From 1f4dba4845249aee9367d6efd62ba71cb3e7d1d3 Mon Sep 17 00:00:00 2001
From: Yudong Jin <krahets@163.com>
Date: Thu, 23 Feb 2023 20:00:40 +0800
Subject: [PATCH] Add the Python codes for the chapter of Graph and Heap (#382)

---
 codes/java/include/Vertex.java                |   4 +-
 .../chapter_graph/graph_adjacency_list.py     | 104 +++++++++++++
 .../chapter_graph/graph_adjacency_matrix.py   | 116 ++++++++++++++
 codes/python/chapter_graph/graph_bfs.py       |  48 ++++++
 codes/python/chapter_graph/graph_dfs.py       |  48 ++++++
 codes/python/chapter_heap/heap.py             |  62 ++++++++
 codes/python/chapter_heap/my_heap.py          | 144 ++++++++++++++++++
 codes/python/include/__init__.py              |   4 +-
 codes/python/include/print_util.py            |  10 +-
 codes/python/include/vertex.py                |  18 +++
 10 files changed, 553 insertions(+), 5 deletions(-)
 create mode 100644 codes/python/chapter_graph/graph_adjacency_list.py
 create mode 100644 codes/python/chapter_graph/graph_adjacency_matrix.py
 create mode 100644 codes/python/chapter_graph/graph_bfs.py
 create mode 100644 codes/python/chapter_graph/graph_dfs.py
 create mode 100644 codes/python/chapter_heap/heap.py
 create mode 100644 codes/python/chapter_heap/my_heap.py
 create mode 100644 codes/python/include/vertex.py

diff --git a/codes/java/include/Vertex.java b/codes/java/include/Vertex.java
index 80c59cfd8..ade707d82 100644
--- a/codes/java/include/Vertex.java
+++ b/codes/java/include/Vertex.java
@@ -1,6 +1,6 @@
 /**
- * File: TreeNode.java
- * Created Time: 2022-11-25
+ * File: Vertex.java
+ * Created Time: 2023-02-15
  * Author: Krahets (krahets@163.com)
  */
 
diff --git a/codes/python/chapter_graph/graph_adjacency_list.py b/codes/python/chapter_graph/graph_adjacency_list.py
new file mode 100644
index 000000000..eed5f98d8
--- /dev/null
+++ b/codes/python/chapter_graph/graph_adjacency_list.py
@@ -0,0 +1,104 @@
+"""
+File: graph_adjacency_list.py
+Created Time: 2023-02-23
+Author: Krahets (krahets@163.com)
+"""
+
+import sys, os.path as osp
+sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
+from include import *
+
+
+""" 基于邻接表实现的无向图类  """                                                               
+class GraphAdjList:
+    # 邻接表，key: 顶点，value：该顶点的所有邻接结点
+    adj_list = {}
+
+    """ 构造方法 """
+    def __init__(self, edges: List[List[Vertex]]) -> None:
+        self.adj_list = {}
+        # 添加所有顶点和边
+        for edge in edges:
+            self.add_vertex(edge[0])
+            self.add_vertex(edge[1])
+            self.add_edge(edge[0], edge[1])
+
+    """ 获取顶点数量 """
+    def size(self) -> int:
+        return len(self.adj_list)
+
+    """ 添加边 """
+    def add_edge(self, vet1: Vertex, vet2: Vertex) -> None:
+        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:
+            raise ValueError
+        # 添加边 vet1 - vet2
+        self.adj_list[vet1].append(vet2)
+        self.adj_list[vet2].append(vet1)
+
+    """ 删除边 """
+    def remove_edge(self, vet1: Vertex, vet2: Vertex) -> None:
