diff --git a/codes/python/chapter_tree/array_binary_tree.py b/codes/python/chapter_tree/array_binary_tree.py index ba7f3bc21..fd8fc032c 100644 --- a/codes/python/chapter_tree/array_binary_tree.py +++ b/codes/python/chapter_tree/array_binary_tree.py @@ -22,7 +22,7 @@ class ArrayBinaryTree: """列表容量""" return len(self._tree) - def val(self, i: int) -> int: + def val(self, i: int) -> int | None: """获取索引为 i 节点的值""" # 若索引越界，则返回 None ，代表空位 if i < 0 or i >= self.size(): diff --git a/docs/chapter_array_and_linkedlist/ram_and_cache.md b/docs/chapter_array_and_linkedlist/ram_and_cache.md index db58b7e1d..e4271f028 100644 --- a/docs/chapter_array_and_linkedlist/ram_and_cache.md +++ b/docs/chapter_array_and_linkedlist/ram_and_cache.md @@ -25,7 +25,7 @@ ![计算机存储系统](ram_and_cache.assets/storage_pyramid.png) -!!! note +!!! tip 计算机的存储层次结构体现了速度、容量和成本三者之间的精妙平衡。实际上，这种权衡普遍存在于所有工业领域，它要求我们在不同的优势和限制之间找到最佳平衡点。 diff --git a/docs/chapter_backtracking/backtracking_algorithm.assets/preorder_find_paths_step11.png b/docs/chapter_backtracking/backtracking_algorithm.assets/preorder_find_paths_step11.png index 32d55a258..128a3c17b 100644 Binary files a/docs/chapter_backtracking/backtracking_algorithm.assets/preorder_find_paths_step11.png and b/docs/chapter_backtracking/backtracking_algorithm.assets/preorder_find_paths_step11.png differ diff --git a/docs/chapter_computational_complexity/time_complexity.md b/docs/chapter_computational_complexity/time_complexity.md index ee2866b7f..c02f91443 100755 --- a/docs/chapter_computational_complexity/time_complexity.md +++ b/docs/chapter_computational_complexity/time_complexity.md @@ -747,7 +747,7 @@ $T(n)$ 是一次函数，说明其运行时间的增长趋势是线性的，因 时间复杂度分析本质上是计算“操作数量 $T(n)$”的渐近上界，它具有明确的数学定义。 -!!! abstract "函数渐近上界" +!!! note "函数渐近上界" 若存在正实数 $c$ 和实数 $n_0$ ，使得对于所有的 $n > n_0$ ，均有 $T(n) \leq c \cdot f(n)$ ，则可认为 $f(n)$ 给出了 $T(n)$ 的一个渐近上界，记为 $T(n) = O(f(n))$ 。 diff --git a/docs/chapter_data_structure/number_encoding.md b/docs/chapter_data_structure/number_encoding.md index 012eefbf8..95cc5ca36 100644 --- a/docs/chapter_data_structure/number_encoding.md +++ b/docs/chapter_data_structure/number_encoding.md @@ -1,6 +1,6 @@ # 数字编码 * -!!! note +!!! tip 在本书中，标题带有 * 符号的是选读章节。如果你时间有限或感到理解困难，可以先跳过，等学完必读章节后再单独攻克。 diff --git a/docs/chapter_graph/graph_traversal.assets/graph_bfs_step1.png b/docs/chapter_graph/graph_traversal.assets/graph_bfs_step1.png index dc91f3de9..f8a1eaa99 100644 Binary files a/docs/chapter_graph/graph_traversal.assets/graph_bfs_step1.png and b/docs/chapter_graph/graph_traversal.assets/graph_bfs_step1.png differ diff --git a/docs/chapter_graph/graph_traversal.assets/graph_bfs_step10.png b/docs/chapter_graph/graph_traversal.assets/graph_bfs_step10.png index e36b33d39..a4f234f48 100644 Binary files a/docs/chapter_graph/graph_traversal.assets/graph_bfs_step10.png and b/docs/chapter_graph/graph_traversal.assets/graph_bfs_step10.png differ diff --git a/docs/chapter_graph/graph_traversal.assets/graph_bfs_step11.png 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b/docs/chapter_graph/graph_traversal.assets/graph_dfs_step8.png index b2d529514..d6cf543ab 100644 Binary files a/docs/chapter_graph/graph_traversal.assets/graph_dfs_step8.png and b/docs/chapter_graph/graph_traversal.assets/graph_dfs_step8.png differ diff --git a/docs/chapter_graph/graph_traversal.assets/graph_dfs_step9.png b/docs/chapter_graph/graph_traversal.assets/graph_dfs_step9.png index 1a114f14c..78759442f 100644 Binary files a/docs/chapter_graph/graph_traversal.assets/graph_dfs_step9.png and b/docs/chapter_graph/graph_traversal.assets/graph_dfs_step9.png differ diff --git a/docs/chapter_introduction/summary.md b/docs/chapter_introduction/summary.md index ad7c42e34..6eb7301a2 100644 --- a/docs/chapter_introduction/summary.md +++ b/docs/chapter_introduction/summary.md @@ -7,3 +7,16 @@ - 算法是在有限时间内解决特定问题的一组指令或操作步骤，而数据结构是计算机中组织和存储数据的方式。 - 数据结构与算法紧密相连。数据结构是算法的基石，而算法是数据结构发挥作用的舞台。 - 我们可以将数据结构与算法类比为拼装积木，积木代表数据，积木的形状和连接方式等代表数据结构，拼装积木的步骤则对应算法。 + +### Q & A + +**Q**：作为一名程序员，我在日常工作中从未用算法解决过问题，常用算法都被编程语言封装好了，直接用就可以了；这是否意味着我们工作中的问题还没有到达需要算法的程度？ + +如果把具体的工作技能比作是武功的“招式”的话，那么基础科目应该更像是“内功”。 + +我认为学算法（以及其他基础科目）的意义不是在于在工作中从零实现它，而是基于学到的知识，在解决问题时能够作出专业的反应和判断，从而提升工作的整体质量。举一个简单例子，每种编程语言都内置了排序函数： + +- 如果我们没有学过数据结构与算法，那么给定任何数据，我们可能都塞给这个排序函数去做了。运行顺畅、性能不错，看上去并没有什么问题。 +- 但如果学过算法，我们就会知道内置排序函数的时间复杂度是 $O(n \log n)$ ；而如果给定的数据是固定位数的整数（例如学号），那么我们就可以用效率更高的“基数排序”来做，将时间复杂度降为 $O(nk)$ ，其中 $k$ 为位数。当数据体量很大时，节省出来的运行时间就能创造较大价值（成本降低、体验变好等）。 + +在工程领域中，大量问题是难以达到最优解的，许多问题只是被“差不多”地解决了。问题的难易程度一方面取决于问题本身的性质，另一方面也取决于观测问题的人的知识储备。人的知识越完备、经验越多，分析问题就会越深入，问题就能被解决得更优雅。 diff --git a/docs/chapter_stack_and_queue/deque.md b/docs/chapter_stack_and_queue/deque.md index 742ffe9a0..a8c5da91a 100644 --- a/docs/chapter_stack_and_queue/deque.md +++ b/docs/chapter_stack_and_queue/deque.md @@ -27,28 +27,28 @@ from collections import deque # 初始化双向队列 - deque: deque[int] = deque() + deq: deque[int] = deque() # 元素入队 - deque.append(2) # 添加至队尾 - deque.append(5) - deque.append(4) - deque.appendleft(3) # 添加至队首 - deque.appendleft(1) + deq.append(2) # 添加至队尾 + deq.append(5) + deq.append(4) + deq.appendleft(3) # 添加至队首 + deq.appendleft(1) # 访问元素 - front: int = deque[0] # 队首元素 - rear: int = deque[-1] # 队尾元素 + front: int = deq[0] # 队首元素 + rear: int = deq[-1] # 队尾元素 # 元素出队 - pop_front: int = deque.popleft() # 队首元素出队 - pop_rear: int = deque.pop() # 队尾元素出队 + pop_front: int = deq.popleft() # 队首元素出队 + pop_rear: int = deq.pop() # 队尾元素出队 # 获取双向队列的长度 - size: int = len(deque) + size: int = len(deq) # 判断双向队列是否为空 - is_empty: bool = len(deque) == 0 + is_empty: bool = len(deq) == 0 ``` === "C++" diff --git a/docs/chapter_tree/avl_tree.md b/docs/chapter_tree/avl_tree.md index e5c2b9810..0b37a726f 100644 --- a/docs/chapter_tree/avl_tree.md +++ b/docs/chapter_tree/avl_tree.md @@ -248,7 +248,7 @@ AVL 树既是二叉搜索树，也是平衡二叉树，同时满足这两类二 [file]{avl_tree}-[class]{avl_tree}-[func]{balance_factor} ``` -!!! note +!!! tip 设平衡因子为 $f$ ，则一棵 AVL 树的任意节点的平衡因子皆满足 $-1 \le f \le 1$ 。 diff --git a/docs/chapter_tree/binary_tree.md b/docs/chapter_tree/binary_tree.md index f856c1ed2..18db2d51c 100644 --- a/docs/chapter_tree/binary_tree.md +++ b/docs/chapter_tree/binary_tree.md @@ -633,7 +633,7 @@ https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E4%B8%8E%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20%E5%9C%A8%20n1%20-%3E%20n2%20%E4%B8%AD%E9%97%B4%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20n2&cumulative=false&curInstr=37&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false -!!! note +!!! tip 需要注意的是，插入节点可能会改变二叉树的原有逻辑结构，而删除节点通常意味着删除该节点及其所有子树。因此，在二叉树中，插入与删除通常是由一套操作配合完成的，以实现有实际意义的操作。 diff --git a/en/README.md b/en/README.md index 7e5ea35d9..06cf1a2da 100644 --- a/en/README.md +++ b/en/README.md @@ -43,7 +43,7 @@ English

