mirror of
https://github.com/krahets/hello-algo.git
synced 2024-12-26 01:06:28 +08:00
330 lines
10 KiB
Markdown
330 lines
10 KiB
Markdown
|
---
|
|||
|
comments: true
|
|||
|
---
|
|||
|
|
|||
|
# 图基础操作
|
|||
|
|
|||
|
图的基础操作分为对「边」的操作和对「顶点」的操作,在「邻接矩阵」和「邻接表」这两种表示下的实现方式不同。
|
|||
|
|
|||
|
## 基于邻接矩阵的实现
|
|||
|
|
|||
|
设图的顶点总数为 $n$ ,则有:
|
|||
|
|
|||
|
- **添加或删除边**:直接在邻接矩阵中修改指定边的对应元素即可,使用 $O(1)$ 时间。而由于是无向图,因此需要同时更新两个方向的边。
|
|||
|
- **添加顶点**:在邻接矩阵的尾部添加一行一列,并全部填 $0$ 即可,使用 $O(n)$ 时间。
|
|||
|
- **删除顶点**:在邻接矩阵中删除一行一列。当删除首行首列时达到最差情况,需要将 $(n-1)^2$ 个元素“向左上移动”,从而使用 $O(n^2)$ 时间。
|
|||
|
- **初始化**:传入 $n$ 个顶点,初始化长度为 $n$ 的顶点列表 `vertices` ,使用 $O(n)$ 时间;初始化 $n \times n$ 大小的邻接矩阵 `adjMat` ,使用 $O(n^2)$ 时间。
|
|||
|
|
|||
|
=== "初始化邻接矩阵"
|
|||
|
![adjacency_matrix_initialization](basic_operation_of_graph.assets/adjacency_matrix_initialization.png)
|
|||
|
|
|||
|
=== "添加边"
|
|||
|
![adjacency_matrix_add_edge](basic_operation_of_graph.assets/adjacency_matrix_add_edge.png)
|
|||
|
|
|||
|
=== "删除边"
|
|||
|
![adjacency_matrix_remove_edge](basic_operation_of_graph.assets/adjacency_matrix_remove_edge.png)
|
|||
|
|
|||
|
=== "添加顶点"
|
|||
|
![adjacency_matrix_add_vertex](basic_operation_of_graph.assets/adjacency_matrix_add_vertex.png)
|
|||
|
|
|||
|
=== "删除顶点"
|
|||
|
![adjacency_matrix_remove_vertex](basic_operation_of_graph.assets/adjacency_matrix_remove_vertex.png)
|
|||
|
|
|||
|
以下是基于邻接矩阵表示图的实现代码。
|
|||
|
|
|||
|
=== "Java"
|
|||
|
|
|||
|
```java title="graph_adjacency_matrix.java"
|
|||
|
/* 基于邻接矩阵实现的无向图类 */
|
|||
|
class GraphAdjMat {
|
|||
|
List<Integer> vertices; // 顶点列表,元素代表“顶点值”,索引代表“顶点索引”
|
|||
|
List<List<Integer>> adjMat; // 邻接矩阵,行列索引对应“顶点索引”
|
|||
|
|
|||
|
/* 构造函数 */
|
|||
|
public GraphAdjMat(int[] vertices, int[][] edges) {
|
|||
|
this.vertices = new ArrayList<>();
|
|||
|
this.adjMat = new ArrayList<>();
|
|||
|
// 添加顶点
|
|||
|
for (int val : vertices) {
|
|||
|
addVertex(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 (int[] e : edges) {
|
|||
|
addEdge(e[0], e[1]);
|
|||
|
}
|
|||
|
}
|
|||
|
|
|||
|
/* 获取顶点数量 */
|
|||
|
public int size() {
|
|||
|
return vertices.size();
|
|||
|
}
|
|||
|
|
|||
|
/* 添加顶点 */
|
|||
|
public void addVertex(int val) {
|
|||
|
int n = size();
|
|||
|
// 向顶点列表中添加新顶点的值
|
|||
|
vertices.add(val);
|
|||
|
// 在邻接矩阵中添加一行
|
|||
|
List<Integer> newRow = new ArrayList<>(n);
|
|||
|
for (int j = 0; j < n; j++) {
|
|||
|
newRow.add(0);
|
|||
|
}
|
|||
|
adjMat.add(newRow);
|
|||
|
// 在邻接矩阵中添加一列
|
|||
|
for (List<Integer> row : adjMat) {
|
|||
|
row.add(0);
|
|||
|
}
|
|||
|
}
|
|||
|
|
|||
|
/* 删除顶点 */
|
|||
|
public void removeVertex(int index) {
|
|||
|
if (index >= size())
|
|||
|
throw new IndexOutOfBoundsException();
|
|||
|
// 在顶点列表中移除索引 index 的顶点
|
|||
|
vertices.remove(index);
|
|||
|
// 在邻接矩阵中删除索引 index 的行
|
|||
|
adjMat.remove(index);
|
|||
|
// 在邻接矩阵中删除索引 index 的列
|
|||
|
for (List<Integer> row : adjMat) {
|
|||
|
row.remove(index);
|
|||
|
}
|
|||
|
}
|
|||
|
|
|||
|
/* 添加边 */
|
|||
|
// 参数 i, j 对应 vertices 元素索引
|
|||
|
public void addEdge(int i, int j) {
|
|||
|
// 索引越界与相等处理
|
|||
|
if (i < 0 || j < 0 || i >= size() || j >= size() || i == j)