+        if vet1 not in self.adj_list or vet2 not in self.adj_list or vet1 == vet2:
+            raise ValueError
+        # 删除边 vet1 - vet2
+        self.adj_list[vet1].remove(vet2)
+        self.adj_list[vet2].remove(vet1)
+
+    """ 添加顶点 """
+    def add_vertex(self, vet: Vertex) -> None:
+        if vet in self.adj_list:
+            return
+        # 在邻接表中添加一个新链表
+        self.adj_list[vet] = []
+
+    """ 删除顶点 """
+    def remove_vertex(self, vet: Vertex) -> None:
+        if vet not in self.adj_list:
+            raise ValueError
+        # 在邻接表中删除顶点 vet 对应的链表
+        self.adj_list.pop(vet)
+        # 遍历其它顶点的链表，删除所有包含 vet 的边
+        for vertex in self.adj_list:
+            if vet in self.adj_list[vertex]:
+                self.adj_list[vertex].remove(vet)
+
+    """ 打印邻接表 """
+    def print(self) -> None:
+        print("邻接表 =")
+        for vertex in self.adj_list:
+            tmp = [v.val for v in self.adj_list[vertex]]
+            print(f"{vertex.val}: {tmp},")
+
+
+""" Driver Code """
+if __name__ == "__main__":
+    """ 初始化无向图 """
+    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(edges)
+    print("\n初始化后，图为")
+    graph.print()
+
+    """ 添加边 """
+    # 顶点 1, 2 即 v[0], v[2]
+    graph.add_edge(v[0], v[2])
+    print("\n添加边 1-2 后，图为")
+    graph.print()
+
+    """ 删除边 """
+    # 顶点 1, 3 即 v[0], v[1]
+    graph.remove_edge(v[0], v[1])
+    print("\n删除边 1-3 后，图为")
+    graph.print()
+
+    """ 添加顶点 """
+    v5 = Vertex(6)
+    graph.add_vertex(v5)
+    print("\n添加顶点 6 后，图为")
+    graph.print()
+
+    """ 删除顶点 """
+    # 顶点 3 即 v[1]
+    graph.remove_vertex(v[1])
+    print("\n删除顶点 3 后，图为")
+    graph.print()
diff --git a/codes/python/chapter_graph/graph_adjacency_matrix.py b/codes/python/chapter_graph/graph_adjacency_matrix.py
new file mode 100644
index 000000000..00af6455b
--- /dev/null
+++ b/codes/python/chapter_graph/graph_adjacency_matrix.py
@@ -0,0 +1,116 @@
+"""
+File: graph_adjacency_matrix.py
+Created Time: 2023-02-23
+Author: Krahets (krahets@163.com)
+"""
+
+import sys, os.path as osp
+sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
+from include import *
+
+""" 基于邻接矩阵实现的无向图类 """
+class GraphAdjMat:
+    # 顶点列表，元素代表“顶点值”，索引代表“顶点索引”
+    vertices = []
+    # 邻接矩阵，行列索引对应“顶点索引”
+    adj_mat = []
+
+    """ 构造方法 """
+    def __init__(self, vertices, edges):
+        self.vertices = []
+        self.adj_mat = []
+        # 添加顶点
+        for val in vertices:
+            self.add_vertex(val)
+        # 添加边
+        # 请注意，edges 元素代表顶点索引，即对应 vertices 元素索引
+        for e in edges:
+            self.add_edge(e[0], e[1])
+
+    """ 获取顶点数量 """
+    def size(self):
+        return len(self.vertices)
+
+    """ 添加顶点 """
+    def add_vertex(self, val):
+        n = self.size()
+        # 向顶点列表中添加新顶点的值
+        self.vertices.append(val)
+        # 在邻接矩阵中添加一行
+        new_row = [0]*n
+        self.adj_mat.append(new_row)
+        # 在邻接矩阵中添加一列
+        for row in self.adj_mat:
+            row.append(0)
+
+
+    """ 删除顶点 """
+    def remove_vertex(self, index):