-## About +## The book This open-source project aims to create a free and beginner-friendly crash course for data structures and algorithms. diff --git a/en/docs/chapter_array_and_linkedlist/ram_and_cache.md b/en/docs/chapter_array_and_linkedlist/ram_and_cache.md index dea2a268f..9e582939f 100644 --- a/en/docs/chapter_array_and_linkedlist/ram_and_cache.md +++ b/en/docs/chapter_array_and_linkedlist/ram_and_cache.md @@ -25,7 +25,7 @@ We can imagine the computer storage system as a pyramid structure shown in the f ![Computer storage system](ram_and_cache.assets/storage_pyramid.png) -!!! note +!!! tip The storage hierarchy of computers reflects a delicate balance between speed, capacity, and cost. In fact, this kind of trade-off is common in all industrial fields, requiring us to find the best balance between different advantages and limitations. diff --git a/en/docs/chapter_computational_complexity/time_complexity.md b/en/docs/chapter_computational_complexity/time_complexity.md index 4a8453cfa..46275ddcf 100644 --- a/en/docs/chapter_computational_complexity/time_complexity.md +++ b/en/docs/chapter_computational_complexity/time_complexity.md @@ -665,7 +665,7 @@ Since $T(n)$ is a linear function, its growth trend is linear, and therefore, it In essence, time complexity analysis is about finding the asymptotic upper bound of the "number of operations $T(n)$". It has a precise mathematical definition. -!!! abstract "Asymptotic Upper Bound" +!!! note "Asymptotic Upper Bound" If there exist positive real numbers $c$ and $n_0$ such that for all $n > n_0$, $T(n) \leq c \cdot f(n)$, then $f(n)$ is considered an asymptotic upper bound of $T(n)$, denoted as $T(n) = O(f(n))$. diff --git a/en/docs/chapter_data_structure/number_encoding.md b/en/docs/chapter_data_structure/number_encoding.md index c8b7366e3..2d21f8dda 100644 --- a/en/docs/chapter_data_structure/number_encoding.md +++ b/en/docs/chapter_data_structure/number_encoding.md @@ -1,6 +1,6 @@ # Number encoding * -!!! note +!!! tip In this book, chapters marked with an asterisk '*' are optional readings. If you are short on time or find them challenging, you may skip these initially and return to them after completing the essential chapters. diff --git a/en/docs/chapter_stack_and_queue/deque.md b/en/docs/chapter_stack_and_queue/deque.md index eaeb1269e..e49e249a4 100644 --- a/en/docs/chapter_stack_and_queue/deque.md +++ b/en/docs/chapter_stack_and_queue/deque.md @@ -27,28 +27,28 @@ Similarly, we can directly use the double-ended queue classes implemented in pro from collections import deque # Initialize the deque - deque: deque[int] = deque() + deq: deque[int] = deque() # Enqueue elements - deque.append(2) # Add to the tail - deque.append(5) - deque.append(4) - deque.appendleft(3) # Add to the head - deque.appendleft(1) + deq.append(2) # Add to the tail + deq.append(5) + deq.append(4) + deq.appendleft(3) # Add to the head + deq.appendleft(1) # Access elements - front: int = deque[0] # The first element - rear: int = deque[-1] # The last element + front: int = deq[0] # The first element + rear: int = deq[-1] # The last element # Dequeue elements - pop_front: int = deque.popleft() # The first element dequeued - pop_rear: int = deque.pop() # The last element dequeued + pop_front: int = deq.popleft() # The first element dequeued + pop_rear: int = deq.pop() # The last element dequeued # Get the length of the deque - size: int = len(deque) + size: int = len(deq) # Check if the deque is empty - is_empty: bool = len(deque) == 0 + is_empty: bool = len(deq) == 0 ``` === "C++" diff --git a/en/docs/chapter_tree/avl_tree.md b/en/docs/chapter_tree/avl_tree.md index f93b43f0a..502a72456 100644 --- a/en/docs/chapter_tree/avl_tree.md +++ b/en/docs/chapter_tree/avl_tree.md @@ -237,7 +237,7 @@ The "balance factor" of a node is defined as the height of the node's left subtr [file]{avl_tree}-[class]{avl_tree}-[func]{balance_factor} ``` -!!! note +!!! tip Let the balance factor be $f$, then the balance factor of any node in an AVL tree satisfies $-1 \le f \le 1$. diff --git a/en/docs/chapter_tree/binary_tree.md b/en/docs/chapter_tree/binary_tree.md index 696e7844f..b9dce3ef0 100644 --- a/en/docs/chapter_tree/binary_tree.md +++ b/en/docs/chapter_tree/binary_tree.md @@ -607,7 +607,7 @@ Similar to a linked list, inserting and removing nodes in a binary tree can be a https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%8F%89%E6%A0%91%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%8F%89%E6%A0%91%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E8%8A%82%E7%82%B9%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E9%92%88%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E4%B8%8E%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20%E5%9C%A8%20n1%20-%3E%20n2%20%E4%B8%AD%E9%97%B4%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%20P%0A%20%20%20%20n1.left%20%3D%20n2&cumulative=false&curInstr=37&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false -!!! note +!!! tip It's important to note that inserting nodes may change the original logical structure of the binary tree, while removing nodes usually means removing the node and all its subtrees. Therefore, in a binary tree, insertion and removal are usually performed through a set of operations to achieve meaningful actions. diff --git a/zh-hant/codes/cpp/chapter_graph/graph_bfs.cpp b/zh-hant/codes/cpp/chapter_graph/graph_bfs.cpp index dd70735a4..9e4b4c485 100644 --- a/zh-hant/codes/cpp/chapter_graph/graph_bfs.cpp +++ b/zh-hant/codes/cpp/chapter_graph/graph_bfs.cpp @@ -12,7 +12,7 @@ vector graphBFS(GraphAdjList &graph, Vertex *startVet) { // 頂點走訪序列 vector res; - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 unordered_set visited = {startVet}; // 佇列用於實現 BFS queue que; diff --git a/zh-hant/codes/cpp/chapter_graph/graph_dfs.cpp b/zh-hant/codes/cpp/chapter_graph/graph_dfs.cpp index f22c851c9..ea78733b1 100644 --- a/zh-hant/codes/cpp/chapter_graph/graph_dfs.cpp +++ b/zh-hant/codes/cpp/chapter_graph/graph_dfs.cpp @@ -25,7 +25,7 @@ void dfs(GraphAdjList &graph, unordered_set &visited, vector vector graphDFS(GraphAdjList &graph, Vertex *startVet) { // 頂點走訪序列 vector res; - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 unordered_set visited; dfs(graph, visited, res, startVet); return res; diff --git a/zh-hant/codes/csharp/chapter_graph/graph_bfs.cs b/zh-hant/codes/csharp/chapter_graph/graph_bfs.cs index 7c1a3d69d..f3a82e524 100644 --- a/zh-hant/codes/csharp/chapter_graph/graph_bfs.cs +++ b/zh-hant/codes/csharp/chapter_graph/graph_bfs.cs @@ -12,7 +12,7 @@ public class graph_bfs { List GraphBFS(GraphAdjList graph, Vertex startVet) { // 頂點走訪序列 List res = []; - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 HashSet visited = [startVet]; // 佇列用於實現 BFS Queue que = new(); diff --git a/zh-hant/codes/csharp/chapter_graph/graph_dfs.cs b/zh-hant/codes/csharp/chapter_graph/graph_dfs.cs index cbeb7eab9..a31040366 100644 --- a/zh-hant/codes/csharp/chapter_graph/graph_dfs.cs +++ b/zh-hant/codes/csharp/chapter_graph/graph_dfs.cs @@ -26,7 +26,7 @@ public class graph_dfs { List GraphDFS(GraphAdjList graph, Vertex startVet) { // 頂點走訪序列 List res = []; - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 HashSet visited = []; DFS(graph, visited, res, startVet); return res; diff --git a/zh-hant/codes/dart/chapter_graph/graph_bfs.dart b/zh-hant/codes/dart/chapter_graph/graph_bfs.dart index 5ee762152..26757114b 100644 --- a/zh-hant/codes/dart/chapter_graph/graph_bfs.dart +++ b/zh-hant/codes/dart/chapter_graph/graph_bfs.dart @@ -14,7 +14,7 @@ List graphBFS(GraphAdjList graph, Vertex startVet) { // 使用鄰接表來表示圖，以便獲取指定頂點的所有鄰接頂點 // 頂點走訪序列 List res = []; - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 Set visited = {}; visited.add(startVet); // 佇列用於實現 BFS diff --git a/zh-hant/codes/dart/chapter_graph/graph_dfs.dart b/zh-hant/codes/dart/chapter_graph/graph_dfs.dart index 0da9c94d3..82b390764 100644 --- a/zh-hant/codes/dart/chapter_graph/graph_dfs.dart +++ b/zh-hant/codes/dart/chapter_graph/graph_dfs.dart @@ -30,7 +30,7 @@ void dfs( List graphDFS(GraphAdjList graph, Vertex startVet) { // 頂點走訪序列 List res = []; - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 Set visited = {}; dfs(graph, visited, res, startVet); return res; diff --git a/zh-hant/codes/go/chapter_graph/graph_bfs.go b/zh-hant/codes/go/chapter_graph/graph_bfs.go index 7928604b2..692365d00 100644 --- a/zh-hant/codes/go/chapter_graph/graph_bfs.go +++ b/zh-hant/codes/go/chapter_graph/graph_bfs.go @@ -13,7 +13,7 @@ import ( func graphBFS(g *graphAdjList, startVet Vertex) []Vertex { // 頂點走訪序列 res := make([]Vertex, 0) - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 visited := make(map[Vertex]struct{}) visited[startVet] = struct{}{} // 佇列用於實現 BFS, 使用切片模擬佇列 diff --git a/zh-hant/codes/go/chapter_graph/graph_dfs.go b/zh-hant/codes/go/chapter_graph/graph_dfs.go index a1561932a..90aa26f1c 100644 --- a/zh-hant/codes/go/chapter_graph/graph_dfs.go +++ b/zh-hant/codes/go/chapter_graph/graph_dfs.go @@ -28,7 +28,7 @@ func dfs(g *graphAdjList, visited map[Vertex]struct{}, res *[]Vertex, vet Vertex func graphDFS(g *graphAdjList, startVet Vertex) []Vertex { // 頂點走訪序列 res := make([]Vertex, 0) - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 visited := make(map[Vertex]struct{}) dfs(g, visited, &res, startVet) // 返回頂點走訪序列 diff --git a/zh-hant/codes/go/chapter_searching/two_sum.go b/zh-hant/codes/go/chapter_searching/two_sum.go index 65a243c7e..314ea6b9d 100644 --- a/zh-hant/codes/go/chapter_searching/two_sum.go +++ b/zh-hant/codes/go/chapter_searching/two_sum.go @@ -9,7 +9,7 @@ func twoSumBruteForce(nums []int, target int) []int { size := len(nums) // 兩層迴圈，時間複雜度為 O(n^2) for i := 0; i < size-1; i++ { - for j := i + 1; i < size; j++ { + for j := i + 1; j < size; j++ { if nums[i]+nums[j] == target { return []int{i, j} } diff --git a/zh-hant/codes/java/chapter_graph/graph_bfs.java b/zh-hant/codes/java/chapter_graph/graph_bfs.java index a1f077d0a..b3eddc77e 100644 --- a/zh-hant/codes/java/chapter_graph/graph_bfs.java +++ b/zh-hant/codes/java/chapter_graph/graph_bfs.java @@ -15,7 +15,7 @@ public class graph_bfs { static List graphBFS(GraphAdjList graph, Vertex startVet) { // 頂點走訪序列 List res = new ArrayList<>(); - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 Set visited = new HashSet<>(); visited.add(startVet); // 佇列用於實現 BFS diff --git a/zh-hant/codes/java/chapter_graph/graph_dfs.java b/zh-hant/codes/java/chapter_graph/graph_dfs.java index 68eff464c..266bf913a 100644 --- a/zh-hant/codes/java/chapter_graph/graph_dfs.java +++ b/zh-hant/codes/java/chapter_graph/graph_dfs.java @@ -28,7 +28,7 @@ public class graph_dfs { static List graphDFS(GraphAdjList graph, Vertex startVet) { // 頂點走訪序列 List res = new ArrayList<>(); - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 Set visited = new HashSet<>(); dfs(graph, visited, res, startVet); return res; diff --git a/zh-hant/codes/javascript/chapter_graph/graph_bfs.js b/zh-hant/codes/javascript/chapter_graph/graph_bfs.js index 2fa20cf91..4e46a624f 100644 --- a/zh-hant/codes/javascript/chapter_graph/graph_bfs.js +++ b/zh-hant/codes/javascript/chapter_graph/graph_bfs.js @@ -12,7 +12,7 @@ const { Vertex } = require('../modules/Vertex'); function graphBFS(graph, startVet) { // 頂點走訪序列 const res = []; - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 const visited = new Set(); visited.add(startVet); // 佇列用於實現 BFS diff --git a/zh-hant/codes/javascript/chapter_graph/graph_dfs.js b/zh-hant/codes/javascript/chapter_graph/graph_dfs.js index ff1a71f5e..1d5ea527c 100644 --- a/zh-hant/codes/javascript/chapter_graph/graph_dfs.js +++ b/zh-hant/codes/javascript/chapter_graph/graph_dfs.js @@ -27,7 +27,7 @@ function dfs(graph, visited, res, vet) { function graphDFS(graph, startVet) { // 頂點走訪序列 const res = []; - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 const visited = new Set(); dfs(graph, visited, res, startVet); return res; diff --git a/zh-hant/codes/javascript/test_all.js b/zh-hant/codes/javascript/test_all.js new file mode 100644 index 000000000..b43ef49f5 --- /dev/null +++ b/zh-hant/codes/javascript/test_all.js @@ -0,0 +1,63 @@ +import { bold, brightRed } from 'jsr:@std/fmt/colors'; +import { expandGlob } from 'jsr:@std/fs'; +import { relative, resolve } from 'jsr:@std/path'; + +/** + * @typedef {import('jsr:@std/fs').WalkEntry} WalkEntry + * @type {WalkEntry[]} + */ +const entries = []; + +for await (const entry of expandGlob( + resolve(import.meta.dirname, './chapter_*/*.js') +)) { + entries.push(entry); +} + +/** @type {{ status: Promise; stderr: ReadableStream; }[]} */ +const processes = []; + +for (const file of entries) { + const execute = new Deno.Command('node', { + args: [relative(import.meta.dirname, file.path)], + cwd: import.meta.dirname, + stdin: 'piped', + stdout: 'piped', + stderr: 'piped', + }); + + const process = execute.spawn(); + processes.push({ status: process.status, stderr: process.stderr }); +} + +const results = await Promise.all( + processes.map(async (item) => { + const status = await item.status; + return { status, stderr: item.stderr }; + }) +); + +/** @type {ReadableStream[]} */ +const errors = []; + +for (const result of results) { + if (!result.status.success) { + errors.push(result.stderr); + } +} + +console.log(`Tested ${entries.length} files`); +console.log(`Found exception in ${errors.length} files`); + +if (errors.length) { + console.log(); + + for (const error of errors) { + const reader = error.getReader(); + const { value } = await reader.read(); + const decoder = new TextDecoder(); + console.log(`${bold(brightRed('error'))}: ${decoder.decode(value)}`); + } + + throw new Error('Test failed'); +} diff --git a/zh-hant/codes/kotlin/chapter_graph/graph_bfs.kt b/zh-hant/codes/kotlin/chapter_graph/graph_bfs.kt index c56eebecc..d76e3d810 100644 --- a/zh-hant/codes/kotlin/chapter_graph/graph_bfs.kt +++ b/zh-hant/codes/kotlin/chapter_graph/graph_bfs.kt @@ -14,7 +14,7 @@ import java.util.* fun graphBFS(graph: GraphAdjList, startVet: Vertex): MutableList { // 頂點走訪序列 val res = mutableListOf() - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 val visited = HashSet() visited.add(startVet) // 佇列用於實現 BFS diff --git a/zh-hant/codes/kotlin/chapter_graph/graph_dfs.kt b/zh-hant/codes/kotlin/chapter_graph/graph_dfs.kt index 4c57f6d60..2b0217035 100644 --- a/zh-hant/codes/kotlin/chapter_graph/graph_dfs.kt +++ b/zh-hant/codes/kotlin/chapter_graph/graph_dfs.kt @@ -31,7 +31,7 @@ fun dfs( fun graphDFS(graph: GraphAdjList, startVet: Vertex?): MutableList { // 頂點走訪序列 val res = mutableListOf() - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 val visited = HashSet() dfs(graph, visited, res, startVet) return res diff --git a/zh-hant/codes/python/chapter_graph/graph_bfs.py b/zh-hant/codes/python/chapter_graph/graph_bfs.py index 6d29057df..0f8b7f415 100644 --- a/zh-hant/codes/python/chapter_graph/graph_bfs.py +++ b/zh-hant/codes/python/chapter_graph/graph_bfs.py @@ -18,7 +18,7 @@ def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -> list[Vertex]: # 使用鄰接表來表示圖，以便獲取指定頂點的所有鄰接頂點 # 頂點走訪序列 res = [] - # 雜湊表，用於記錄已被訪問過的頂點 + # 雜湊集合，用於記錄已被訪問過的頂點 visited = set[Vertex]([start_vet]) # 佇列用於實現 BFS que = deque[Vertex]([start_vet]) diff --git a/zh-hant/codes/python/chapter_graph/graph_dfs.py b/zh-hant/codes/python/chapter_graph/graph_dfs.py index 1ef97c63e..c37b5d6f0 100644 --- a/zh-hant/codes/python/chapter_graph/graph_dfs.py +++ b/zh-hant/codes/python/chapter_graph/graph_dfs.py @@ -29,7 +29,7 @@ def graph_dfs(graph: GraphAdjList, start_vet: Vertex) -> list[Vertex]: # 使用鄰接表來表示圖，以便獲取指定頂點的所有鄰接頂點 # 頂點走訪序列 res = [] - # 雜湊表，用於記錄已被訪問過的頂點 + # 雜湊集合，用於記錄已被訪問過的頂點 visited = set[Vertex]() dfs(graph, visited, res, start_vet) return res diff --git a/zh-hant/codes/python/chapter_stack_and_queue/array_stack.py b/zh-hant/codes/python/chapter_stack_and_queue/array_stack.py index 5e81c9907..bc4b74d69 100644 --- a/zh-hant/codes/python/chapter_stack_and_queue/array_stack.py +++ b/zh-hant/codes/python/chapter_stack_and_queue/array_stack.py @@ -18,7 +18,7 @@ class ArrayStack: def is_empty(self) -> bool: """判斷堆疊是否為空""" - return self._size == 0 + return self.size() == 0 def push(self, item: int): """入堆疊""" diff --git a/zh-hant/codes/python/chapter_tree/array_binary_tree.py b/zh-hant/codes/python/chapter_tree/array_binary_tree.py index a7d272756..d46338351 100644 --- a/zh-hant/codes/python/chapter_tree/array_binary_tree.py +++ b/zh-hant/codes/python/chapter_tree/array_binary_tree.py @@ -22,7 +22,7 @@ class ArrayBinaryTree: """串列容量""" return len(self._tree) - def val(self, i: int) -> int: + def val(self, i: int) -> int | None: """獲取索引為 i 節點的值""" # 若索引越界，則返回 None ，代表空位 if i < 0 or i >= self.size(): diff --git a/zh-hant/codes/ruby/chapter_graph/graph_adjacency_list.rb b/zh-hant/codes/ruby/chapter_graph/graph_adjacency_list.rb new file mode 100644 index 000000000..48108fe48 --- /dev/null +++ b/zh-hant/codes/ruby/chapter_graph/graph_adjacency_list.rb @@ -0,0 +1,116 @@ +=begin +File: graph_adjacency_list.rb +Created Time: 2024-04-25 +Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com) +=end + +require_relative '../utils/vertex' + +### 基於鄰接表實現的無向圖類別 ### +class GraphAdjList + attr_reader :adj_list + + ### 建構子 ### + def initialize(edges) + # 鄰接表，key：頂點，value：該頂點的所有鄰接頂點 + @adj_list = {} + # 新增所有頂點和邊 + for edge in edges + add_vertex(edge[0]) + add_vertex(edge[1]) + add_edge(edge[0], edge[1]) + end + end + + ### 獲取頂點數量 ### + def size + @adj_list.length + end + + ### 新增邊 ### + def add_edge(vet1, vet2) + raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2) + + @adj_list[vet1] << vet2 + @adj_list[vet2] << vet1 + end + + ### 刪除邊 ### + def remove_edge(vet1, vet2) + raise ArgumentError if !@adj_list.include?(vet1) || !@adj_list.include?(vet2) + + # 刪除邊 vet1 - vet2 + @adj_list[vet1].delete(vet2) + @adj_list[vet2].delete(vet1) + end + + ### 新增頂點 ### + def add_vertex(vet) + return if @adj_list.include?(vet) + + # 在鄰接表中新增一個新鏈結串列 + @adj_list[vet] = [] + end + + ### 刪除頂點 ### + def remove_vertex(vet) + raise ArgumentError unless @adj_list.include?(vet) + + # 在鄰接表中刪除頂點 vet 對應的鏈結串列 + @adj_list.delete(vet) + # 走訪其他頂點的鏈結串列，刪除所有包含 vet 的邊 + for vertex in @adj_list + @adj_list[vertex.first].delete(vet) if @adj_list[vertex.first].include?(vet) + end + end + + ### 列印鄰接表 ### + def __print__ + puts '鄰接表 =' + for vertex in @adj_list + tmp = @adj_list[vertex.first].map { |v| v.val } + puts "#{vertex.first.val}: #{tmp}," + end + end +end + +### Driver Code ### +if __FILE__ == $0 + # 初始化無向圖 + v = vals_to_vets([1, 3, 2, 5, 4]) + edges = [ + [v[0], v[1]], + [v[0], v[3]], + [v[1], v[2]], + [v[2], v[3]], + [v[2], v[4]], + [v[3], v[4]], + ] + graph = GraphAdjList.new(edges) + puts "\n初始化後，圖為" + graph.__print__ + + # 新增邊 + # 頂點 1，2 即 v[0]，v[2] + graph.add_edge(v[0], v[2]) + puts "\n新增邊 1-2 後，圖為" + graph.__print__ + + # 刪除邊 + # 頂點 1，3 即 v[0]，v[1] + graph.remove_edge(v[0], v[1]) + puts "\n刪除邊 1-3 後，圖為" + graph.__print__ + + # 新增頂點 + v5 = Vertex.new(6) + graph.add_vertex(v5) + puts "\n新增頂點 6 後，圖為" + graph.__print__ + + # 刪除頂點 + # 頂點 3 即 v[1] + graph.remove_vertex(v[1]) + puts "\n刪除頂點 3 後，圖為" + graph.