图算法总览

1. 图的基本算法：

    A.  图的表示：邻接表和邻接矩阵；

    B.  广度优先搜索：搜索计算出来的s到各顶点的距离为最短路径。

    BFS(G, s) for each vertex u 属于 V[G] - {s}do color[u] = WHITE d[u] = 无穷大 p[u] = NILcolor[s] = GRAYd[s] = 0p[s] = NILQ = NULLENQUEUE(Q, s)while Q != NULLdo u = DEQUEUE(Q) for each v 属于 Adj[u] do if color[v] = WHITE then color[v] = GRAY d[v] = d[u] + 1 p[v] = u ENQUEUE(Q, v) color[u] = BLACK  

   C. 深度优先搜索：http://blog.csdn.net/liyongjin2009/archive/2011/02/16/6188556.aspx

   D. 拓扑排序： 有向无环图进行拓扑排序。说明事件发生的先后顺序。一个图的拓扑排序可以看成是图中的所有顶点排列而成的一个序列，

       使得所有的有向边均从左指向右。

       TOPOLOGICAL_SORT(G) call DFS(G) to compute finishing times f[v] for each vertex vas each vertex is finished, insert it onto the front of a linked listreturn the linked list of vertices 

    E. 强连通分支：http://baike.baidu.com/view/1976645.htm。有一种算法求强连通分支。对图和它的转置图分别调用DFS。

 

2. 最小生成树

    A. 通用算法

    GENERIC_MST(G, w) A = NULLwhile A does not form a spanning treedo find an edge(u, v) that is safe for A A = A 并上 {(u, v)}return A 

    B. Kruskal 算法 和 Prim 算法

        Kruskal算法：

        MST_KRUSKAL(G, w) A = NULLfor each vetex v 属于 V[G]do MAKE_SET(v)sort the edges of E into nondecreasing order by weight wfor each edge(u, v) 属于 E, taken in nondecreasing order by weithtdo if FIND_SET(u) != FIND_SET(v) then A = A + {(u, v)}return A 

       Prim算法：

       MST_PRIM(G, w, r) for each u 属于 V[G]do key[u] = 无穷大 p[u] = NILkey[r] = 0Q = V[G]while Q != NULLdo u = EXTRACT_MIN(Q) for each v 属于 Adj[u] do if v 属于 Q and w(u, v)<key[v] then p[v] = u key[v] = w(u, v) 

3. 单源最短径

    A. Bellman_Ford 算法

    INITIALIZE_SINGLE_SOURCE(G, s) for each vertex v belongs to V[G]do d[v] = INFINITY p[v] = NILd[s] = 0RELAX(u, v, w) if d[v] > d[u]+w(u,v)then d[v] = d[u]+w(u,v) p[v] = uBELLMAN_FORD(G, w, s) INITIALIZE_SINGLE_SOURCE(G, s)for i=1 to |V[G]|-1do for each edge(u, v) belongs to E[G] do RELAX(u, v, w)for each edge(u, v) belongs to E[G]do if d[v] > d[u]+w(u, v) then return FALSEreturn TRUE

B. 有向无环图中的单源最短路径算法

DAG_SHORTEST_PATHS(G, w, s) topologically sort the verteices of GINITIALIZE_SINGLE_SOURCE(G, s)for each vertex u, taken in topologically sorted orded do for each vertex v belongs to Adj[u] do RELAX(u, v, w)

C. Dijkstra 算法

DIJKSTRA(G, w, s) INITIALIZE_SINGLE_SOURCE(G, s)S = NULLQ = V[G]while Q != NULLdo u = EXTRACT_MIN(Q) S = S + {u} for each vertex v belongs to Adj[u] do RELAX(u, v, w)

4. 每对顶点间的最短路径

A.最短路径与矩阵乘法

B.Floyd-Warshall 算法

3.稀疏图上的Johnson 算法

 

5. 最大流

A. Ford-Fulkerson 算法