|
|||
|
throw new IndexOutOfBoundsException();
|
|||
|
// 在无向图中,邻接矩阵沿主对角线对称,即满足 (i, j) == (j, i)
|
|||
|
adjMat.get(i).set(j, 1);
|
|||
|
adjMat.get(j).set(i, 1);
|
|||
|
}
|
|||
|
|
|||
|
/* 删除边 */
|
|||
|
// 参数 i, j 对应 vertices 元素索引
|
|||
|
public void removeEdge(int i, int j) {
|
|||
|
// 索引越界与相等处理
|
|||
|
if (i < 0 || j < 0 || i >= size() || j >= size() || i == j)
|
|||
|
throw new IndexOutOfBoundsException();
|
|||
|
adjMat.get(i).set(j, 0);
|
|||
|
adjMat.get(j).set(i, 0);
|
|||
|
}
|
|||
|
}
|
|||
|
```
|
|||
|
|
|||
|
=== "C++"
|
|||
|
|
|||
|
```cpp title="graph_adjacency_matrix.cpp"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "Python"
|
|||
|
|
|||
|
```python title="graph_adjacency_matrix.py"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "Go"
|
|||
|
|
|||
|
```go title="graph_adjacency_matrix.go"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "JavaScript"
|
|||
|
|
|||
|
```js title="graph_adjacency_matrix.js"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "TypeScript"
|
|||
|
|
|||
|
```typescript title="graph_adjacency_matrix.ts"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "C"
|
|||
|
|
|||
|
```c title="graph_adjacency_matrix.c"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "C#"
|
|||
|
|
|||
|
```csharp title="graph_adjacency_matrix.cs"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "Swift"
|
|||
|
|
|||
|
```swift title="graph_adjacency_matrix.swift"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
## 基于邻接表的实现
|
|||
|
|
|||
|
设图的顶点总数为 $n$ 、边总数为 $m$ ,则有:
|
|||
|
|
|||
|
- **添加边**:在顶点对应链表的尾部添加边即可,使用 $O(1)$ 时间。因为是无向图,所以需要同时添加两个方向的边。
|
|||
|
- **删除边**:在顶点对应链表中查询与删除指定边,使用 $O(m)$ 时间。与添加边一样,需要同时删除两个方向的边。
|
|||
|
- **添加顶点**:在邻接表中添加一个链表即可,并以新增顶点为链表头结点,使用 $O(1)$ 时间。
|
|||
|
- **删除顶点**:需要遍历整个邻接表,删除包含指定顶点的所有边,使用 $O(n + m)$ 时间。
|
|||
|
- **初始化**:需要在邻接表中建立 $n$ 个结点和 $2m$ 条边,使用 $O(n + m)$ 时间。
|
|||
|
|
|||
|
=== "初始化邻接表"
|
|||
|
![adjacency_list_initialization](basic_operation_of_graph.assets/adjacency_list_initialization.png)
|
|||
|
|
|||
|
=== "添加边"
|
|||
|
![adjacency_list_add_edge](basic_operation_of_graph.assets/adjacency_list_add_edge.png)
|
|||
|
|
|||
|
=== "删除边"
|
|||
|
![adjacency_list_remove_edge](basic_operation_of_graph.assets/adjacency_list_remove_edge.png)
|
|||
|
|
|||
|
=== "添加顶点"
|
|||
|
![adjacency_list_add_vertex](basic_operation_of_graph.assets/adjacency_list_add_vertex.png)
|
|||
|
|
|||
|
=== "删除顶点"
|
|||
|
![adjacency_list_remove_vertex](basic_operation_of_graph.assets/adjacency_list_remove_vertex.png)
|
|||
|
|
|||
|
基于邻接表实现图的代码如下所示。
|
|||
|
|
|||
|
=== "Java"
|
|||
|
|
|||
|
```java title="graph_adjacency_list.java"
|
|||
|
/* 顶点类 */
|
|||
|
class Vertex {
|
|||
|
int val;
|
|||
|
public Vertex(int val) {
|
|||
|
this.val = val;
|
|||
|
}
|
|||
|
}
|
|||
|
|
|||
|
/* 基于邻接表实现的无向图类 */
|
|||
|
class GraphAdjList {
|
|||
|
// 请注意,vertices 和 adjList 中存储的都是 Vertex 对象
|
|||
|
Map<Vertex, Set<Vertex>> adjList; // 邻接表(使用哈希表实现)
|
|||
|
|
|||
|
/* 构造函数 */
|
|||
|
public GraphAdjList(Vertex[][] edges) {
|
|||
|
this.adjList = new HashMap<>();
|
|||
|
// 添加所有顶点和边
