListNode(i32){.val = 1}; diff --git a/codes/zig/chapter_array_and_linkedlist/list.zig b/codes/zig/chapter_array_and_linkedlist/list.zig index c27a641bd..608bce96c 100644 --- a/codes/zig/chapter_array_and_linkedlist/list.zig +++ b/codes/zig/chapter_array_and_linkedlist/list.zig @@ -7,10 +7,6 @@ const inc = @import("include"); // Driver Code pub fn main() !void { - // 查看本地CPU架构和操作系统信息 - var native_target_info = try std.zig.system.NativeTargetInfo.detect(std.zig.CrossTarget{}); - std.debug.print("Native Info: CPU Arch = {}, OS = {}\n", .{native_target_info.target.cpu.arch, native_target_info.target.os.tag}); - // 初始化列表 var list = std.ArrayList(i32).init(std.heap.page_allocator); // 延迟释放内存 diff --git a/codes/zig/chapter_array_and_linkedlist/my_list.zig b/codes/zig/chapter_array_and_linkedlist/my_list.zig index 21aa921f3..fccd5ac65 100644 --- a/codes/zig/chapter_array_and_linkedlist/my_list.zig +++ b/codes/zig/chapter_array_and_linkedlist/my_list.zig @@ -126,10 +126,6 @@ pub fn MyList(comptime T: type) type { // Driver Code pub fn main() !void { - // 查看本地CPU架构和操作系统信息 - var native_target_info = try std.zig.system.NativeTargetInfo.detect(std.zig.CrossTarget{}); - std.debug.print("Native Info: CPU Arch = {}, OS = {}\n", .{native_target_info.target.cpu.arch, native_target_info.target.os.tag}); - // 初始化列表 var list = MyList(i32){}; try list.init(std.heap.page_allocator); From 2367318a28f33c45e77cdeae034a32b09c371f30 Mon Sep 17 00:00:00 2001 From: sjinzh <99076655+sjinzh@users.noreply.github.com> Date: Wed, 11 Jan 2023 18:05:42 +0800 Subject: [PATCH 12/23] add zig codes for Section 'Array', 'LinkedList' and 'List' --- codes/zig/chapter_array_and_linkedlist/array.zig | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/codes/zig/chapter_array_and_linkedlist/array.zig b/codes/zig/chapter_array_and_linkedlist/array.zig index 55000bf9b..52ebc8606 100644 --- a/codes/zig/chapter_array_and_linkedlist/array.zig +++ b/codes/zig/chapter_array_and_linkedlist/array.zig @@ -137,7 +137,7 @@ pub fn main() !void { // 查找元素 var index = find(nums, 3); - std.debug.print("\n在 nums_ext 中查找元素 3 ,得到索引 = {}\n", .{index}); + std.debug.print("\n在 nums 中查找元素 3 ,得到索引 = {}\n", .{index}); const getchar = try std.io.getStdIn().reader().readByte(); _ = getchar; From d75a3582b06ddd76494bfbb54a017ee18df02b63 Mon Sep 17 00:00:00 2001 From: sjinzh <99076655+sjinzh@users.noreply.github.com> Date: Wed, 11 Jan 2023 19:56:25 +0800 Subject: [PATCH 13/23] add zig codes for Section 'Array', 'LinkedList' and 'List' --- codes/zig/chapter_array_and_linkedlist/my_list.zig | 1 - 1 file changed, 1 deletion(-) diff --git a/codes/zig/chapter_array_and_linkedlist/my_list.zig b/codes/zig/chapter_array_and_linkedlist/my_list.zig index fccd5ac65..f018e61ca 100644 --- a/codes/zig/chapter_array_and_linkedlist/my_list.zig +++ b/codes/zig/chapter_array_and_linkedlist/my_list.zig @@ -120,7 +120,6 @@ pub fn MyList(comptime T: type) type { } return nums; } - }; } From 08afeb2981cb01e450c2138dcef480ef79ec2b94 Mon Sep 17 00:00:00 2001 From: ubuntu Date: Wed, 11 Jan 2023 22:50:23 +0800 Subject: [PATCH 14/23] fix(codes/cpp): fix worst_best_time_complexity.cpp compile error. --- codes/cpp/include/include.hpp | 2 ++ 1 file changed, 2 insertions(+) diff --git a/codes/cpp/include/include.hpp b/codes/cpp/include/include.hpp index 8e4a8070d..03bf9b503 100644 --- a/codes/cpp/include/include.hpp +++ b/codes/cpp/include/include.hpp @@ -17,6 +17,8 @@ #include #include #include +#include +#include #include "ListNode.hpp" #include "TreeNode.hpp" From 40b5be53021d459ad22263b4dcc8e56aff5784fa Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Thu, 12 Jan 2023 00:31:11 +0800 Subject: [PATCH 15/23] Unify the naming of the C codes. --- codes/c/chapter_array_and_linkedlist/array.c | 10 +++++----- .../worst_best_time_complexity.c | 2 +- codes/c/chapter_sorting/bubble_sort.c | 8 ++++---- codes/c/chapter_tree/binary_tree.c | 19 +++++++++---------- codes/c/chapter_tree/binary_tree_bfs.c | 6 +++--- codes/c/chapter_tree/binary_tree_dfs.c | 12 ++++++------ codes/c/include/include_test.c | 14 +++++++------- codes/c/include/list_node.h | 10 +++++----- codes/c/include/print_util.h | 6 +++--- codes/c/include/tree_node.h | 12 ++++++------ docs/chapter_sorting/bubble_sort.md | 4 ++-- 11 files changed, 51 insertions(+), 52 deletions(-) diff --git a/codes/c/chapter_array_and_linkedlist/array.c b/codes/c/chapter_array_and_linkedlist/array.c index 11fcb67f9..33dc99903 100644 --- a/codes/c/chapter_array_and_linkedlist/array.c +++ b/codes/c/chapter_array_and_linkedlist/array.c @@ -74,11 +74,11 @@ int main() { int size = 5; int arr[5]; printf("数组 arr = "); - PrintArray(arr, size); + printArray(arr, size); int nums[5] = { 1, 3, 2, 5, 4 }; printf("数组 nums = "); - PrintArray(nums, size); + printArray(nums, size); /* 随机访问 */ int randomNum = randomAccess(nums, size); @@ -89,17 +89,17 @@ int main() { int* res = extend(nums, size, enlarge); size += enlarge; printf("将数组长度扩展至 8 ,得到 nums = "); - PrintArray(res, size); + printArray(res, size); /* 插入元素 */ insert(res, size, 6, 3); printf("在索引 3 处插入数字 6 ,得到 nums = "); - PrintArray(res, size); + printArray(res, size); /* 删除元素 */ removeItem(res, size, 2); printf("删除索引 2 处的元素,得到 nums = "); - PrintArray(res, size); + printArray(res, size); /* 遍历数组 */ traverse(res, size); diff --git a/codes/c/chapter_computational_complexity/worst_best_time_complexity.c b/codes/c/chapter_computational_complexity/worst_best_time_complexity.c index 531a76e51..e5cee821f 100644 --- a/codes/c/chapter_computational_complexity/worst_best_time_complexity.c +++ b/codes/c/chapter_computational_complexity/worst_best_time_complexity.c @@ -41,7 +41,7 @@ int main(int argc, char *argv[]) { int *nums = randomNumbers(n); int index = findOne(nums, n); printf("\n数组 [ 1, 2, ..., n ] 被打乱后 = "); - PrintArray(nums, n); + printArray(nums, n); printf("数字 1 的索引为 %d\n", index); // 释放堆区内存 if (nums != NULL) { diff --git a/codes/c/chapter_sorting/bubble_sort.c b/codes/c/chapter_sorting/bubble_sort.c index 4a2a24b44..51bab6c64 100644 --- a/codes/c/chapter_sorting/bubble_sort.c +++ b/codes/c/chapter_sorting/bubble_sort.c @@ -7,7 +7,7 @@ #include "../include/include.h" /* 冒泡排序 */ -void bubble_sort(int nums[], int size) { +void bubbleSort(int nums[], int size) { // 外循环:待排序元素数量为 n-1, n-2, ..., 1 for (int i = 0; i < size - 1; i++) { @@ -25,7 +25,7 @@ void bubble_sort(int nums[], int size) { } /* 冒泡排序(标志优化)*/ -void bubble_sort_with_flag(int nums[], int size) { +void bubbleSortWithFlag(int nums[], int size) { // 外循环:待排序元素数量为 n-1, n-2, ..., 1 for (int i = 0; i < size - 1; i++) { @@ -51,14 +51,14 @@ void bubble_sort_with_flag(int nums[], int size) { int main() { int nums[6] = {4, 1, 3, 1, 5, 2}; printf("冒泡排序后: "); - bubble_sort(nums, 6); + bubbleSort(nums, 6); for (int i = 0; i < 6; i++) { printf("%d ", nums[i]); } printf("\n优化版冒泡排序后: "); - bubble_sort_with_flag(nums, 6); + bubbleSortWithFlag(nums, 6); for (int i = 0; i < 6; i++) { printf("%d ", nums[i]); diff --git a/codes/c/chapter_tree/binary_tree.c b/codes/c/chapter_tree/binary_tree.c index 0a2f946a7..1eb8e846a 100644 --- a/codes/c/chapter_tree/binary_tree.c +++ b/codes/c/chapter_tree/binary_tree.c @@ -10,34 +10,33 @@ int main() { /* 初始化二叉树 */ // 初始化结点 - TreeNode* n1 = NewTreeNode(1); - TreeNode* n2 = NewTreeNode(2); - TreeNode* n3 = NewTreeNode(3); - TreeNode* n4 = NewTreeNode(4); - TreeNode* n5 = NewTreeNode(5); + TreeNode* n1 = newTreeNode(1); + TreeNode* n2 = newTreeNode(2); + TreeNode* n3 = newTreeNode(3); + TreeNode* n4 = newTreeNode(4); + TreeNode* n5 = newTreeNode(5); // 构建引用指向(即指针) n1->left = n2; n1->right = n3; n2->left = n4; n2->right = n5; printf("初始化二叉树\n"); - PrintTree(n1); + printTree(n1); /* 插入与删除结点 */ - TreeNode* P = NewTreeNode(0); + TreeNode* P = newTreeNode(0); // 在 n1 -> n2 中间插入结点 P n1->left = P; P->left = n2; printf("插入结点 P 后\n"); - PrintTree(n1); + printTree(n1); // 删除结点 P n1->left = n2; // 释放内存 free(P); printf("删除结点 P 后\n"); - PrintTree(n1); + printTree(n1); return 0; } - diff --git a/codes/c/chapter_tree/binary_tree_bfs.c b/codes/c/chapter_tree/binary_tree_bfs.c index abad3735f..c7b83b912 100644 --- a/codes/c/chapter_tree/binary_tree_bfs.c +++ b/codes/c/chapter_tree/binary_tree_bfs.c @@ -52,15 +52,15 @@ int main() { // 这里借助了一个从数组直接生成二叉树的函数 int nums[] = {1, 2, 3, NIL, 5, 6, NIL}; int size = sizeof(nums) / sizeof(int); - TreeNode *root = ArrayToTree(nums, size); + TreeNode *root = arrToTree(nums, size); printf("初始化二叉树\n"); - PrintTree(root); + printTree(root); /* 层序遍历 */ // 需要传入数组的长度 int *arr = levelOrder(root, &size); printf("层序遍历的结点打印序列 = "); - PrintArray(arr, size); + printArray(arr, size); return 0; } \ No newline at end of file diff --git a/codes/c/chapter_tree/binary_tree_dfs.c b/codes/c/chapter_tree/binary_tree_dfs.c index f72e6687b..bafc90fd1 100644 --- a/codes/c/chapter_tree/binary_tree_dfs.c +++ b/codes/c/chapter_tree/binary_tree_dfs.c @@ -43,10 +43,10 @@ int main() { /* 初始化二叉树 */ // 这里借助了一个从数组直接生成二叉树的函数 int nums[] = {1, 2, 3, 4, 5, 6, 7}; - int size = sizeof(nums) / sizt ceof(int); - TreeNode *root = ArrayToTree(nums, size); + int size = sizeof(nums) / sizeof(int); + TreeNode *root = arrToTree(nums, size); printf("初始化二叉树\n"); - PrintTree(root); + printTree(root); /* 前序遍历 */ // 初始化辅助数组 @@ -54,19 +54,19 @@ int main() { size = 0; preOrder(root, &size); printf("前序遍历的结点打印序列 = "); - PrintArray(arr, size); + printArray(arr, size); /* 中序遍历 */ size = 0; inOrder(root, &size); printf("中序遍历的结点打印序列 = "); - PrintArray(arr, size); + printArray(arr, size); /* 后序遍历 */ size = 