222 lines
5.4 KiB
Markdown
222 lines
5.4 KiB
Markdown
## 图论常见算法
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### Dijkstra算法
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用于求解单个源最短路问题：
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1. 初始化：将所有顶点的距离设为正无穷，源顶点设为0；
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2. 迭代：从源顶点出发，遍历所有边，更新顶点的距离；
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3. 检验：检查是否存在负权重环，如果有则返回错误；
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4. 返回：返回所有顶点的距离。
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### Bellman-Ford算法
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#### 数学公式
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对于图 $G=(V,E)$，Bellman-Ford算法通过迭代松弛更新每个顶点的最短路径估计：
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$$d(v) = \min\{d(u) + w(u,v) | (u,v) \in E\}$$
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其中：
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- $d(v)$ 是从源顶点到顶点 $v$ 的最短路径估计
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- $w(u,v)$ 是边 $(u,v)$ 的权重
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- 迭代次数最多为 $|V|-1$ 次
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#### Python实现
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```python
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def bellman_ford(graph, source):
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"""
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Bellman-Ford算法求单源最短路径
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参数:
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graph: 邻接表表示的图，格式 {u: [(v, w), ...]}
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source: 源顶点
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返回:
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distances: 各顶点到源顶点的最短距离
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"""
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# 初始化距离
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distances = {v: float('inf') for v in graph}
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distances[source] = 0
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# 获取所有顶点
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vertices = list(graph.keys())
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# 最多进行 |V|-1 次迭代
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for _ in range(len(vertices) - 1):
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# 对每条边进行松弛操作
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for u in graph:
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for v, w in graph[u]:
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if distances[u] != float('inf') and distances[u] + w < distances[v]:
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distances[v] = distances[u] + w
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# 检测负权重环
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for u in graph:
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for v, w in graph[u]:
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if distances[u] != float('inf') and distances[u] + w < distances[v]:
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raise ValueError("图中存在负权重环")
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return distances
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# 示例使用
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if __name__ == "__main__":
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# 构建图 (邻接表)
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graph = {
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'A': [('B', 4), ('C', 2)],
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'B': [('C', 3), ('D', 2), ('E', 3)],
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'C': [('B', 1), ('D', 4)],
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'D': [('E', -5)], # 负权重边
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'E': []
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}
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try:
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distances = bellman_ford(graph, 'A')
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print("Bellman-Ford 结果:", distances)
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except ValueError as e:
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print(e)
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```
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---
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## 常用算法Python实现
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### 1. Dijkstra算法
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#### 数学公式
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Dijkstra算法维护一个优先队列，每次选取具有最小距离估计的顶点：
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$$d(v) = \min_{u \in S} \{ d(u) + w(u,v) \}$$
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其中 $S$ 是已确定最短路径的顶点集合。
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#### Python实现
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```python
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import heapq
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def dijkstra(graph, source):
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"""
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Dijkstra算法求单源最短路径
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参数:
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graph: 邻接表表示的图，格式 {u: [(v, w), ...]}
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source: 源顶点
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返回:
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distances: 各顶点到源顶点的最短距离
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"""
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# 初始化距离
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distances = {v: float('inf') for v in graph}
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distances[source] = 0
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# 优先队列: (距离, 顶点)
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pq = [(0, source)]
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visited = set()
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while pq:
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# 取出最小距离的顶点
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current_dist, u = heapq.heappop(pq)
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# 如果已访问则跳过
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if u in visited:
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continue
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visited.add(u)
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# 更新邻接顶点的距离
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for v, w in graph.get(u, []):
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if v not in visited and current_dist + w < distances[v]:
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distances[v] = current_dist + w
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heapq.heappush(pq, (distances[v], v))
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return distances
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# 示例使用
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if __name__ == "__main__":
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# 构建图 (邻接表)
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graph = {
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'A': [('B', 4), ('C', 2)],
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'B': [('C', 3), ('D', 2), ('E', 3)],
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'C': [('B', 1), ('D', 4)],
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'D': [('E', 5)],
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'E': []
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}
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distances = dijkstra(graph, 'A')
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print("Dijkstra 结果:", distances)
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# 输出: {'A': 0, 'B': 3, 'C': 2, 'D': 5, 'E': 8}
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```
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### 2. Bellman-Ford算法
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#### 数学公式
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对于图 $G=(V,E)$，Bellman-Ford算法通过迭代松弛更新每个顶点的最短路径估计：
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$$d(v) = \min\{d(u) + w(u,v) | (u,v) \in E\}$$
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其中：
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- $d(v)$ 是从源顶点到顶点 $v$ 的最短路径估计
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- $w(u,v)$ 是边 $(u,v)$ 的权重
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- 迭代次数最多为 $|V|-1$ 次
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#### Python实现
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```python
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def bellman_ford(graph, source):
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"""
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Bellman-Ford算法求单源最短路径
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参数:
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graph: 邻接表表示的图，格式 {u: [(v, w), ...]}
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source: 源顶点
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返回:
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distances: 各顶点到源顶点的最短距离
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"""
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# 初始化距离
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distances = {v: float('inf') for v in graph}
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distances[source] = 0
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# 获取所有顶点
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vertices = list(graph.keys())
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# 最多进行 |V|-1 次迭代
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for _ in range(len(vertices) - 1):
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# 对每条边进行松弛操作
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for u in graph:
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for v, w in graph[u]:
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if distances[u] != float('inf') and distances[u] + w < distances[v]:
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distances[v] = distances[u] + w
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# 检测负权重环
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for u in graph:
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for v, w in graph[u]:
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if distances[u] != float('inf') and distances[u] + w < distances[v]:
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raise ValueError("图中存在负权重环")
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return distances
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# 示例使用
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if __name__ == "__main__":
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# 构建图 (邻接表)
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graph = {
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'A': [('B', 4), ('C', 2)],
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'B': [('C', 3), ('D', 2), ('E', 3)],
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'C': [('B', 1), ('D', 4)],
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'D': [('E', -5)], # 负权重边
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'E': []
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}
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try:
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distances = bellman_ford(graph, 'A')
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print("Bellman-Ford 结果:", distances)
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except ValueError as e:
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print(e)
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```
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