__print__ +end diff --git a/zh-hant/codes/ruby/chapter_graph/graph_adjacency_matrix.rb b/zh-hant/codes/ruby/chapter_graph/graph_adjacency_matrix.rb new file mode 100644 index 000000000..a343fee21 --- /dev/null +++ b/zh-hant/codes/ruby/chapter_graph/graph_adjacency_matrix.rb @@ -0,0 +1,116 @@ +=begin +File: graph_adjacency_matrix.rb +Created Time: 2024-04-25 +Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com) +=end + +require_relative '../utils/print_util' + +### 基於鄰接矩陣實現的無向圖類別 ### +class GraphAdjMat + def initialize(vertices, edges) + ### 建構子 ### + # 頂點串列，元素代表“頂點值”，索引代表“頂點索引” + @vertices = [] + # 鄰接矩陣，行列索引對應“頂點索引” + @adj_mat = [] + # 新增頂點 + vertices.each { |val| add_vertex(val) } + # 新增邊 + # 請注意，edges 元素代表頂點索引，即對應 vertices 元素索引 + edges.each { |e| add_edge(e[0], e[1]) } + end + + ### 獲取頂點數量 ### + def size + @vertices.length + end + + ### 新增頂點 ### + def add_vertex(val) + n = size + # 向頂點串列中新增新頂點的值 + @vertices << val + # 在鄰接矩陣中新增一行 + new_row = Array.new(n, 0) + @adj_mat << new_row + # 在鄰接矩陣中新增一列 + @adj_mat.each { |row| row << 0 } + end + + ### 刪除頂點 ### + def remove_vertex(index) + raise IndexError if index >= size + + # 在頂點串列中移除索引 index 的頂點 + @vertices.delete_at(index) + # 在鄰接矩陣中刪除索引 index 的行 + @adj_mat.delete_at(index) + # 在鄰接矩陣中刪除索引 index 的列 + @adj_mat.each { |row| row.delete_at(index) } + end + + ### 新增邊 ### + def add_edge(i, j) + # 參數 i, j 對應 vertices 元素索引 + # 索引越界與相等處理 + if i < 0 || j < 0 || i >= size || j >= size || i == j + raise IndexError + end + # 在無向圖中，鄰接矩陣關於主對角線對稱，即滿足 (i, j) == (j, i) + @adj_mat[i][j] = 1 + @adj_mat[j][i] = 1 + end + + ### 刪除邊 ### + def remove_edge(i, j) + # 參數 i, j 對應 vertices 元素索引 + # 索引越界與相等處理 + if i < 0 || j < 0 || i >= size || j >= size || i == j + raise IndexError + end + @adj_mat[i][j] = 0 + @adj_mat[j][i] = 0 + end + + ### 列印鄰接矩陣 ### + def __print__ + puts "頂點串列 = #{@vertices}" + puts '鄰接矩陣 =' + print_matrix(@adj_mat) + end +end + +### Driver Code ### +if __FILE__ == $0 + # 初始化無向圖 + # 請注意，edges 元素代表頂點索引，即對應 vertices 元素索引 + vertices = [1, 3, 2, 5, 4] + edges = [[0, 1], [0, 3], [1, 2], [2, 3], [2, 4], [3, 4]] + graph = GraphAdjMat.new(vertices, edges) + puts "\n初始化後，圖為" + graph.__print__ + + # 新增邊 + # 頂點 1, 2 的索引分別為 0, 2 + graph.add_edge(0, 2) + puts "\n新增邊 1-2 後，圖為" + graph.__print__ + + # 刪除邊 + # 定點 1, 3 的索引分別為 0, 1 + graph.remove_edge(0, 1) + puts "\n刪除邊 1-3 後，圖為" + graph.__print__ + + # 新增頂點 + graph.add_vertex(6) + puts "\n新增頂點 6 後，圖為" + graph.__print__ + + # 刪除頂點 + # 頂點 3 的索引為 1 + graph.remove_vertex(1) + puts "\n刪除頂點 3 後，圖為" + graph.__print__ +end diff --git a/zh-hant/codes/ruby/chapter_graph/graph_bfs.rb b/zh-hant/codes/ruby/chapter_graph/graph_bfs.rb new file mode 100644 index 000000000..337bc7879 --- /dev/null +++ b/zh-hant/codes/ruby/chapter_graph/graph_bfs.rb @@ -0,0 +1,61 @@ +=begin +File: graph_bfs.rb +Created Time: 2024-04-25 +Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com) +=end + +require 'set' +require_relative './graph_adjacency_list' +require_relative '../utils/vertex' + +### 廣度優先走訪 ### +def graph_bfs(graph, start_vet) + # 使用鄰接表來表示圖，以便獲取指定頂點的所有鄰接頂點 + # 頂點走訪序列 + res = [] + # 雜湊集合，用於記錄已被訪問過的頂點 + visited = Set.new([start_vet]) + # 佇列用於實現 BFS + que = [start_vet] + # 以頂點 vet 為起點，迴圈直至訪問完所有頂點 + while que.length > 0 + vet = que.shift # 佇列首頂點出隊 + res << vet # 記錄訪問頂點 + # 走訪該頂點的所有鄰接頂點 + for adj_vet in graph.adj_list[vet] + next if visited.include?(adj_vet) # 跳過已被訪問的頂點 + que << adj_vet # 只入列未訪問的頂點 + visited.add(adj_vet) # 標記該頂點已被訪問 + end + end + # 返回頂點走訪序列 + res +end + +### Driver Code ### +if __FILE__ == $0 + # 初始化無向圖 + v = vals_to_vets([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]) + edges = [ + [v[0], v[1]], + [v[0], v[3]], + [v[1], v[2]], + [v[1], v[4]], + [v[2], v[5]], + [v[3], v[4]], + [v[3], v[6]], + [v[4], v[5]], + [v[4], v[7]], + [v[5], v[8]], + [v[6], v[7]], + [v[7], v[8]], + ] + graph = GraphAdjList.new(edges) + puts "\n初始化後，圖為" + graph.__print__ + + # 廣度優先走訪 + res = graph_bfs(graph, v.first) + puts "\n廣度優先便利（BFS）頂點序列為" + p vets_to_vals(res) +end diff --git a/zh-hant/codes/ruby/chapter_graph/graph_dfs.rb b/zh-hant/codes/ruby/chapter_graph/graph_dfs.rb new file mode 100644 index 000000000..77eb709aa --- /dev/null +++ b/zh-hant/codes/ruby/chapter_graph/graph_dfs.rb @@ -0,0 +1,54 @@ +=begin +File: graph_dfs.rb +Created Time: 2024-04-25 +Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com) +=end + +require 'set' +require_relative './graph_adjacency_list' +require_relative '../utils/vertex' + +### 深度優先走訪輔助函式 ### +def dfs(graph, visited, res, vet) + res << vet # 記錄訪問頂點 + visited.add(vet) # 標記該頂點已被訪問 + # 走訪該頂點的所有鄰接頂點 + for adj_vet in graph.adj_list[vet] + next if visited.include?(adj_vet) # 跳過已被訪問的頂點 + # 遞迴訪問鄰接頂點 + dfs(graph, visited, res, adj_vet) + end +end + +### 深度優先走訪 ### +def graph_dfs(graph, start_vet) + # 使用鄰接表來表示圖，以便獲取指定頂點的所有鄰接頂點 + # 頂點走訪序列 + res = [] + # 雜湊集合，用於記錄已被訪問過的頂點 + visited = Set.new + dfs(graph, visited, res, start_vet) + res +end + +### Driver Code ### +if __FILE__ == $0 + # 初始化無向圖 + v = vals_to_vets([0, 1, 2, 3, 4, 5, 6]) + edges = [ + [v[0], v[1]], + [v[0], v[3]], + [v[1], v[2]], + [v[2], v[5]], + [v[4], v[5]], + [v[5], v[6]], + ] + graph = GraphAdjList.new(edges) + puts "\n初始化後，圖為" + graph.__print__ + + # 深度優先走訪 + res = graph_dfs(graph, v[0]) + puts "\n深度優先走訪（DFS）頂點序列為" + p vets_to_vals(res) +end diff --git a/zh-hant/codes/ruby/chapter_heap/my_heap.rb b/zh-hant/codes/ruby/chapter_heap/my_heap.rb new file mode 100644 index 000000000..90f61badb --- /dev/null +++ b/zh-hant/codes/ruby/chapter_heap/my_heap.rb @@ -0,0 +1,147 @@ +=begin +File: my_heap.rb +Created Time: 2024-04-19 +Author: Blue Bean (lonnnnnnner@gmail.com) +=end + +require_relative '../utils/print_util' + +### 大頂堆積 ### +class MaxHeap + attr_reader :max_heap + + ### 建構子，根據輸入串列建堆積 ### + def initialize(nums) + # 將串列元素原封不動新增進堆積 + @max_heap = nums + # 堆積化除葉節點以外的其他所有節點 + parent(size - 1).downto(0) do |i| + sift_down(i) + end + end + + ### 獲取左子節點的索引 ### + def left(i) + 2 * i + 1 + end + + ### 獲取右子節點的索引 ### + def right(i) + 2 * i + 2 + end + + ### 獲取父節點的索引 ### + def parent(i) + (i - 1) / 2 # 向下整除 + end + + ### 交換元素 ### + def swap(i, j) + @max_heap[i], @max_heap[j] = @max_heap[j], @max_heap[i] + end + + ### 獲取堆積大小 ### + def size + @max_heap.length + end + + ### 判斷堆積是否為空 ### + def is_empty? + size == 0 + end + + ### 訪問堆積頂元素 ### + def peek + @max_heap[0] + end + + ### 元素入堆積 ### + def push(val) + # 新增節點 + @max_heap << val + # 從底至頂堆積化 + sift_up(size - 1) + end + + ### 從節點 i 開始，從底至頂堆積化 ### + def sift_up(i) + loop do + # 獲取節點 i 的父節點 + p = parent(i) + # 當“越過根節點”或“節點無須修復”時，結束堆積化 + break if p < 0 || @max_heap[i] <= @max_heap[p] + # 交換兩節點 + swap(i, p) + # 迴圈向上堆積化 + i = p + end + end + + ### 元素出堆積 ### + def pop + # 判空處理 + raise IndexError, "堆積為空" if is_empty? + # 交換根節點與最右葉節點（交換首元素與尾元素） + swap(0, size - 1) + # 刪除節點 + val = @max_heap.pop + # 從頂至底堆積化 + sift_down(0) + # 返回堆積頂元素 + val + end + + ### 從節點 i 開始，從頂至底堆積化 ### + def sift_down(i) + loop do + # 判斷節點 i, l, r 中值最大的節點，記為 ma + l, r, ma = left(i), right(i), i + ma = l if l < size && @max_heap[l] > @max_heap[ma] + ma = r if r < size && @max_heap[r] > @max_heap[ma] + + # 若節點 i 最大或索引 l, r 越界，則無須繼續堆積化，跳出 + break if ma == i + + # 交換兩節點 + swap(i, ma) + # 迴圈向下堆積化 + i = ma + end + end + + ### 列印堆積（二元樹）### + def __print__ + print_heap(@max_heap) + end +end + +### Driver Code ### +if __FILE__ == $0 + # 初始化大頂堆積 + max_heap = MaxHeap.new([9, 8, 6, 6, 7, 5, 2, 1, 4, 3, 6, 2]) + puts "\n輸入串列並建堆積後" + max_heap.__print__ + + # 獲取堆積頂元素 + peek = max_heap.peek + puts "\n堆積頂元素為 #{peek}" + + # 元素入堆積 + val = 7 + max_heap.push(val) + puts "\n元素 #{val} 入堆積後" + max_heap.__print__ + + # 堆積頂元素出堆積 + peek = max_heap.pop + puts "\n堆積頂元素 #{peek} 出堆積後" + max_heap.__print__ + + # 獲取堆積大小 + size = max_heap.size + puts "\n堆積元素數量為 #{size}" + + # 判斷堆積是否為空 + is_empty = max_heap.is_empty? + puts "\n堆積是否為空 #{is_empty}" +end diff --git a/zh-hant/codes/ruby/chapter_heap/top_k.rb b/zh-hant/codes/ruby/chapter_heap/top_k.rb new file mode 100644 index 000000000..fb22062ba --- /dev/null +++ b/zh-hant/codes/ruby/chapter_heap/top_k.rb @@ -0,0 +1,64 @@ +=begin +File: top_k.rb +Created Time: 2024-04-19 +Author: Blue Bean (lonnnnnnner@gmail.com) +=end + +require_relative "./my_heap" + +### 元素入堆積 ### +def push_min_heap(heap, val) + # 元素取反 + heap.push(-val) +end + +### 元素出堆積 ### +def pop_min_heap(heap) + # 元素取反 + -heap.pop +end + +### 訪問堆積頂元素 ### +def peek_min_heap(heap) + # 元素取反 + -heap.peek +end + +### 取出堆積中元素 ### +def get_min_heap(heap) + # 將堆積中所有元素取反 + heap.max_heap.map { |x| -x } +end + +### 基於堆積查詢陣列中最大的 k 個元素 ### +def top_k_heap(nums, k) + # 初始化小頂堆積 + # 請注意：我們將堆積中所有元素取反，從而用大頂堆積來模擬小頂堆積 + max_heap = MaxHeap.new([]) + + # 將陣列的前 k 個元素入堆積 + for i in 0...k + push_min_heap(max_heap, nums[i]) + end + + # 從第 k+1 個元素開始，保持堆積的長度為 k + for i in k...nums.length + # 若當前元素大於堆積頂元素，則將堆積頂元素出堆積、當前元素入堆積 + if nums[i] > peek_min_heap(max_heap) + pop_min_heap(max_heap) + push_min_heap(max_heap, nums[i]) + end + end + + get_min_heap(max_heap) +end + +### Driver Code ### +if __FILE__ == $0 + nums = [1, 7, 6, 3, 2] + k = 3 + + res = top_k_heap(nums, k) + puts "最大的 #{k} 個元素為" + print_heap(res) +end diff --git a/zh-hant/codes/ruby/utils/print_util.rb b/zh-hant/codes/ruby/utils/print_util.rb index 78ab6621a..1e2cb6db5 100644 --- a/zh-hant/codes/ruby/utils/print_util.rb +++ b/zh-hant/codes/ruby/utils/print_util.rb @@ -4,6 +4,15 @@ Created Time: 2024-03-18 Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com) =end +require_relative "./tree_node" + +### 列印矩陣 ### +def print_matrix(mat) + s = [] + mat.each { |arr| s << " #{arr.to_s}" } + puts "[\n#{s.join(",\n")}\n]" +end + ### 列印鏈結串列 ### def print_linked_list(head) list = [] @@ -61,3 +70,11 @@ end def print_hash_map(hmap) hmap.entries.each { |key, value| puts "#{key} -> #{value}" } end + +### 列印堆積 ### +def print_heap(heap) + puts "堆積的陣列表示：#{heap}" + puts "堆積的樹狀表示：" + root = arr_to_tree(heap) + print_tree(root) +end diff --git a/zh-hant/codes/ruby/utils/vertex.rb b/zh-hant/codes/ruby/utils/vertex.rb new file mode 100644 index 000000000..43f688a40 --- /dev/null +++ b/zh-hant/codes/ruby/utils/vertex.rb @@ -0,0 +1,24 @@ +=begin +File: vertex.rb +Created Time: 2024-04-25 +Author: Xuan Khoa Tu Nguyen (ngxktuzkai2000@gmail.com) +=end + +### 頂點類別 ### +class Vertex + attr_accessor :val + + def initialize(val) + @val = val + end +end + +### 輸入值串列 vals ，返回頂點串列 vets ### +def vals_to_vets(vals) + Array.new(vals.length) { |i| Vertex.new(vals[i]) } +end + +### 輸入頂點串列 vets, 返回值串列 vals ### +def vets_to_vals(vets) + Array.new(vets.length) { |i| vets[i].val } +end diff --git a/zh-hant/codes/rust/chapter_graph/graph_bfs.rs b/zh-hant/codes/rust/chapter_graph/graph_bfs.rs index f2ea5b099..4c0fbccfd 100644 --- a/zh-hant/codes/rust/chapter_graph/graph_bfs.rs +++ b/zh-hant/codes/rust/chapter_graph/graph_bfs.rs @@ -15,7 +15,7 @@ use std::collections::{HashSet, VecDeque}; fn graph_bfs(graph: GraphAdjList, start_vet: Vertex) -> Vec { // 頂點走訪序列 let mut res = vec![]; - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 let mut visited = HashSet::new(); visited.insert(start_vet); // 佇列用於實現 BFS diff --git a/zh-hant/codes/rust/chapter_graph/graph_dfs.rs b/zh-hant/codes/rust/chapter_graph/graph_dfs.rs index a1bfc4d1c..b6e3c4c7d 100644 --- a/zh-hant/codes/rust/chapter_graph/graph_dfs.rs +++ b/zh-hant/codes/rust/chapter_graph/graph_dfs.rs @@ -31,7 +31,7 @@ fn dfs(graph: &GraphAdjList, visited: &mut HashSet, res: &mut Vec Vec { // 頂點走訪序列 let mut res = vec![]; - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 let mut visited = HashSet::new(); dfs(&graph, &mut visited, &mut res, start_vet); diff --git a/zh-hant/codes/rust/chapter_heap/my_heap.rs b/zh-hant/codes/rust/chapter_heap/my_heap.rs index 989d2754e..7a3652a8d 100644 --- a/zh-hant/codes/rust/chapter_heap/my_heap.rs +++ b/zh-hant/codes/rust/chapter_heap/my_heap.rs @@ -96,7 +96,7 @@ impl MaxHeap { // 交換根節點與最右葉節點（交換首元素與尾元素） self.swap(0, self.size() - 1); // 刪除節點 - let val = self.max_heap.remove(self.size() - 1); + let val = self.max_heap.pop().unwrap(); // 從頂至底堆積化 self.sift_down(0); // 返回堆積頂元素 diff --git a/zh-hant/codes/swift/chapter_graph/graph_bfs.swift b/zh-hant/codes/swift/chapter_graph/graph_bfs.swift index 1358588ad..63bcc0082 100644 --- a/zh-hant/codes/swift/chapter_graph/graph_bfs.swift +++ b/zh-hant/codes/swift/chapter_graph/graph_bfs.swift @@ -12,7 +12,7 @@ import utils func graphBFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] { // 頂點走訪序列 var res: [Vertex] = [] - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 var visited: Set = [startVet] // 佇列用於實現 BFS var que: [Vertex] = [startVet] diff --git a/zh-hant/codes/swift/chapter_graph/graph_dfs.swift b/zh-hant/codes/swift/chapter_graph/graph_dfs.swift index c808d017a..308e05f6f 100644 --- a/zh-hant/codes/swift/chapter_graph/graph_dfs.swift +++ b/zh-hant/codes/swift/chapter_graph/graph_dfs.swift @@ -26,7 +26,7 @@ func dfs(graph: GraphAdjList, visited: inout