0; postOrder(root, &size); printf("后序遍历的结点打印序列 = "); - PrintArray(arr, size); + printArray(arr, size); return 0; } diff --git a/codes/c/include/include_test.c b/codes/c/include/include_test.c index baa8e649d..602670d68 100644 --- a/codes/c/include/include_test.c +++ b/codes/c/include/include_test.c @@ -9,24 +9,24 @@ void testListNode() { int nums[] = {2, 3, 5, 6, 7}; int size = sizeof(nums) / sizeof(int); - ListNode *head = ArrayToLinkedList(nums, size); - PrintLinkedList(head); + ListNode *head = arrToLinkedList(nums, size); + printLinkedList(head); - ListNode *node = GetListNode(head, 5); + ListNode *node = getListNode(head, 5); printf("find node: %d\n", node->val); } void testTreeNode() { int nums[] = {1, 2, 3, NIL, 5, 6, NIL}; int size = sizeof(nums) / sizeof(int); - TreeNode *root = ArrayToTree(nums, size); + TreeNode *root = arrToTree(nums, size); // print tree - PrintTree(root); + printTree(root); // tree to arr - int *arr = TreeToArray(root); - PrintArray(arr, size); + int *arr = treeToArr(root); + printArray(arr, size); } int main(int argc, char *argv[]) { diff --git a/codes/c/include/list_node.h b/codes/c/include/list_node.h index 5597d2102..1845ffafd 100644 --- a/codes/c/include/list_node.h +++ b/codes/c/include/list_node.h @@ -22,7 +22,7 @@ struct ListNode { // typedef 为 C 语言的关键字,作用是为一种数据类型定义一个新名字 typedef struct ListNode ListNode; -ListNode *NewListNode(int val) { +ListNode *newListNode(int val) { ListNode *node, *next; node = (ListNode *) malloc(sizeof(ListNode)); node->val = val; @@ -37,15 +37,15 @@ ListNode *NewListNode(int val) { * @return ListNode* */ -ListNode *ArrayToLinkedList(const int *arr, size_t size) { +ListNode *arrToLinkedList(const int *arr, size_t size) { if (size <= 0) { return NULL; } - ListNode *dummy = NewListNode(0); + ListNode *dummy = newListNode(0); ListNode *node = dummy; for (int i = 0; i < size; i++) { - node->next = NewListNode(arr[i]); + node->next = newListNode(arr[i]); node = node->next; } return dummy->next; @@ -58,7 +58,7 @@ ListNode *ArrayToLinkedList(const int *arr, size_t size) { * @param val * @return ListNode* */ -ListNode *GetListNode(ListNode *head, int val) { +ListNode *getListNode(ListNode *head, int val) { while (head != NULL && head->val != val) { head = head->next; } diff --git a/codes/c/include/print_util.h b/codes/c/include/print_util.h index fcb0a4ca6..a0c3a1500 100644 --- a/codes/c/include/print_util.h +++ b/codes/c/include/print_util.h @@ -25,7 +25,7 @@ extern "C" { * @param arr * @param size */ -static void PrintArray(int *arr, int size) { +static void printArray(int *arr, int size) { printf("["); for (int i = 0; i < size - 1; i++) { if (arr[i] != NIL) { @@ -46,7 +46,7 @@ static void PrintArray(int *arr, int size) { * * @param head */ -static void PrintLinkedList(ListNode *node) { +static void printLinkedList(ListNode *node) { if (node == NULL) { return; } @@ -123,7 +123,7 @@ static void printTreeHelper(TreeNode *node, Trunk *prev, bool isLeft) { * * @param head */ -static void PrintTree(TreeNode *root) { +static void printTree(TreeNode *root) { printTreeHelper(root, NULL, false); } diff --git a/codes/c/include/tree_node.h b/codes/c/include/tree_node.h index 4b0e70187..c0438c437 100644 --- a/codes/c/include/tree_node.h +++ b/codes/c/include/tree_node.h @@ -25,7 +25,7 @@ struct TreeNode { typedef struct TreeNode TreeNode; -TreeNode *NewTreeNode(int val) { +TreeNode *newTreeNode(int val) { TreeNode *node; node = (TreeNode *) malloc(sizeof(TreeNode)); @@ -43,7 +43,7 @@ TreeNode *NewTreeNode(int val) { * @param size * @return TreeNode * */ -TreeNode *ArrayToTree(const int *arr, size_t size) { +TreeNode *arrToTree(const int *arr, size_t size) { if (size <= 0) { return NULL; } @@ -53,7 +53,7 @@ TreeNode *ArrayToTree(const int *arr, size_t size) { TreeNode **queue; /* 根结点 */ - root = NewTreeNode(arr[0]); + root = newTreeNode(arr[0]); /* 辅助队列 */ queue = (TreeNode **) malloc(sizeof(TreeNode) * MAX_NODE_SIZE); // 队列指针 @@ -68,14 +68,14 @@ TreeNode *ArrayToTree(const int *arr, size_t size) { index++; if (index < size) { if (arr[index] != NIL) { - node->left = NewTreeNode(arr[index]); + node->left = newTreeNode(arr[index]); queue[rear++] = node->left; } } index++; if (index < size) { if (arr[index] != NIL) { - node->right = NewTreeNode(arr[index]); + node->right = newTreeNode(arr[index]); queue[rear++] = node->right; } } @@ -91,7 +91,7 @@ TreeNode *ArrayToTree(const int *arr, size_t size) { * @param size * @return TreeNode * */ -int *TreeToArray(TreeNode *root) { +int *treeToArr(TreeNode *root) { if (root == NULL) { return NULL; } diff --git a/docs/chapter_sorting/bubble_sort.md b/docs/chapter_sorting/bubble_sort.md index e3e394b20..3bb0bc994 100644 --- a/docs/chapter_sorting/bubble_sort.md +++ b/docs/chapter_sorting/bubble_sort.md @@ -170,7 +170,7 @@ comments: true ```c title="bubble_sort.c" /* 冒泡排序 */ - void bubble_sort(int nums[], int size) { + void bubbleSort(int nums[], int size) { // 外循环:待排序元素数量为 n-1, n-2, ..., 1 for (int i = 0; i < size - 1; i++) { @@ -373,7 +373,7 @@ comments: true ```c title="bubble_sort.c" /* 冒泡排序 */ - void bubble_sort(int nums[], int size) { + void bubbleSortWithFlag(int nums[], int size) { // 外循环:待排序元素数量为 n-1, n-2, ..., 1 for (int i = 0; i < size - 1; i++) { From 274d6a5882cd01cc2c473fac0640a413e433d790 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Thu, 12 Jan 2023 00:39:08 +0800 Subject: [PATCH 16/23] Remove a IDE-related file. --- .prettierrc | 4 ---- 1 file changed, 4 deletions(-) delete mode 100644 .prettierrc diff --git a/.prettierrc b/.prettierrc deleted file mode 100644 index 5a938ce18..000000000 --- a/.prettierrc +++ /dev/null @@ -1,4 +0,0 @@ -{ - "tabWidth": 4, - "useTabs": false -} From 3abcefae99165efd78c007a1737e8fb156e5f56e Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Thu, 12 Jan 2023 04:08:45 +0800 Subject: [PATCH 17/23] Update heap. --- .../java/chapter_hashing/array_hash_map.java | 1 + codes/java/chapter_hashing/hash_map.java | 1 + codes/java/chapter_heap/heap.java | 9 +- codes/java/chapter_heap/my_heap.java | 32 ++--- .../heap.assets/heap_poll_step1.png | Bin 0 -> 70281 bytes .../heap.assets/heap_poll_step10.png | Bin 0 -> 90820 bytes .../heap.assets/heap_poll_step2.png | Bin 0 -> 67583 bytes .../heap.assets/heap_poll_step3.png | Bin 0 -> 70220 bytes .../heap.assets/heap_poll_step4.png | Bin 0 -> 83162 bytes .../heap.assets/heap_poll_step5.png | Bin 0 -> 91437 bytes .../heap.assets/heap_poll_step6.png | Bin 0 -> 87190 bytes .../heap.assets/heap_poll_step7.png | Bin 0 -> 96058 bytes .../heap.assets/heap_poll_step8.png | Bin 0 -> 89271 bytes .../heap.assets/heap_poll_step9.png | Bin 0 -> 97988 bytes .../heap.assets/heap_push_step1.png | Bin 0 -> 70363 bytes .../heap.assets/heap_push_step2.png | Bin 0 -> 67080 bytes .../heap.assets/heap_push_step3.png | Bin 0 -> 83246 bytes .../heap.assets/heap_push_step4.png | Bin 0 -> 85784 bytes .../heap.assets/heap_push_step5.png | Bin 0 -> 86726 bytes .../heap.assets/heap_push_step6.png | Bin 0 -> 86922 bytes .../heap.assets/heapify_count.png | Bin 0 -> 65310 bytes .../heap.assets/min_heap_and_max_heap.png | Bin 0 -> 81457 bytes .../heap.assets/representation_of_heap.png | Bin 0 -> 115505 bytes docs/chapter_heap/heap.md | 130 ++++++++++++------ 24 files changed, 112 insertions(+), 61 deletions(-) create mode 100644 docs/chapter_heap/heap.assets/heap_poll_step1.png create mode 100644 docs/chapter_heap/heap.assets/heap_poll_step10.png create mode 100644 docs/chapter_heap/heap.assets/heap_poll_step2.png create mode 100644 docs/chapter_heap/heap.assets/heap_poll_step3.png create mode 100644 docs/chapter_heap/heap.assets/heap_poll_step4.png create mode 100644 docs/chapter_heap/heap.assets/heap_poll_step5.png create mode 100644 docs/chapter_heap/heap.assets/heap_poll_step6.png create mode 100644 docs/chapter_heap/heap.assets/heap_poll_step7.png create mode 100644 docs/chapter_heap/heap.assets/heap_poll_step8.png create mode 100644 docs/chapter_heap/heap.assets/heap_poll_step9.png create mode 100644 docs/chapter_heap/heap.assets/heap_push_step1.png create mode 100644 docs/chapter_heap/heap.assets/heap_push_step2.png create mode 100644 docs/chapter_heap/heap.assets/heap_push_step3.png create mode 100644 docs/chapter_heap/heap.assets/heap_push_step4.png create mode 100644 docs/chapter_heap/heap.assets/heap_push_step5.png create mode 100644 docs/chapter_heap/heap.assets/heap_push_step6.png create mode 100644 docs/chapter_heap/heap.assets/heapify_count.png create mode 100644 docs/chapter_heap/heap.assets/min_heap_and_max_heap.png create mode 100644 docs/chapter_heap/heap.assets/representation_of_heap.png diff --git a/codes/java/chapter_hashing/array_hash_map.java b/codes/java/chapter_hashing/array_hash_map.java index 68df1f3b9..980a00ab7 100644 --- a/codes/java/chapter_hashing/array_hash_map.java +++ b/codes/java/chapter_hashing/array_hash_map.java @@ -5,6 +5,7 @@ */ package chapter_hashing; + import java.util.*; /* 键值对 int->String */ diff --git a/codes/java/chapter_hashing/hash_map.java b/codes/java/chapter_hashing/hash_map.java index fe3f822f4..47f4823bf 100644 --- a/codes/java/chapter_hashing/hash_map.java +++ b/codes/java/chapter_hashing/hash_map.java @@ -5,6 +5,7 @@ */ package chapter_hashing; + import java.util.*; import include.*; diff --git a/codes/java/chapter_heap/heap.java b/codes/java/chapter_heap/heap.java index b964830fd..1a0c53b94 100644 --- a/codes/java/chapter_heap/heap.java +++ b/codes/java/chapter_heap/heap.java @@ -13,13 +13,13 @@ import java.util.