+        if index >= self.size():
+            raise IndexError()
+        # 在顶点列表中移除索引 index 的顶点
+        self.vertices.pop(index)
+        # 在邻接矩阵中删除索引 index 的行
+        self.adj_mat.pop(index)
+        # 在邻接矩阵中删除索引 index 的列
+        for row in self.adj_mat:
+            row.pop(index)
+
+    """ 添加边 """
+    # 参数 i, j 对应 vertices 元素索引
+    def add_edge(self, i, j):
+        # 索引越界与相等处理
+        if i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:
+            raise IndexError()
+        # 在无向图中，邻接矩阵沿主对角线对称，即满足 (i, j) == (j, i)
+        self.adj_mat[i][j] = 1
+        self.adj_mat[j][i] = 1
+
+    """ 删除边 """
+    # 参数 i, j 对应 vertices 元素索引
+    def remove_edge(self, i, j):
+        # 索引越界与相等处理
+        if i < 0 or j < 0 or i >= self.size() or j >= self.size() or i == j:
+            raise IndexError()
+        self.adj_mat[i][j] = 0
+        self.adj_mat[j][i] = 0
+
+    # 打印邻接矩阵
+    def print(self):
+        print("顶点列表 =", self.vertices)
+        print("邻接矩阵 =")
+        print_matrix(self.adj_mat)
+
+
+""" Driver Code """
+if __name__ == "__main__":
+    """ 初始化无向图 """
+    # 请注意，edges 元素代表顶点索引，即对应 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(vertices, edges)
+    print("\n初始化后，图为")
+    graph.print()
+
+    """ 添加边 """
+    # 顶点 1, 2 的索引分别为 0, 2
+    graph.add_edge(0, 2)
+    print("\n添加边 1-2 后，图为")
+    graph.print()
+
+    """ 删除边 """
+    # 顶点 1, 3 的索引分别为 0, 1
+    graph.remove_edge(0, 1)
+    print("\n删除边 1-3 后，图为")
+    graph.print()
+
+    """ 添加顶点 """
+    graph.add_vertex(6)
+    print("\n添加顶点 6 后，图为")
+    graph.print()
+
+    """ 删除顶点 """
+    # 顶点 3 的索引为 1
+    graph.remove_vertex(1)
+    print("\n删除顶点 3 后，图为")
+    graph.print()
diff --git a/codes/python/chapter_graph/graph_bfs.py b/codes/python/chapter_graph/graph_bfs.py
new file mode 100644
index 000000000..cb4d6c67c
--- /dev/null
+++ b/codes/python/chapter_graph/graph_bfs.py
@@ -0,0 +1,48 @@
+"""
+File: graph_bfs.py
+Created Time: 2023-02-23
+Author: Krahets (krahets@163.com)
+"""
+
+import sys, os.path as osp
+sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
+from include import *
+from graph_adjacency_list import GraphAdjList
+
+""" 广度优先遍历 BFS """
+# 使用邻接表来表示图，以便获取指定顶点的所有邻接顶点
+def graph_bfs(graph: GraphAdjList, start_vet: Vertex) -> List[Vertex]:
+    # 顶点遍历序列
+    res = []
+    # 哈希表，用于记录已被访问过的顶点
+    visited = set([start_vet])
+    # 队列用于实现 BFS
+    que = collections.deque([start_vet])
+    # 以顶点 vet 为起点，循环直至访问完所有顶点
+    while len(que) > 0:
+        vet = que.popleft()  # 队首顶点出队
+        res.append(vet)      # 记录访问顶点
+        # 遍历该顶点的所有邻接顶点
+        for adj_vet in graph.adj_list[vet]:
+            if adj_vet in visited:
+                continue         # 跳过已被访问过的顶点
+            que.append(adj_vet)  # 只入队未访问的顶点
+            visited.add(adj_vet) # 标记该顶点已被访问
+    # 返回顶点遍历序列
+    return res
+
+
+if __name__ == "__main__":
+    """初始化无向图"""
+    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(edges)
+    print("\n初始化后，图为")
+    graph.print()
+
+    """广度优先遍历 BFS"""
+    res = graph_bfs(graph, v[0])