Set, res: inout [Vertex], v func graphDFS(graph: GraphAdjList, startVet: Vertex) -> [Vertex] { // 頂點走訪序列 var res: [Vertex] = [] - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 var visited: Set = [] dfs(graph: graph, visited: &visited, res: &res, vet: startVet) return res diff --git a/zh-hant/codes/typescript/chapter_graph/graph_bfs.ts b/zh-hant/codes/typescript/chapter_graph/graph_bfs.ts index 8095f98c2..916b4f1c4 100644 --- a/zh-hant/codes/typescript/chapter_graph/graph_bfs.ts +++ b/zh-hant/codes/typescript/chapter_graph/graph_bfs.ts @@ -12,7 +12,7 @@ import { Vertex } from '../modules/Vertex'; function graphBFS(graph: GraphAdjList, startVet: Vertex): Vertex[] { // 頂點走訪序列 const res: Vertex[] = []; - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 const visited: Set = new Set(); visited.add(startVet); // 佇列用於實現 BFS diff --git a/zh-hant/codes/typescript/chapter_graph/graph_dfs.ts b/zh-hant/codes/typescript/chapter_graph/graph_dfs.ts index 59f546899..846e21200 100644 --- a/zh-hant/codes/typescript/chapter_graph/graph_dfs.ts +++ b/zh-hant/codes/typescript/chapter_graph/graph_dfs.ts @@ -31,7 +31,7 @@ function dfs( function graphDFS(graph: GraphAdjList, startVet: Vertex): Vertex[] { // 頂點走訪序列 const res: Vertex[] = []; - // 雜湊表，用於記錄已被訪問過的頂點 + // 雜湊集合，用於記錄已被訪問過的頂點 const visited: Set = new Set(); dfs(graph, visited, res, startVet); return res; diff --git a/zh-hant/docs/chapter_array_and_linkedlist/linked_list.md b/zh-hant/docs/chapter_array_and_linkedlist/linked_list.md index b954e73e2..263ccb02d 100755 --- a/zh-hant/docs/chapter_array_and_linkedlist/linked_list.md +++ b/zh-hant/docs/chapter_array_and_linkedlist/linked_list.md @@ -745,7 +745,7 @@ 單向鏈結串列通常用於實現堆疊、佇列、雜湊表和圖等資料結構。 -- **堆疊與佇列**：當插入和刪除操作都在鏈結串列的一端進行時，它表現出先進後出的特性，對應堆疊；當插入操作在鏈結串列的一端進行，刪除操作在鏈結串列的另一端進行，它表現出先進先出的特性，對應佇列。 +- **堆疊與佇列**：當插入和刪除操作都在鏈結串列的一端進行時，它表現的特性為先進後出，對應堆疊；當插入操作在鏈結串列的一端進行，刪除操作在鏈結串列的另一端進行，它表現的特性為先進先出，對應佇列。 - **雜湊表**：鏈式位址是解決雜湊衝突的主流方案之一，在該方案中，所有衝突的元素都會被放到一個鏈結串列中。 - **圖**：鄰接表是表示圖的一種常用方式，其中圖的每個頂點都與一個鏈結串列相關聯，鏈結串列中的每個元素都代表與該頂點相連的其他頂點。 diff --git a/zh-hant/docs/chapter_array_and_linkedlist/ram_and_cache.md b/zh-hant/docs/chapter_array_and_linkedlist/ram_and_cache.md index 65d90ddf8..8f315c5dd 100644 --- a/zh-hant/docs/chapter_array_and_linkedlist/ram_and_cache.md +++ b/zh-hant/docs/chapter_array_and_linkedlist/ram_and_cache.md @@ -25,7 +25,7 @@ ![計算機儲存系統](ram_and_cache.assets/storage_pyramid.png) -!!! note +!!! tip 計算機的儲存層次結構體現了速度、容量和成本三者之間的精妙平衡。實際上，這種權衡普遍存在於所有工業領域，它要求我們在不同的優勢和限制之間找到最佳平衡點。 diff --git a/zh-hant/docs/chapter_backtracking/backtracking_algorithm.assets/preorder_find_paths_step11.png b/zh-hant/docs/chapter_backtracking/backtracking_algorithm.assets/preorder_find_paths_step11.png index 1bcdf7dfa..195d58fe4 100644 Binary files a/zh-hant/docs/chapter_backtracking/backtracking_algorithm.assets/preorder_find_paths_step11.png and b/zh-hant/docs/chapter_backtracking/backtracking_algorithm.assets/preorder_find_paths_step11.png differ diff --git a/zh-hant/docs/chapter_backtracking/permutations_problem.md b/zh-hant/docs/chapter_backtracking/permutations_problem.md index 9539b9096..58fac34f7 100644 --- a/zh-hant/docs/chapter_backtracking/permutations_problem.md +++ b/zh-hant/docs/chapter_backtracking/permutations_problem.md @@ -59,7 +59,7 @@ ![重複排列](permutations_problem.assets/permutations_ii.png) -那麼如何去除重複的排列呢？最直接地，考慮藉助一個雜湊表，直接對排列結果進行去重。然而這樣做不夠優雅，**因為生成重複排列的搜尋分支沒有必要，應當提前識別並剪枝**，這樣可以進一步提升演算法效率。 +那麼如何去除重複的排列呢？最直接地，考慮藉助一個雜湊集合，直接對排列結果進行去重。然而這樣做不夠優雅，**因為生成重複排列的搜尋分支沒有必要，應當提前識別並剪枝**，這樣可以進一步提升演算法效率。 ### 相等元素剪枝 @@ -73,7 +73,7 @@ ### 程式碼實現 -在上一題的程式碼的基礎上，我們考慮在每一輪選擇中開啟一個雜湊表 `duplicated` ，用於記錄該輪中已經嘗試過的元素，並將重複元素剪枝： +在上一題的程式碼的基礎上，我們考慮在每一輪選擇中開啟一個雜湊集合 `duplicated` ，用於記錄該輪中已經嘗試過的元素，並將重複元素剪枝： ```src [file]{permutations_ii}-[class]{}-[func]{permutations_ii} diff --git a/zh-hant/docs/chapter_backtracking/summary.md b/zh-hant/docs/chapter_backtracking/summary.md index cee5f4427..96ab6e545 100644 --- a/zh-hant/docs/chapter_backtracking/summary.md +++ b/zh-hant/docs/chapter_backtracking/summary.md @@ -7,7 +7,7 @@ - 回溯問題通常包含多個約束條件，它們可用於實現剪枝操作。剪枝可以提前結束不必要的搜尋分支，大幅提升搜尋效率。 - 回溯演算法主要可用於解決搜尋問題和約束滿足問題。組合最佳化問題雖然可以用回溯演算法解決，但往往存在效率更高或效果更好的解法。 - 全排列問題旨在搜尋給定集合元素的所有可能的排列。我們藉助一個陣列來記錄每個元素是否被選擇，剪掉重複選擇同一元素的搜尋分支，確保每個元素只被選擇一次。 -- 在全排列問題中，如果集合中存在重複元素，則最終結果會出現重複排列。我們需要約束相等元素在每輪中只能被選擇一次，這通常藉助一個雜湊表來實現。 +- 在全排列問題中，如果集合中存在重複元素，則最終結果會出現重複排列。我們需要約束相等元素在每輪中只能被選擇一次，這通常藉助一個雜湊集合來實現。 - 子集和問題的目標是在給定集合中找到和為目標值的所有子集。集合不區分元素順序，而搜尋過程會輸出所有順序的結果，產生重複子集。我們在回溯前將資料進行排序，並設定一個變數來指示每一輪的走訪起始點，從而將生成重複子集的搜尋分支進行剪枝。 - 對於子集和問題，陣列中的相等元素會產生重複集合。我們利用陣列已排序的前置條件，透過判斷相鄰元素是否相等實現剪枝，從而確保相等元素在每輪中只能被選中一次。 - $n$ 皇后問題旨在尋找將 $n$ 個皇后放置到 $n \times n$ 尺寸棋盤上的方案，要求所有皇后兩兩之間無法攻擊對方。該問題的約束條件有行約束、列約束、主對角線和次對角線約束。為滿足行約束，我們採用按行放置的策略，保證每一行放置一個皇后。 diff --git a/zh-hant/docs/chapter_computational_complexity/time_complexity.md b/zh-hant/docs/chapter_computational_complexity/time_complexity.md index 927dcf49c..4c23d4002 100755 --- a/zh-hant/docs/chapter_computational_complexity/time_complexity.md +++ b/zh-hant/docs/chapter_computational_complexity/time_complexity.md @@ -747,7 +747,7 @@ $T(n)$ 是一次函式，說明其執行時間的增長趨勢是線性的，因 時間複雜度分析本質上是計算“操作數量 $T(n)$”的漸近上界，它具有明確的數學定義。 -!!! abstract "函式漸近上界" +!!! note "函式漸近上界" 若存在正實數 $c$ 和實數 $n_0$ ，使得對於所有的 $n > n_0$ ，均有 $T(n) \leq c \cdot f(n)$ ，則可認為 $f(n)$ 給出了 $T(n)$ 的一個漸近上界，記為 $T(n) = O(f(n))$ 。 diff --git a/zh-hant/docs/chapter_data_structure/number_encoding.md b/zh-hant/docs/chapter_data_structure/number_encoding.md index 4a050d695..ae64c83a1 100644 --- a/zh-hant/docs/chapter_data_structure/number_encoding.md +++ b/zh-hant/docs/chapter_data_structure/number_encoding.md @@ -1,6 +1,6 @@ # 數字編碼 * -!!! note +!!! tip 在本書中，標題帶有 * 符號的是選讀章節。如果你時間有限或感到理解困難，可以先跳過，等學完必讀章節後再單獨攻克。 diff --git a/zh-hant/docs/chapter_graph/graph_traversal.assets/graph_bfs_step1.png b/zh-hant/docs/chapter_graph/graph_traversal.assets/graph_bfs_step1.png index 8b70e7ac5..35fbc71a1 100644 Binary files a/zh-hant/docs/chapter_graph/graph_traversal.assets/graph_bfs_step1.png and b/zh-hant/docs/chapter_graph/graph_traversal.assets/graph_bfs_step1.png differ diff --git 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b/zh-hant/docs/chapter_graph/graph_traversal.assets/graph_dfs_step8.png index 42b3fcb36..e3eb4ceac 100644 Binary files a/zh-hant/docs/chapter_graph/graph_traversal.assets/graph_dfs_step8.png and b/zh-hant/docs/chapter_graph/graph_traversal.assets/graph_dfs_step8.png differ diff --git a/zh-hant/docs/chapter_graph/graph_traversal.assets/graph_dfs_step9.png b/zh-hant/docs/chapter_graph/graph_traversal.assets/graph_dfs_step9.png index afbd16f39..7f5d4c488 100644 Binary files a/zh-hant/docs/chapter_graph/graph_traversal.assets/graph_dfs_step9.png and b/zh-hant/docs/chapter_graph/graph_traversal.assets/graph_dfs_step9.png differ diff --git a/zh-hant/docs/chapter_graph/graph_traversal.md b/zh-hant/docs/chapter_graph/graph_traversal.md index 3a55288a7..f611b734c 100644 --- a/zh-hant/docs/chapter_graph/graph_traversal.md +++ b/zh-hant/docs/chapter_graph/graph_traversal.md @@ -18,7 +18,11 @@ BFS 通常藉助佇列來實現，程式碼如下所示。