连通性1 求无向图的low值

这是DFS系列的第一篇 。

首先给出一个重要的定理。该定理来自《算法导论》。

An undirected graph may entail some ambiguity in how we classify edges,
since (u, v) and (v, u) are really the same edge. In such a case, we classify the edge according to whichever of (u, v) or (v, u) the search encounters first.

Introduction to Algorithm 3ed. edition p.610

Theorem 22.10
In a depth-first search of an undirected graph G, every edge of G is either a tree
edge or a back edge.

Proof Let (u, v) be an arbitrary edge of G, and suppose without loss of generality
that u.d < v.d. Then the search must discover and finish v before it finishes u
(while u is gray), since v is on u’s adjacency list. If the first time that the search
explores edge (u, v), it is in the direction from u to v, then v is undiscovered
(white) until that time, for otherwise the search would have explored this edge
already in the direction from v to u. Thus, (u, v) becomes a tree edge. If the
search explores (u, v) first in the direction from v to u, then (u, v) is a back edge,
since u is still gray at the time the edge is first explored.

low值大概是Robert Taryan在论文 Depth-first search and linear graph algorithms  SIAM J. Comput. Vol. 1, No. 2, June 1972给出的概念。

(p.150)"..., LOWPT(v) is the smallest vertex reachable from v by traversing zero or more tree arcs followed by at most one frond."

代码如下

 

 1 #define set0(a) memset(a, 0, sizeof(a))

 2 typedef vector<int> vi;

 3 vi G[MAX_N];

 4 int ts; //time stamp

 5 int dfn[MAX_N], low[MAX_N];

 6 void dfs(int u, int f){

 7     dfn[u]=low[u]=++ts;

 8     for(int i=0; i<G[u].size(); i++){

 9         int &v=G[u][i];

10         if(!dfn[v]){    //tree edge

11             dfs(v, u);

12             low[u]=min(low[u], low[v]);

13         }

14         else if(dfn[v]<dfn[u]&&v!=f){    //back edge

15             low[u]=min(low[u], dfn[v]);

16         }

17     }

18 }

19 void solve(int N){

20     set0(dfn);

21     ts=0;

22     for(int i=1; i<=N; i++)

23         if(!dfn[i]) dfs(i, i);

24 }