+    print("\n广度优先遍历（BFS）顶点序列为")
+    print(vets_to_vals(res))
\ No newline at end of file
diff --git a/codes/python/chapter_graph/graph_dfs.py b/codes/python/chapter_graph/graph_dfs.py
new file mode 100644
index 000000000..f4a81ed56
--- /dev/null
+++ b/codes/python/chapter_graph/graph_dfs.py
@@ -0,0 +1,48 @@
+"""
+File: graph_dfs.py
+Created Time: 2023-02-23
+Author: Krahets (krahets@163.com)
+"""
+
+import sys, os.path as osp
+sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
+from include import *
+from graph_adjacency_list import GraphAdjList
+
+
+""" 深度优先遍历 DFS 辅助函数 """
+def dfs(graph: GraphAdjList, visited: Set[Vertex], res: List[Vertex], vet: Vertex):
+    res.append(vet)     # 记录访问顶点
+    visited.add(vet) # 标记该顶点已被访问
+    # 遍历该顶点的所有邻接顶点
+    for adjVet in graph.adj_list[vet]:
+        if adjVet in visited:
+            continue # 跳过已被访问过的顶点
+        # 递归访问邻接顶点
+        dfs(graph, visited, res, adjVet)
+
+""" 深度优先遍历 DFS """
+# 使用邻接表来表示图，以便获取指定顶点的所有邻接顶点
+def graph_dfs(graph: GraphAdjList, start_vet: Vertex) -> List[Vertex]:
+    # 顶点遍历序列
+    res = []
+    # 哈希表，用于记录已被访问过的顶点
+    visited = set()
+    dfs(graph, visited, res, start_vet)
+    return res
+
+
+""" Driver Code """
+if __name__ == "__main__":
+    # 初始化无向图
+    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(edges)
+    print("\n初始化后，图为")
+    graph.print()
+    
+    # 深度优先遍历 BFS
+    res = graph_dfs(graph, v[0])
+    print("\n深度优先遍历（DFS）顶点序列为")
+    print(vets_to_vals(res))
diff --git a/codes/python/chapter_heap/heap.py b/codes/python/chapter_heap/heap.py
new file mode 100644
index 000000000..e6bc0c7f1
--- /dev/null
+++ b/codes/python/chapter_heap/heap.py
@@ -0,0 +1,62 @@
+"""
+File: heap.py
+Created Time: 2023-02-23
+Author: Krahets (krahets@163.com)
+"""
+
+import sys, os.path as osp
+sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
+from include import *
+
+def test_push(heap, val, flag=1):
+    heapq.heappush(heap, flag * val)  # 元素入堆
+    print(f"\n元素 {val} 入堆后")
+    print_heap([flag * val for val in heap])
+
+def test_pop(heap, flag=1):
+    val = flag * heapq.heappop(heap)  # 堆顶元素出堆
+    print(f"\n堆顶元素 {val} 出堆后")
+    print_heap([flag * val for val in heap])
+
+if __name__ == "__main__":
+    # 初始化小顶堆
+    min_heap, flag = [], 1
+    # 初始化大顶堆
+    max_heap, flag = [], -1
+
+    print("\n以下测试样例为大顶堆")
+    # Python 的 heapq 模块默认实现小顶堆
+    # 考虑将“元素取负”后再入堆，这样就可以将大小关系颠倒，从而实现大顶堆
+    # 在本示例中，flag = 1 时对应小顶堆，flag = -1 时对应大顶堆
+    """ 元素入堆 """
+    test_push(max_heap, 1, flag)
+    test_push(max_heap, 3, flag)
+    test_push(max_heap, 2, flag)
+    test_push(max_heap, 5, flag)
+    test_push(max_heap, 4, flag)
+
+    """ 获取堆顶元素 """
+    peek = flag * max_heap[0]
+    print(f"\n堆顶元素为 {peek}")
+
+    """ 堆顶元素出堆 """
+    test_pop(max_heap, flag)
+    test_pop(max_heap, flag)
+    test_pop(max_heap, flag)
+    test_pop(max_heap, flag)
+    test_pop(max_heap, flag)
+
+    """ 获取堆大小 """
+    size = len(max_heap)
+    print(f"\n堆元素数量为 {size}")
+
+    """ 判断堆是否为空 """
+    is_empty = not max_heap
+    print(f"\n堆是否为空 {is_empty}")