*; public class heap { public static void testPush(Queue heap, int val) { heap.add(val); // 元素入堆 - System.out.format("\n添加元素 %d 后\n", val); + System.out.format("\n元素 %d 入堆后\n", val); PrintUtil.printHeap(heap); } public static void testPoll(Queue heap) { int val = heap.poll(); // 堆顶元素出堆 - System.out.format("\n出堆元素为 %d\n", val); + System.out.format("\n堆顶元素 %d 出堆后\n", val); PrintUtil.printHeap(heap); } @@ -46,6 +46,9 @@ public class heap { /* 堆顶元素出堆 */ testPoll(maxHeap); testPoll(maxHeap); + testPoll(maxHeap); + testPoll(maxHeap); + testPoll(maxHeap); /* 获取堆大小 */ int size = maxHeap.size(); @@ -58,7 +61,7 @@ public class heap { /* 输入列表并建堆 */ // 时间复杂度为 O(n) ,而非 O(nlogn) minHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4)); - System.out.println("\n输入 [1, 3, 2, 5, 4] ,建立小顶堆"); + System.out.println("\n输入列表并建立小顶堆后"); PrintUtil.printHeap(minHeap); } } diff --git a/codes/java/chapter_heap/my_heap.java b/codes/java/chapter_heap/my_heap.java index 8fd748217..b80ce6d08 100644 --- a/codes/java/chapter_heap/my_heap.java +++ b/codes/java/chapter_heap/my_heap.java @@ -146,26 +146,25 @@ public class my_heap { } public static void main(String[] args) { - /* 初始化堆 */ - // 初始化大顶堆 - MaxHeap maxHeap = new MaxHeap(); - - System.out.println("\n以下测试样例为大顶堆"); - - /* 元素入堆 */ - testPush(maxHeap, 1); - testPush(maxHeap, 3); - testPush(maxHeap, 2); - testPush(maxHeap, 5); - testPush(maxHeap, 4); + /* 初始化大顶堆 */ + MaxHeap maxHeap = new MaxHeap(Arrays.asList(9, 8, 6, 6, 7, 5, 2, 1, 4, 3, 6, 2)); + System.out.println("\n输入列表并建堆后"); + maxHeap.print(); /* 获取堆顶元素 */ int peek = maxHeap.peek(); System.out.format("\n堆顶元素为 %d\n", peek); + /* 元素入堆 */ + int val = 7; + maxHeap.push(val); + System.out.format("\n元素 %d 入堆后\n", val); + maxHeap.print(); + /* 堆顶元素出堆 */ - testPoll(maxHeap); - testPoll(maxHeap); + peek = maxHeap.poll(); + System.out.format("\n堆顶元素 %d 出堆后\n", peek); + maxHeap.print(); /* 获取堆大小 */ int size = maxHeap.size(); @@ -174,10 +173,5 @@ public class my_heap { /* 判断堆是否为空 */ boolean isEmpty = maxHeap.isEmpty(); System.out.format("\n堆是否为空 %b\n", isEmpty); - - /* 将输入列表堆化 */ - maxHeap = new MaxHeap(Arrays.asList(1, 3, 2, 5, 4)); - System.out.println("\n输入 [1, 3, 2, 5, 4] ,建立大顶堆"); - maxHeap.print(); } } diff --git a/docs/chapter_heap/heap.assets/heap_poll_step1.png b/docs/chapter_heap/heap.assets/heap_poll_step1.png new file mode 100644 index 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z(Nl3_C$ET-5jS$v8sq#VV>tx?|A-H?TNoC8!G68XH3J#GizW&`8Mt$}zrWx6P*LHm z^E4Z6#y17w*YDpSCpJ%n(>@FE39M{1GtM6-gkCcLi^?@7hQCP4$M(g|eY~Kwe&`$A r9Dwk6c`odmNN&CI)e_JAAvwXx<@2OwFFspB0Dtn(2ar-Jlc4_rHR>vm literal 0 HcmV?d00001 diff --git a/docs/chapter_heap/heap.md b/docs/chapter_heap/heap.md index f41826159..37e5aaf66 100644 --- a/docs/chapter_heap/heap.md +++ b/docs/chapter_heap/heap.md @@ -9,15 +9,13 @@ comments: true - 「大顶堆 Max Heap」,任意结点的值 $\geq$ 其子结点的值; - 「小顶堆 Min Heap」,任意结点的值 $\leq$ 其子结点的值; -由于堆是完全二叉树,因此最底层结点靠左填充,其它层结点皆被填满。 +![min_heap_and_max_heap](heap.assets/min_heap_and_max_heap.png) -对于大顶堆(小顶堆),其根结点的值最大(最小)。根结点被称为「堆顶」。 +## 堆术语与性质 -(图) - -!!! tip - - 大顶堆和小顶堆的定义、性质、操作本质上是相同的,区别只是大、小顶堆分别在求最大、最小值。若无特别说明,本文将使用大顶堆来举例。 +- 由于堆是完全二叉树,因此最底层结点靠左填充,其它层结点皆被填满。 +- 二叉树中的根结点对应「堆顶」,底层最靠右结点对应「堆底」。 +- 对于大顶堆 / 小顶堆,其堆顶元素(即根结点)的值最大 / 最小。 ## 堆常用操作 @@ -43,6 +41,10 @@ comments: true 我们可以直接使用编程语言提供的堆类(或优先队列类)。 +!!! tip + + 类似于排序中“从小到大排列”和“从大到小排列”,“大顶堆”和“小顶堆”可仅通过修改 Comparator 来互相转换。 + === "Java" ```java title="heap.java" @@ -60,10 +62,15 @@ comments: true maxHeap.add(4); /* 获取堆顶元素 */ - int peek = maxHeap.peek(); + int peek = maxHeap.peek(); // 5 /* 堆顶元素出堆 */ - int val = heap.poll(); + // 出堆元素会形成一个从大到小的序列 + peek = heap.poll(); // 5 + peek = heap.poll(); // 4 + peek = heap.poll(); // 3 + peek = heap.poll(); // 2 + peek = heap.poll(); // 1 /* 获取堆大小 */ int size = maxHeap.size(); @@ -77,19 +84,19 @@ comments: true ## 堆的实现 -!!! tip - - 下文使用「大顶堆」来举例,将所有 $>$ ($<$) 替换为 $<$ ($>$) 即可实现「小顶堆」。 +下文实现的是「大顶堆」,若想转换为「小顶堆」,将所有大小逻辑判断取逆(例如将 $\geq$ 替换为 $\leq$ )即可,有兴趣的同学可自行实现。 ### 堆的存储与表示 -在二叉树章节我们学过,「完全二叉树」非常适合使用「数组」来表示,而堆恰好是一颗完全二叉树,因而我们一般使用「数组」来存储「堆」。 +在二叉树章节我们学过,「完全二叉树」非常适合使用「数组」来表示,而堆恰好是一颗完全二叉树,**因而我们采用「数组」来存储「堆」**。 **二叉树指针**。使用数组表示二叉树时,元素代表结点值,索引代表结点在二叉树中的位置,**而结点指针通过索引映射公式来实现**。 -具体地,给定索引 $i$ ,那么其左子结点索引为 $2i + 1$ 、右子结点索引为 $2i + 2$ 、父结点索引为 $(i - 1) / 2$ (向下整除)。当索引越界时,代表空结点或结点不存在。我们将以上映射公式封装成函数,以便后续使用。 +具体地,给定索引 $i$ ,那么其左子结点索引为 $2i + 1$ 、右子结点索引为 $2i + 2$ 、父结点索引为 $(i - 1) / 2$ (向下整除)。当索引越界时,代表空结点或结点不存在。 + +![representation_of_heap](heap.assets/representation_of_heap.png) -(图) +我们将索引映射公式封装成函数,以便后续使用。 === "Java" @@ -133,13 +140,29 @@ comments: true ### 元素入堆 -给定元素 `val` ,我们先将其添加到堆的末尾。由于 `val` 可能大于堆中其它元素,此时堆的成立条件可能已经被破坏,**因此需要修复从插入结点到根结点这条路径上的各个结点**,该操作被称为「堆化 Heapify」。 +给定元素 `val` ,我们先将其添加到堆底。添加后,由于 `val` 可能大于堆中其它元素,此时堆的成立条件可能已经被破坏,**因此需要修复从插入结点到根结点这条路径上的各个结点**,该操作被称为「堆化 Heapify」。 