佇列具有“先入 2. 在迴圈的每輪迭代中，彈出佇列首頂點並記錄訪問，然後將該頂點的所有鄰接頂點加入到佇列尾部。 3. 迴圈步驟 `2.` ，直到所有頂點被訪問完畢後結束。 -為了防止重複走訪頂點，我們需要藉助一個雜湊表 `visited` 來記錄哪些節點已被訪問。 +為了防止重複走訪頂點，我們需要藉助一個雜湊集合 `visited` 來記錄哪些節點已被訪問。 + +!!! tip + + 雜湊集合可以看作一個只儲存 `key` 而不儲存 `value` 的雜湊表，它可以在 $O(1)$ 時間複雜度下進行 `key` 的增刪查改操作。根據 `key` 的唯一性，雜湊集合通常用於資料去重等場景。 ```src [file]{graph_bfs}-[class]{}-[func]{graph_bfs} @@ -67,7 +71,7 @@ BFS 通常藉助佇列來實現，程式碼如下所示。佇列具有“先入 **時間複雜度**：所有頂點都會入列並出隊一次，使用 $O(|V|)$ 時間；在走訪鄰接頂點的過程中，由於是無向圖，因此所有邊都會被訪問 $2$ 次，使用 $O(2|E|)$ 時間；總體使用 $O(|V| + |E|)$ 時間。 -**空間複雜度**：串列 `res` ，雜湊表 `visited` ，佇列 `que` 中的頂點數量最多為 $|V|$ ，使用 $O(|V|)$ 空間。 +**空間複雜度**：串列 `res` ，雜湊集合 `visited` ，佇列 `que` 中的頂點數量最多為 $|V|$ ，使用 $O(|V|)$ 空間。 ## 深度優先走訪 @@ -77,7 +81,7 @@ BFS 通常藉助佇列來實現，程式碼如下所示。佇列具有“先入 ### 演算法實現 -這種“走到盡頭再返回”的演算法範式通常基於遞迴來實現。與廣度優先走訪類似，在深度優先走訪中，我們也需要藉助一個雜湊表 `visited` 來記錄已被訪問的頂點，以避免重複訪問頂點。 +這種“走到盡頭再返回”的演算法範式通常基於遞迴來實現。與廣度優先走訪類似，在深度優先走訪中，我們也需要藉助一個雜湊集合 `visited` 來記錄已被訪問的頂點，以避免重複訪問頂點。 ```src [file]{graph_dfs}-[class]{}-[func]{graph_dfs} @@ -133,4 +137,4 @@ BFS 通常藉助佇列來實現，程式碼如下所示。佇列具有“先入 **時間複雜度**：所有頂點都會被訪問 $1$ 次，使用 $O(|V|)$ 時間；所有邊都會被訪問 $2$ 次，使用 $O(2|E|)$ 時間；總體使用 $O(|V| + |E|)$ 時間。 -**空間複雜度**：串列 `res` ，雜湊表 `visited` 頂點數量最多為 $|V|$ ，遞迴深度最大為 $|V|$ ，因此使用 $O(|V|)$ 空間。 +**空間複雜度**：串列 `res` ，雜湊集合 `visited` 頂點數量最多為 $|V|$ ，遞迴深度最大為 $|V|$ ，因此使用 $O(|V|)$ 空間。 diff --git a/zh-hant/docs/chapter_introduction/summary.md b/zh-hant/docs/chapter_introduction/summary.md index d33fe6eb1..501abf9e3 100644 --- a/zh-hant/docs/chapter_introduction/summary.md +++ b/zh-hant/docs/chapter_introduction/summary.md @@ -7,3 +7,16 @@ - 演算法是在有限時間內解決特定問題的一組指令或操作步驟，而資料結構是計算機中組織和儲存資料的方式。 - 資料結構與演算法緊密相連。資料結構是演算法的基石，而演算法是資料結構發揮作用的舞臺。 - 我們可以將資料結構與演算法類比為拼裝積木，積木代表資料，積木的形狀和連線方式等代表資料結構，拼裝積木的步驟則對應演算法。 + +### Q & A + +**Q**：作為一名程式設計師，我在日常工作中從未用演算法解決過問題，常用演算法都被程式語言封裝好了，直接用就可以了；這是否意味著我們工作中的問題還沒有到達需要演算法的程度？ + +如果把具體的工作技能比作是武功的“招式”的話，那麼基礎科目應該更像是“內功”。 + +我認為學演算法（以及其他基礎科目）的意義不是在於在工作中從零實現它，而是基於學到的知識，在解決問題時能夠作出專業的反應和判斷，從而提升工作的整體質量。舉一個簡單例子，每種程式語言都內建了排序函式： + +- 如果我們沒有學過資料結構與演算法，那麼給定任何資料，我們可能都塞給這個排序函式去做了。執行順暢、效能不錯，看上去並沒有什麼問題。 +- 但如果學過演算法，我們就會知道內建排序函式的時間複雜度是 $O(n \log n)$ ；而如果給定的資料是固定位數的整數（例如學號），那麼我們就可以用效率更高的“基數排序”來做，將時間複雜度降為 $O(nk)$ ，其中 $k$ 為位數。當資料體量很大時，節省出來的執行時間就能創造較大價值（成本降低、體驗變好等）。 + +在工程領域中，大量問題是難以達到最優解的，許多問題只是被“差不多”地解決了。問題的難易程度一方面取決於問題本身的性質，另一方面也取決於觀測問題的人的知識儲備。人的知識越完備、經驗越多，分析問題就會越深入，問題就能被解決得更優雅。 diff --git a/zh-hant/docs/chapter_stack_and_queue/deque.md b/zh-hant/docs/chapter_stack_and_queue/deque.md index d54e02f91..f82e04544 100644 --- a/zh-hant/docs/chapter_stack_and_queue/deque.md +++ b/zh-hant/docs/chapter_stack_and_queue/deque.md @@ -27,28 +27,28 @@ from collections import deque # 初始化雙向佇列 - deque: deque[int] = deque() + deq: deque[int] = deque() # 元素入列 - deque.append(2) # 新增至佇列尾 - deque.append(5) - deque.append(4) - deque.appendleft(3) # 新增至佇列首 - deque.appendleft(1) + deq.append(2) # 新增至佇列尾 + deq.append(5) + deq.append(4) + deq.appendleft(3) # 新增至佇列首 + deq.appendleft(1) # 訪問元素 - front: int = deque[0] # 佇列首元素 - rear: int = deque[-1] # 佇列尾元素 + front: int = deq[0] # 佇列首元素 + rear: int = deq[-1] # 佇列尾元素 # 元素出列 - pop_front: int = deque.popleft() # 佇列首元素出列 - pop_rear: int = deque.pop() # 佇列尾元素出列 + pop_front: int = deq.popleft() # 佇列首元素出列 + pop_rear: int = deq.pop() # 佇列尾元素出列 # 獲取雙向佇列的長度 - size: int = len(deque) + size: int = len(deq) # 判斷雙向佇列是否為空 - is_empty: bool = len(deque) == 0 + is_empty: bool = len(deq) == 0 ``` === "C++" diff --git a/zh-hant/docs/chapter_tree/avl_tree.md b/zh-hant/docs/chapter_tree/avl_tree.md index 45d3b07e2..38da5155b 100644 --- a/zh-hant/docs/chapter_tree/avl_tree.md +++ b/zh-hant/docs/chapter_tree/avl_tree.md @@ -248,7 +248,7 @@ AVL 樹既是二元搜尋樹，也是平衡二元樹，同時滿足這兩類二 [file]{avl_tree}-[class]{avl_tree}-[func]{balance_factor} ``` -!!! note +!!! tip 設平衡因子為 $f$ ，則一棵 AVL 樹的任意節點的平衡因子皆滿足 $-1 \le f \le 1$ 。 diff --git a/zh-hant/docs/chapter_tree/binary_tree.md b/zh-hant/docs/chapter_tree/binary_tree.md index e70a0440a..0e0e0de27 100644 --- a/zh-hant/docs/chapter_tree/binary_tree.md +++ b/zh-hant/docs/chapter_tree/binary_tree.md @@ -633,7 +633,7 @@ https://pythontutor.com/render.html#code=class%20TreeNode%3A%0A%20%20%20%20%22%22%22%E4%BA%8C%E5%85%83%E6%A8%B9%E7%AF%80%E9%BB%9E%E9%A1%9E%E5%88%A5%22%22%22%0A%20%20%20%20def%20__init__%28self%2C%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%20%23%20%E7%AF%80%E9%BB%9E%E5%80%BC%0A%20%20%20%20%20%20%20%20self.left%3A%20TreeNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%B7%A6%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%20%20%20%20%20%20%20%20self.right%3A%20TreeNode%20%7C%20None%20%3D%20None%20%23%20%E5%8F%B3%E5%AD%90%E7%AF%80%E9%BB%9E%E5%BC%95%E7%94%A8%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E4%BA%8C%E5%85%83%E6%A8%B9%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E7%AF%80%E9%BB%9E%0A%20%20%20%20n1%20%3D%20TreeNode%28val%3D1%29%0A%20%20%20%20n2%20%3D%20TreeNode%28val%3D2%29%0A%20%20%20%20n3%20%3D%20TreeNode%28val%3D3%29%0A%20%20%20%20n4%20%3D%20TreeNode%28val%3D4%29%0A%20%20%20%20n5%20%3D%20TreeNode%28val%3D5%29%0A%20%20%20%20%23%20%E6%A7%8B%E5%BB%BA%E7%AF%80%E9%BB%9E%E4%B9%8B%E9%96%93%E7%9A%84%E5%BC%95%E7%94%A8%EF%BC%88%E6%8C%87%E6%A8%99%EF%BC%89%0A%20%20%20%20n1.left%20%3D%20n2%0A%20%20%20%20n1.right%20%3D%20n3%0A%20%20%20%20n2.left%20%3D%20n4%0A%20%20%20%20n2.right%20%3D%20n5%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E8%88%87%E5%88%AA%E9%99%A4%E7%AF%80%E9%BB%9E%0A%20%20%20%20p%20%3D%20TreeNode%280%29%0A%20%20%20%20%23%20%E5%9C%A8%20n1%20-%3E%20n2%20%E4%B8%AD%E9%96%93%E6%8F%92%E5%85%A5%E7%AF%80%E9%BB%9E%20P%0A%20%20%20%20n1.left%20%3D%20p%0A%20%20%20%20p.left%20%3D%20n2%0A%20%20%20%20%23%20%E5%88%AA%E9%99%A4%E7%AF%80%E9%BB%9E%20P%0A%20%20%20%20n1.left%20%3D%20n2&cumulative=false&curInstr=37&heapPrimitives=nevernest&mode=display&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false -!!! note +!!! tip 需要注意的是，插入節點可能會改變二元樹的原有邏輯結構，而刪除節點通常意味著刪除該節點及其所有子樹。因此，在二元樹中，插入與刪除通常是由一套操作配合完成的，以實現有實際意義的操作。