+
+    """ 输入列表并建堆 """
+    # 时间复杂度为 O(n) ，而非 O(nlogn)
+    min_heap = [1, 3, 2, 5, 4]
+    heapq.heapify(min_heap)
+    print("\n输入列表并建立小顶堆后")
+    print_heap(min_heap)
diff --git a/codes/python/chapter_heap/my_heap.py b/codes/python/chapter_heap/my_heap.py
new file mode 100644
index 000000000..f7adc508d
--- /dev/null
+++ b/codes/python/chapter_heap/my_heap.py
@@ -0,0 +1,144 @@
+"""
+File: my_heap.py
+Created Time: 2023-02-23
+Author: Krahets (krahets@163.com)
+"""
+
+import sys, os.path as osp
+sys.path.append(osp.dirname(osp.dirname(osp.abspath(__file__))))
+from include import *
+
+# 大顶堆
+class MaxHeap:
+    # 使用列表而非数组，这样无需考虑扩容问题
+    def __init__(self, nums: List[int]):
+        # 将列表元素原封不动添加进堆
+        self.max_heap = nums
+        # 堆化除叶结点以外的其他所有结点
+        for i in range(self.parent(self.size() - 1), -1, -1):
+            self.sift_down(i)
+
+    # 获取左子结点索引
+    def left(self, i: int) -> int:
+        return 2 * i + 1
+
+    # 获取右子结点索引
+    def right(self, i: int) -> int:
+        return 2 * i + 2
+
+    # 获取父结点索引
+    def parent(self, i: int) -> int:
+        return (i - 1) // 2  # 向下整除
+
+    # 交换元素
+    def swap(self, i: int, j: int):
+        a, b = self.max_heap[i], self.max_heap[j]
+        self.max_heap[i], self.max_heap[j] = b, a
+
+    # 获取堆大小
+    def size(self) -> int:
+        return len(self.max_heap)
+
+    # 判断堆是否为空
+    def is_empty(self) -> bool:
+        return self.size() == 0
+
+    # 访问堆顶元素
+    def peek(self) -> int:
+        return self.max_heap[0]
+
+    # 元素入堆
+    def push(self, val: int):
+        # 添加结点
+        self.max_heap.append(val)
+        # 从底至顶堆化
+        self.sift_up(self.size() - 1)
+
+    # 从结点 i 开始，从底至顶堆化
+    def sift_up(self, i: int):
+        while True:
+            # 获取结点 i 的父结点
+            p = self.parent(i)
+            # 当“越过根结点”或“结点无需修复”时，结束堆化
+            if p < 0 or self.max_heap[i] <= self.max_heap[p]:
+                break
+            # 交换两结点
+            self.swap(i, p)
+            # 循环向上堆化
+            i = p
+
+    # 元素出堆
+    def poll(self) -> int:
+        # 判空处理
+        assert not self.is_empty()
+        # 交换根结点与最右叶结点（即交换首元素与尾元素）
+        self.swap(0, self.size() - 1)
+        # 删除结点
+        val = self.max_heap.pop()
+        # 从顶至底堆化
+        self.sift_down(0)
+        # 返回堆顶元素
+        return val
+
+    # 从结点 i 开始，从顶至底堆化
+    def sift_down(self, i: int):
+        while True:
+            # 判断结点 i, l, r 中值最大的结点，记为 ma
+            l, r, ma = self.left(i), self.right(i), i
+            if l < self.size() and self.max_heap[l] > self.max_heap[ma]:
+                ma = l
+            if r < self.size() and self.max_heap[r] > self.max_heap[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
+            # 交换两结点
+            self.swap(i, ma)
+            # 循环向下堆化
+            i = ma
+
+    # 打印堆（二叉树）
+    def print(self):
+        print_heap(self.max_heap)
+
+
+def test_push(max_heap: MaxHeap, val: int):
+    max_heap.push(val)  # 元素入堆
+    print(f"\n添加元素 {val} 后\n")
+    max_heap.print()
+
+
+def test_poll(max_heap: MaxHeap):
