feat：增加部分图论算法

subjects/major/图论常见算法.md

@@ -12,4 +12,210 @@
 
 ### Bellman-Ford算法
 
-### Bellman-Ford算法
+#### 数学公式
+
+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：
+
+$$d(v) = \min\{d(u) + w(u,v) | (u,v) \in E\}$$
+
+其中：
+- $d(v)$ 是从源顶点到顶点 $v$ 的最短路径估计
+- $w(u,v)$ 是边 $(u,v)$ 的权重
+- 迭代次数最多为 $|V|-1$ 次
+
+#### Python实现
+
+```python
+def bellman_ford(graph, source):
+    """
+    Bellman-Ford算法求单源最短路径
+    
+    参数:
+        graph: 邻接表表示的图，格式 {u: [(v, w), ...]}
+        source: 源顶点
+    
+    返回:
+        distances: 各顶点到源顶点的最短距离
+    """
+    # 初始化距离
+    distances = {v: float('inf') for v in graph}
+    distances[source] = 0
+    
+    # 获取所有顶点
+    vertices = list(graph.keys())
+    
+    # 最多进行 |V|-1 次迭代
+    for _ in range(len(vertices) - 1):
+        # 对每条边进行松弛操作
+        for u in graph:
+            for v, w in graph[u]:
+                if distances[u] != float('inf') and distances[u] + w < distances[v]:
+                    distances[v] = distances[u] + w
+    
+    # 检测负权重环
+    for u in graph:
+        for v, w in graph[u]:
+            if distances[u] != float('inf') and distances[u] + w < distances[v]:
+                raise ValueError("图中存在负权重环")
+    
+    return distances
+
+
+# 示例使用
+if __name__ == "__main__":
+    # 构建图 (邻接表)
+    graph = {
+        'A': [('B', 4), ('C', 2)],
+        'B': [('C', 3), ('D', 2), ('E', 3)],
+        'C': [('B', 1), ('D', 4)],
+        'D': [('E', -5)],  # 负权重边
+        'E': []
+    }
+    
+    try:
+        distances = bellman_ford(graph, 'A')
+        print("Bellman-Ford 结果:", distances)
+    except ValueError as e:
+        print(e)
+```
+
+---
+
+## 常用算法Python实现
+
+### 1. Dijkstra算法
+
+#### 数学公式
+
+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：
+
+$$d(v) = \min_{u \in S} \{ d(u) + w(u,v) \}$$
+
+其中 $S$ 是已确定最短路径的顶点集合。
+
+#### Python实现
+
+```python
+import heapq
+
+def dijkstra(graph, source):
+    """
+    Dijkstra算法求单源最短路径
+    
+    参数:
+        graph: 邻接表表示的图，格式 {u: [(v, w), ...]}
+        source: 源顶点
+    
+    返回:
+        distances: 各顶点到源顶点的最短距离
+    """
+    # 初始化距离
+    distances = {v: float('inf') for v in graph}
+    distances[source] = 0
+    
+    # 优先队列: (距离, 顶点)
+    pq = [(0, source)]
+    visited = set()
+    
+    while pq:
+        # 取出最小距离的顶点
+        current_dist, u = heapq.heappop(pq)
+        
+        # 如果已访问则跳过
+        if u in visited:
+            continue
+        visited.add(u)
+        
+        # 更新邻接顶点的距离
+        for v, w in graph.get(u, []):
+            if v not in visited and current_dist + w < distances[v]:
+                distances[v] = current_dist + w
+                heapq.heappush(pq, (distances[v], v))
+    
+    return distances
+
+
+# 示例使用
+if __name__ == "__main__":
+    # 构建图 (邻接表)
+    graph = {
+        'A': [('B', 4), ('C', 2)],
+        'B': [('C', 3), ('D', 2), ('E', 3)],
+        'C': [('B', 1), ('D', 4)],
+        'D': [('E', 5)],
+        'E': []
+    }
+    
+    distances = dijkstra(graph, 'A')
+    print("Dijkstra 结果:", distances)
+    # 输出: {'A': 0, 'B': 3, 'C': 2, 'D': 5, 'E': 8}
+```
+
+### 2. Bellman-Ford算法
+
+#### 数学公式
+
+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：
+
+$$d(v) = \min\{d(u) + w(u,v) | (u,v) \in E\}$$
+
+其中：
+- $d(v)$ 是从源顶点到顶点 $v$ 的最短路径估计
+- $w(u,v)$ 是边 $(u,v)$ 的权重
+- 迭代次数最多为 $|V|-1$ 次
+
+#### Python实现
+
+```python
+def bellman_ford(graph, source):
+    """
+    Bellman-Ford算法求单源最短路径
+    
+    参数:
+        graph: 邻接表表示的图，格式 {u: [(v, w), ...]}
+        source: 源顶点
+    
+    返回:
+        distances: 各顶点到源顶点的最短距离
+    """
+    # 初始化距离
+    distances = {v: float('inf') for v in graph}
+    distances[source] = 0
+    
+    # 获取所有顶点
+    vertices = list(graph.keys())
+    
+    # 最多进行 |V|-1 次迭代
+    for _ in range(len(vertices) - 1):
+        # 对每条边进行松弛操作
+        for u in graph:
+            for v, w in graph[u]:
+                if distances[u] != float('inf') and distances[u] + w < distances[v]:
+                    distances[v] = distances[u] + w
+    
+    # 检测负权重环
+    for u in graph:
+        for v, w in graph[u]:
+            if distances[u] != float('inf') and distances[u] + w < distances[v]:
+                raise ValueError("图中存在负权重环")
+    
+    return distances
+
+
+# 示例使用
+if __name__ == "__main__":
+    # 构建图 (邻接表)
+    graph = {
+        'A': [('B', 4), ('C', 2)],
+        'B': [('C', 3), ('D', 2), ('E', 3)],
+        'C': [('B', 1), ('D', 4)],
+        'D': [('E', -5)],  # 负权重边
+        'E': []
+    }
+    
+    try:
+        distances = bellman_ford(graph, 'A')
+        print("Bellman-Ford 结果:", distances)
+    except ValueError as e:
+        print(e)
+```