|
|||
|
for (Vertex[] edge : edges) {
|
|||
|
addVertex(edge[0]);
|
|||
|
addVertex(edge[1]);
|
|||
|
addEdge(edge[0], edge[1]);
|
|||
|
}
|
|||
|
}
|
|||
|
|
|||
|
/* 获取顶点数量 */
|
|||
|
public int size() {
|
|||
|
return adjList.size();
|
|||
|
}
|
|||
|
|
|||
|
/* 添加边 */
|
|||
|
public void addEdge(Vertex vet1, Vertex vet2) {
|
|||
|
if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)
|
|||
|
throw new IllegalArgumentException();
|
|||
|
// 添加边 vet1 - vet2
|
|||
|
adjList.get(vet1).add(vet2);
|
|||
|
adjList.get(vet2).add(vet1);
|
|||
|
}
|
|||
|
|
|||
|
/* 删除边 */
|
|||
|
public void removeEdge(Vertex vet1, Vertex vet2) {
|
|||
|
if (!adjList.containsKey(vet1) || !adjList.containsKey(vet2) || vet1 == vet2)
|
|||
|
throw new IllegalArgumentException();
|
|||
|
// 删除边 vet1 - vet2
|
|||
|
adjList.get(vet1).remove(vet2);
|
|||
|
adjList.get(vet2).remove(vet1);
|
|||
|
}
|
|||
|
|
|||
|
/* 添加顶点 */
|
|||
|
public void addVertex(Vertex vet) {
|
|||
|
if (adjList.containsKey(vet))
|
|||
|
return;
|
|||
|
// 在邻接表中添加一个新链表(即 HashSet)
|
|||
|
adjList.put(vet, new HashSet<>());
|
|||
|
}
|
|||
|
|
|||
|
/* 删除顶点 */
|
|||
|
public void removeVertex(Vertex vet) {
|
|||
|
if (!adjList.containsKey(vet))
|
|||
|
throw new IllegalArgumentException();
|
|||
|
// 在邻接表中删除顶点 vet 对应的链表(即 HashSet)
|
|||
|
adjList.remove(vet);
|
|||
|
// 遍历其它顶点的链表(即 HashSet),删除所有包含 vet 的边
|
|||
|
for (Set<Vertex> set : adjList.values()) {
|
|||
|
set.remove(vet);
|
|||
|
}
|
|||
|
}
|
|||
|
}
|
|||
|
```
|
|||
|
|
|||
|
=== "C++"
|
|||
|
|
|||
|
```cpp title="graph_adjacency_list.cpp"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "Python"
|
|||
|
|
|||
|
```python title="graph_adjacency_list.py"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "Go"
|
|||
|
|
|||
|
```go title="graph_adjacency_list.go"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "JavaScript"
|
|||
|
|
|||
|
```js title="graph_adjacency_list.js"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "TypeScript"
|
|||
|
|
|||
|
```typescript title="graph_adjacency_list.ts"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "C"
|
|||
|
|
|||
|
```c title="graph_adjacency_list.c"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "C#"
|
|||
|
|
|||
|
```csharp title="graph_adjacency_list.cs"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
=== "Swift"
|
|||
|
|
|||
|
```swift title="graph_adjacency_list.swift"
|
|||
|
|
|||
|
```
|
|||
|
|
|||
|
## 效率对比
|
|||
|
|
|||
|
设图中共有 $n$ 个顶点和 $m$ 条边,下表为邻接矩阵和邻接表的时间和空间效率对比。
|
|||
|
|
|||
|
<div class="center-table" markdown>
|
|||
|
|
|||
|
| | 邻接矩阵 | 邻接表(链表) | 邻接表(哈希表) |
|
|||
|
| ------------ | -------- | -------------- | ---------------- |
|
|||
|
| 判断是否邻接 | $O(1)$ | $O(m)$ | $O(1)$ |
|
|||
|
| 添加边 | $O(1)$ | $O(1)$ | $O(1)$ |
|
|||
|
| 删除边 | $O(1)$ | $O(m)$ | $O(1)$ |
|
|||
|
| 添加顶点 | $O(n)$ | $O(1)$ | $O(1)$ |
|
|||
|
| 删除顶点 | $O(n^2)$ | $O(n + m)$ | $O(n)$ |
|
|||
|
| 内存空间占用 | $O(n^2)$ | $O(n + m)$ | $O(n + m)$ |
|
|||
|
|
|||
|
</div>
|
|||
|
|
|||
|
观察上表,貌似邻接表(哈希表)的时间与空间效率最优。但实际上,在邻接矩阵中操作边的效率更高,只需要一次数组访问或赋值操作即可。总结以上,**邻接矩阵体现“以空间换时间”,邻接表体现“以时间换空间”**。
|