+    val = max_heap.poll()  # 堆顶元素出堆
+    print(f"\n出堆元素为 {val}\n")
+    max_heap.print()
+
+
+if __name__ == "__main__":
+    # 初始化大顶堆
+    max_heap = MaxHeap([9, 8, 6, 6, 7, 5, 2, 1, 4, 3, 6, 2])
+    print("\n输入列表并建堆后")
+    max_heap.print()
+
+    # 获取堆顶元素
+    peek = max_heap.peek()
+    print(f"\n堆顶元素为 {peek}")
+
+    # 元素入堆
+    val = 7
+    max_heap.push(val)
+    print(f"\n元素 {val} 入堆后")
+    max_heap.print()
+
+    # 堆顶元素出堆
+    peek = max_heap.poll()
+    print(f"\n堆顶元素 {peek} 出堆后")
+    max_heap.print()
+
+    # 获取堆大小
+    size = max_heap.size()
+    print(f"\n堆元素数量为 {size}")
+
+    # 判断堆是否为空
+    is_empty = max_heap.is_empty()
+    print(f"\n堆是否为空 {is_empty}")
diff --git a/codes/python/include/__init__.py b/codes/python/include/__init__.py
index 9d4e90bee..0c899941e 100644
--- a/codes/python/include/__init__.py
+++ b/codes/python/include/__init__.py
@@ -1,5 +1,6 @@
 import copy
 import math
+import heapq
 import queue
 import random
 import functools
@@ -7,4 +8,5 @@ import collections
 from typing import Optional, List, Dict, DefaultDict, OrderedDict, Set, Deque
 from .linked_list import ListNode, list_to_linked_list, linked_list_to_list, get_list_node
 from .binary_tree import TreeNode, list_to_tree, tree_to_list, get_tree_node
-from .print_util import print_matrix, print_linked_list, print_tree, print_dict
\ No newline at end of file
+from .vertex import Vertex, vals_to_vets, vets_to_vals
+from .print_util import print_matrix, print_linked_list, print_tree, print_dict, print_heap
\ No newline at end of file
diff --git a/codes/python/include/print_util.py b/codes/python/include/print_util.py
index 6aa5f7a4e..c9b5504f3 100644
--- a/codes/python/include/print_util.py
+++ b/codes/python/include/print_util.py
@@ -6,7 +6,7 @@ Author: Krahets (krahets@163.com), msk397 (machangxinq@gmail.com)
 
 import copy
 import queue
-from .binary_tree import TreeNode, tree_to_list
+from .binary_tree import TreeNode, tree_to_list, list_to_tree
 from .linked_list import ListNode, linked_list_to_list
 
 def print_matrix(mat):
@@ -80,4 +80,10 @@ def print_dict(d):
         d ([type]): [description]
     """
     for key, value in d.items():
-        print(key, '->', value)
\ No newline at end of file
+        print(key, '->', value)
+
+def print_heap(heap):
+    print("堆的数组表示：", heap);
+    print("堆的树状表示：");
+    root = list_to_tree(heap)
+    print_tree(root);
diff --git a/codes/python/include/vertex.py b/codes/python/include/vertex.py
new file mode 100644
index 000000000..63d08dc35
--- /dev/null
+++ b/codes/python/include/vertex.py
@@ -0,0 +1,18 @@
+# File: vertex.py
+# Created Time: 2023-02-23
+# Author: Krahets (krahets@163.com)
+
+from typing import List
+
+# 顶点类
+class Vertex:
+    def __init__(self, val: int) -> None:
+        self.val = val
+
+# 输入值列表 vals ，返回顶点列表 vets
+def vals_to_vets(vals: List[int]) -> List['Vertex']:
+    return [Vertex(val) for val in vals]
+
+# 输入顶点列表 vets ，返回值列表 vals
+def vets_to_vals(vets: List['Vertex']) -> List[int]:
+    return [vet.val for vet in vets]