考虑从入堆结点开始,**从底至顶执行堆化**。具体地,比较插入结点与其父结点的值,若插入结点更大则将它们交换;并循环以上操作,从底至顶地修复堆中的各个结点;直至越过根结点时结束,或当遇到无需交换的结点时提前结束。 -设结点总数为 $n$ ,则树的高度为 $O(\log n)$ ,易得堆化操作的循环轮数最多为 $O(\log n)$ ,**因而元素入堆操作的时间复杂度为 $O(\log n)$** 。 +=== "Step 1" + ![heap_push_step1](heap.assets/heap_push_step1.png) + +=== "Step 2" + ![heap_push_step2](heap.assets/heap_push_step2.png) + +=== "Step 3" + ![heap_push_step3](heap.assets/heap_push_step3.png) + +=== "Step 4" + ![heap_push_step4](heap.assets/heap_push_step4.png) + +=== "Step 5" + ![heap_push_step5](heap.assets/heap_push_step5.png) -(图) +=== "Step 6" + ![heap_push_step6](heap.assets/heap_push_step6.png) + +设结点总数为 $n$ ,则树的高度为 $O(\log n)$ ,易得堆化操作的循环轮数最多为 $O(\log n)$ ,**因而元素入堆操作的时间复杂度为 $O(\log n)$** 。 === "Java" @@ -172,13 +195,43 @@ comments: true 堆顶元素是二叉树根结点,即列表首元素,如果我们直接将首元素从列表中删除,则二叉树中所有结点都会随之发生移位(索引发生变化),这样后续使用堆化修复就很麻烦了。为了尽量减少元素索引变动,采取以下操作步骤: -1. 交换列表首元素与尾元素(即交换根结点与最右叶结点); -2. 交换完成后,将尾元素从列表中删除(此时堆顶元素已被删除); -3. 从根结点开始,从顶至底堆化; +1. 交换堆顶元素与堆底元素(即交换根结点与最右叶结点); +2. 交换完成后,将堆底从列表中删除(注意,因为已经交换,实际上删除的是原来的堆顶元素); +3. 从根结点开始,**从顶至底执行堆化**; 顾名思义,**从顶至底堆化的操作方向与从底至顶堆化相反**,我们比较根结点的值与其两个子结点的值,将最大的子结点与根结点执行交换,并循环以上操作,直到越过叶结点时结束,或当遇到无需交换的结点时提前结束。 -(图) +=== "Step 1" + ![heap_poll_step1](heap.assets/heap_poll_step1.png) + +=== "Step 2" + ![heap_poll_step2](heap.assets/heap_poll_step2.png) + +=== "Step 3" + ![heap_poll_step3](heap.assets/heap_poll_step3.png) + +=== "Step 4" + ![heap_poll_step4](heap.assets/heap_poll_step4.png) + +=== "Step 5" + ![heap_poll_step5](heap.assets/heap_poll_step5.png) + +=== "Step 6" + ![heap_poll_step6](heap.assets/heap_poll_step6.png) + +=== "Step 7" + ![heap_poll_step7](heap.assets/heap_poll_step7.png) + +=== "Step 8" + ![heap_poll_step8](heap.assets/heap_poll_step8.png) + +=== "Step 9" + ![heap_poll_step9](heap.assets/heap_poll_step9.png) + +=== "Step 10" + ![heap_poll_step10](heap.assets/heap_poll_step10.png) + +与元素入堆操作类似,**堆顶元素出堆操作的时间复杂度为 $O(\log n)$** 。 === "Java" @@ -237,48 +290,47 @@ comments: true } ``` -!!! tip - - 完全二叉树中,设结点总数为 $n$ ,则叶结点数量为 $(n + 1) / 2$ ,其中 $/$ 为向下整除。 +那么,第二种建堆方法的时间复杂度时多少呢?我们来做一下简单推算。 -那么,第二种建堆方法的时间复杂度时多少呢?我们来做一下简单推算。排除叶结点后,需要堆化结点数量为 $(n - 1)/2$ ,即为 $O(n)$ ,而二叉树高度为 $O(\log n)$ ,因此可得时间复杂度为 $O(n \log n)$ 。 +- 完全二叉树中,设结点总数为 $n$ ,则叶结点数量为 $(n + 1) / 2$ ,其中 $/$ 为向下整除。因此在排除叶结点后,需要堆化结点数量为 $(n - 1)/2$ ,即为 $O(n)$ ; +- 从顶至底堆化中,每个结点最多堆化至叶结点,因此最大迭代次数为二叉树高度 $O(\log n)$ ; -然而,该估算结果仍不够准确,因为我们没有考虑到 **二叉树底层结点远多于顶层结点** 的特点。下面我们来换种方法推导。 +将上述两者相乘,可得时间复杂度为 $O(n \log n)$ 。然而,该估算结果仍不够准确,因为我们没有考虑到 **二叉树底层结点远多于顶层结点** 的性质。 -设二叉树(即堆)结点数量为 $n$ ,树高度为 $h$ 。如下图所示,我们将各层的“结点数量 $\times$ 子树高度”进行求和,即可得到准确的操作数量。 +下面我们来尝试展开计算。为了减小计算难度,我们假设树是一个「完美二叉树」,该假设不会影响计算结果的正确性。设二叉树(即堆)结点数量为 $n$ ,树高度为 $h$ 。上文提到,**结点堆化最大迭代次数等于该结点到叶结点的距离,而这正是“结点高度”**。因此,我们将各层的“结点数量 $\times$ 结点高度”求和,即可得到所有结点的堆化的迭代次数总和。 $$ -S = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{(h-1)}\times1 +T(h) = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{(h-1)}\times1 $$ -(图) +![heapify_count](heap.assets/heapify_count.png) -求解上式需要借助中学的数列知识,先对 $S$ 乘以 $2$ ,可得 +化简上式需要借助中学的数列知识,先对 $T(h)$ 乘以 $2$ ,易得 $$ \begin{aligned} -S & = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{h-1}\times1 \newline -2S & = 2^1h + 2^2(h-1) + 2^3(h-2) + \cdots + 2^{h}\times1 \newline +T(h) & = 2^0h + 2^1(h-1) + 2^2(h-2) + \cdots + 2^{h-1}\times1 \newline +2 T(h) & = 2^1h + 2^2(h-1) + 2^3(h-2) + \cdots + 2^{h}\times1 \newline \end{aligned} $$ -令下式 $2S$ 与上式 $S$ 错位相减,易得 +**使用错位相减法**,令下式 $2 T(h)$ 减去上式 $T(h)$ ,可得 $$ -2S - S = S = -2^0h + 2^1 + 2^2 + \cdots + 2^{h-1} + 2^h +2T(h) - T(h) = T(h) = -2^0h + 2^1 + 2^2 + \cdots + 2^{h-1} + 2^h $$ -观察发现,$S$ 是一个等比数列,可直接借助公式求和。并且,对于高度为 $h$ 的完全二叉树,结点数量范围为 $n \in [2^h, 2^{h+1} - 1]$ ,复杂度为 $n = O(n) = O(2^h)$。 +观察上式,$T(h)$ 是一个等比数列,可直接使用求和公式,得到时间复杂度为 $$ \begin{aligned} -S & = 2 \frac{1 - 2^h}{1 - 2} - h \newline +T(h) & = 2 \frac{1 - 2^h}{1 - 2} - h \newline & = 2^{h+1} - h \newline -& = O(2^h) = O(n) +& = O(2^h) \end{aligned} $$ -以上推算表明,输入列表并建堆的时间复杂度为 $O(n)$ ,非常高效。 +进一步地,高度为 $h$ 的完美二叉树的结点数量为 $n = 2^{h+1} - 1$ ,易得复杂度为 $O(2^h) = O(n)$。以上推算表明,**输入列表并建堆的时间复杂度为 $O(n)$ ,非常高效**。 ## 堆常见应用 From d1d2c429b704faa7b5dcf2d1b3895d8860371648 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Thu, 12 Jan 2023 04:19:59 +0800 Subject: [PATCH 18/23] Add multi-language code blocks to heap. --- docs/chapter_heap/heap.md | 288 ++++++++++++++++++++++++++++++++++++++ 1 file changed, 288 insertions(+) diff --git a/docs/chapter_heap/heap.md b/docs/chapter_heap/heap.md index 37e5aaf66..73f425ff7 100644 --- a/docs/chapter_heap/heap.md +++ b/docs/chapter_heap/heap.md @@ -82,6 +82,54 @@ comments: true minHeap = new PriorityQueue<>(Arrays.asList(1, 3, 2, 5, 4)); ``` +=== "C++" + + ```cpp title="heap.cpp" + + ``` + +=== "Python" + + ```python title="heap.py" + + ``` + +=== "Go" + + ```go title="heap.go" + + ``` + +=== "JavaScript" + + ```js title="heap.js" + + ``` + +=== "TypeScript" + + ```typescript title="heap.ts" + + ``` + +=== "C" + + ```c title="heap.c" + + ``` + +=== "C#" + + ```csharp title="heap.cs" + + ``` + +=== "Swift" + + ```swift title="heap.swift" + + ``` + ## 堆的实现 下文实现的是「大顶堆」,若想转换为「小顶堆」,将所有大小逻辑判断取逆(例如将 $\geq$ 替换为 $\leq$ )即可,有兴趣的同学可自行实现。 @@ -125,6 +173,54 @@ comments: true } ``` +=== "C++" + + ```cpp title="my_heap.cpp" + + ``` + +=== "Python" + + ```python title="my_heap.py" + + ``` + +=== "Go" + + ```go title="my_heap.go" + + ``` + +=== "JavaScript" + + ```js title="my_heap.js" + + ``` + +=== "TypeScript" + + ```typescript title="my_heap.ts" + + ``` + +=== "C" + + ```c title="my_heap.c" + + ``` + +=== "C#" + + ```csharp title="my_heap.cs" + + ``` + +=== "Swift" + + ```swift title="my_heap.swift" + + ``` + ### 访问堆顶元素 堆顶元素是二叉树的根结点,即列表首元素。 @@ -138,6 +234,54 @@ comments: true } ``` +=== "C++" + + ```cpp title="my_heap.cpp" + + ``` + +=== "Python" + + ```python title="my_heap.py" + + ``` + +=== "Go" + + ```go title="my_heap.go" + + ``` + +=== "JavaScript" + + ```js title="my_heap.js" + + ``` + +=== "TypeScript" + + ```typescript title="my_heap.ts" + + ``` + +=== "C" + + ```c title="my_heap.c" + + ``` + +=== "C#" + + ```csharp title="my_heap.cs" + + ``` + +=== "Swift" + + ```swift title="my_heap.swift" + + ``` + ### 元素入堆 给定元素 `val` ,我们先将其添加到堆底。添加后,由于 `val` 可能大于堆中其它元素,此时堆的成立条件可能已经被破坏,**因此需要修复从插入结点到根结点这条路径上的各个结点**,该操作被称为「堆化 Heapify」。 @@ -191,6 +335,54 @@ comments: true } ``` +=== "C++" + + ```cpp title="my_heap.cpp" + + ``` + +=== "Python" + + ```python title="my_heap.py" + + ``` + +=== "Go" + + ```go title="my_heap.go" + + ``` + +=== "JavaScript" + + ```js title="my_heap.js" + + ``` + +=== "TypeScript" + + ```typescript title="my_heap.ts" + + ``` + +=== "C" + + ```c title="my_heap.c" + + ``` + +=== "C#" + + ```csharp title="my_heap.cs" + + ``` + +=== "Swift" + + ```swift title="my_heap.swift" + + ``` + ### 堆顶元素出堆 堆顶元素是二叉树根结点,即列表首元素,如果我们直接将首元素从列表中删除,则二叉树中所有结点都会随之发生移位(索引发生变化),这样后续使用堆化修复就很麻烦了。为了尽量减少元素索引变动,采取以下操作步骤: @@ -270,6 +462,54 @@ comments: true } ``` +=== "C++" + + ```cpp title="my_heap.cpp" + + ``` + +=== "Python" + + ```python title="my_heap.py" + + ``` + +=== "Go" + + ```go title="my_heap.go" + + ``` + +=== "JavaScript" + + ```js title="my_heap.js" + + ``` + +=== "TypeScript" + + ```typescript title="my_heap.ts" + + ``` + +=== "C" + + ```c title="my_heap.c" + + ``` + +=== "C#" + + ```csharp title="my_heap.cs" + + ``` + +=== "Swift" + + ```swift title="my_heap.swift" + + ``` + ### 输入数据并建堆 * 如果我们想要直接输入一个列表并将其建堆,那么该怎么做呢?最直接地,考虑使用「元素入堆」方法,将列表元素依次入堆。元素入堆的时间复杂度为 $O(n)$ ,而平均长度为 $\frac{n}{2}$ ,因此该方法的总体时间复杂度为 $O(n \log n)$ 。 @@ -290,6 +530,54 @@ comments: true } ``` +=== "C++" + + ```cpp title="my_heap.cpp" + + ``` + +=== "Python" + + ```python title="my_heap.py" + + ``` + +=== "Go" + + ```go title="my_heap.go" + + ``` + +=== "JavaScript" + + ```js title="my_heap.js" + + ``` + +=== "TypeScript" + + ```typescript title="my_heap.ts" + + ``` + +=== "C" + + ```c title="my_heap.c" + + ``` + +=== "C#" + + ```csharp title="my_heap.cs" + + ``` + +=== "Swift" + + ```swift title="my_heap.swift" + + ``` + 那么,第二种建堆方法的时间复杂度时多少呢?我们来做一下简单推算。 - 完全二叉树中,设结点总数为 $n$ ,则叶结点数量为 $(n + 1) / 2$ ,其中 $/$ 为向下整除。因此在排除叶结点后,需要堆化结点数量为 $(n - 1)/2$ ,即为 $O(n)$ ; From e318c495f734f952c257a73386db35d52f857a53 Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Thu, 12 Jan 2023 04:23:17 +0800 Subject: [PATCH 19/23] Update deploy.sh --- deploy.sh | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/deploy.sh b/deploy.sh index 96a633cba..975d7b2f9 100644 --- a/deploy.sh +++ b/deploy.sh @@ -3,6 +3,6 @@ mkdocs build --clean cd site git init git add -A -git commit -m 'deploy' +git commit -m "deploy" git push -f git@github.com:krahets/hello-algo.git master:gh-pages cd - From e6d1f2fd10fc67bb6b9ed722c95043e2b56c6e48 Mon Sep 17 00:00:00 2001 From: xiongsp <37035570+xiongsp@users.noreply.github.com> Date: Thu, 12 Jan 2023 10:36:54 +0800 Subject: [PATCH 20/23] Move tutorial to docs --- README.md | 22 ---------------------- docs/chapter_preface/contribution.md | 22 ++++++++++++++++++++++ 2 files changed, 22 insertions(+), 22 deletions(-) diff --git a/README.md b/README.md index 8bb7b345d..560e73953 100644 --- a/README.md +++ b/README.md @@ -41,28 +41,6 @@ 如果感觉本书对你有所帮助,请点个 Star :star: 支持一下,谢谢! -### 本地部署 - -#### Docker - -使用本教程前,请确保Docker已经安装并启动。 - -根据如下命令离线部署。 - -```bash -git clone https://github.com/krahets/hello-algo.git -cd hello-algo -docker-compose up -d -``` - -稍等片刻,请访问`http://localhost:8000`即可访问本教程。 - -使用如下命令删除部署。 - -```bash -docker-compose down -``` - ## 推荐语 > “一本通俗易懂的数据结构与算法入门书,引导读者手脑并用地学习,强烈推荐算法初学者阅读。” diff --git a/docs/chapter_preface/contribution.md b/docs/chapter_preface/contribution.md index c710e6de5..892da0b4c 100644 --- a/docs/chapter_preface/contribution.md +++ b/docs/chapter_preface/contribution.md @@ -41,4 +41,26 @@ comments: true 非常欢迎您和我一同来创作本书! +## 离线部署 + +### Docker + +使用本教程前,请确保Docker已经安装并启动。 + +根据如下命令离线部署。 + +```bash +git clone https://github.com/krahets/hello-algo.git +cd hello-algo +docker-compose up -d +``` + +稍等片刻,请访问`http://localhost:8000`即可访问本教程。 + +使用如下命令删除部署。 + +```bash +docker-compose down +``` + (TODO:教学视频) From dea3d0fa921a495bc2e9f8f4b537635e49d7e727 Mon Sep 17 00:00:00 2001 From: fbigm Date: Thu, 12 Jan 2023 11:32:32 +0800 Subject: [PATCH 21/23] =?UTF-8?q?=E6=8F=8F=E8=BF=B0=E4=BF=AE=E6=94=B9?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit 若将上图的「结点 2」看作父结点-->若将下图的「结点 2」看作父结点 --- docs/chapter_tree/binary_tree.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/chapter_tree/binary_tree.md b/docs/chapter_tree/binary_tree.md index 1061e019d..0201f6a2a 100644 --- a/docs/chapter_tree/binary_tree.md +++ b/docs/chapter_tree/binary_tree.md @@ -114,7 +114,7 @@ comments: true 结点的两个指针分别指向「左子结点 Left Child Node」和「右子结点 Right Child Node」,并且称该结点为两个子结点的「父结点 Parent Node」。给定二叉树某结点,将左子结点以下的树称为该结点的「左子树 Left Subtree」,右子树同理。 -除了叶结点外,每个结点都有子结点和子树。例如,若将上图的「结点 2」看作父结点,那么其左子结点和右子结点分别为「结点 4」和「结点 5」,左子树和右子树分别为「结点 4 及其以下结点形成的树」和「结点 5 及其以下结点形成的树」。 +除了叶结点外,每个结点都有子结点和子树。例如,若将下图的「结点 2」看作父结点,那么其左子结点和右子结点分别为「结点 4」和「结点 5」,左子树和右子树分别为「结点 4 及其以下结点形成的树」和「结点 5 及其以下结点形成的树」。 ![binary_tree_definition](binary_tree.assets/binary_tree_definition.png) From ebf049f2c6ebb903f67cd1e8c7cb47edfc5a58ef Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Thu, 12 Jan 2023 13:50:02 +0800 Subject: [PATCH 22/23] Update contribution.md --- docs/chapter_preface/contribution.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/chapter_preface/contribution.md b/docs/chapter_preface/contribution.md index 892da0b4c..4e5f3dc85 100644 --- a/docs/chapter_preface/contribution.md +++ b/docs/chapter_preface/contribution.md @@ -45,7 +45,7 @@ comments: true ### Docker -使用本教程前,请确保Docker已经安装并启动。 +使用本教程前,请确保 Docker 已经安装并启动。 根据如下命令离线部署。 @@ -55,7 +55,7 @@ cd hello-algo docker-compose up -d ``` -稍等片刻,请访问`http://localhost:8000`即可访问本教程。 +稍等片刻,即可通过浏览器打开 `http://localhost:8000` 访问本教程。 使用如下命令删除部署。 From 8325ce44748bea5a366e10619c8828083ced959c Mon Sep 17 00:00:00 2001 From: sjinzh <99076655+sjinzh@users.noreply.github.com> Date: Thu, 12 Jan 2023 17:08:33 +0800 Subject: [PATCH 23/23] update zig codes for Section 'Array', 'LinkedList', 'List' and 'Stack' --- .../chapter_array_and_linkedlist/array.zig | 32 ++----------------- 1 file changed, 2 insertions(+), 30 deletions(-) diff --git a/codes/zig/chapter_array_and_linkedlist/array.zig b/codes/zig/chapter_array_and_linkedlist/array.zig index 52ebc8606..8c830cf4d 100644 --- a/codes/zig/chapter_array_and_linkedlist/array.zig +++ b/codes/zig/chapter_array_and_linkedlist/array.zig @@ -14,7 +14,7 @@ pub fn randomAccess(nums: []i32) i32 { return randomNum; } -// 扩展数组长度(运行时方法) +// 扩展数组长度 pub fn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![]i32 { // 初始化一个扩展长度后的数组 var res = try mem_allocator.alloc(i32, nums.len + enlarge); @@ -25,26 +25,6 @@ pub fn extend(mem_allocator: std.mem.Allocator, nums: []i32, enlarge: usize) ![] return res; } -// 扩展数组长度(编译期方法A:初始化新数组并进行元素拷贝) -pub fn extendComptimeA(comptime nums: anytype, comptime enlarge: i32) [nums.len + enlarge]i32 { - // 初始化一个扩展长度后的数组 - var res = [_]i32{0} ** (nums.len + enlarge); - // 将原数组中的所有元素复制到新数组 - for (nums) |num, i| { - res[i] = num; - } - // 返回扩展后的新数组 - return res; -} - -// 扩展数组长度(编译期方法B: 通过数组拼接运算符“++”) -pub fn extendComptimeB(comptime nums: anytype, comptime enlarge: i32) [nums.len + enlarge]i32 { - // 数组拼接操作 - var res = nums ++ [_]i32{0} ** enlarge; - // 返回拼接后的新数组 - return res; -} - // 在数组的索引 index 处插入元素 num pub fn insert(nums: []i32, num: i32, index: usize) void { // 把索引 index 以及之后的所有元素向后移动一位 @@ -104,7 +84,7 @@ pub fn main() !void { var randomNum = randomAccess(&array); std.debug.print("\n在 nums 中获取随机元素 {}", .{randomNum}); - // 长度扩展(运行时方法) + // 长度扩展 var known_at_runtime_zero: usize = 0; var nums: []i32 = array[known_at_runtime_zero..array.len]; var mem_arena = std.heap.ArenaAllocator.init(std.heap.page_allocator); @@ -113,14 +93,6 @@ pub fn main() !void { nums = try extend(mem_allocator, nums, 3); std.debug.print("\n将数组长度扩展至 8 ,得到 nums = ", .{}); inc.PrintUtil.printArray(i32, nums); - // { - // // 长度扩展(编译期方法) - // comptime var array_comptime = [_]i32{ 1, 3, 2, 5, 4 }; - // var nums_comptime = extendComptimeA(array_comptime, 3); - // // var nums_comptime = extendComptimeB(array_comptime, 3); - // std.debug.print("\n将数组长度扩展至 8 ,得到 nums_comptime = ", .{}); - // inc.PrintUtil.printArray(i32, &nums_comptime); - // } // 插入元素 insert(nums, 6, 3);

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