POJ - 3352 Road Construction(边双连通分量)

题目大意：给出一张无向图，问添加多少边才能使得这张无向图变成边双连通分量

解题思路：先求出所有的边双连通分量，再将边双连通缩成一个点，通过桥连接起来，这样就形成了一棵无根树了
现在的问题是，将这颗无根树变成边双连通分量

网上的解释是：统计出树中度为1的节点的个数，即为叶节点的个数，记为leaf。则至少在树上添加(leaf+1)/2条边，就能使树达到边二连通，所以至少添加的边数就是(leaf+1)/2。具体方法为，首先把两个最近公共祖先最远的两个叶节点之间连接一条边，这样可以把这两个点到祖先的路径上所有点收缩到一起，因为一个形成的环一定是双连通的。然后再找两个最近公共祖先最远的两个叶节点，这样一对一对找完，恰好是(leaf+1)/2次，把所有点收缩到了一起。

附上大神的详解
和相关的连通分量的概念

#include <cstdio>
#include <cstring>

#define N 1010
#define min(a,b) ((a)<(b) ?(a):(b))

struct Edge{
    int to, next;
}E[N*2];

int n, m, tot, dfs_clock, bcc_cnt, top, bnum;;
int head[N], pre[N], belong[N], degree[N], stack[N], bridge[N][2];

void Addegreedge(int u, int v) {
    E[tot].to = v; E[tot].next = head[u]; head[u] = tot++;
    u = u ^ v; v = v ^ u; u = u ^ v;
    E[tot].to = v; E[tot].next = head[u]; head[u] = tot++;
}

void init() {
    memset(head, -1, sizeof(head));
    tot = 0;

    int u, v;
    for (int i = 0; i < m; i++) {
        scanf("%d%d", &u, &v);
        Addegreedge(u, v);
    }
}

int dfs(int u, int fa) {
    int lowu = pre[u]  = ++dfs_clock;
    stack[++top] = u;
    for (int i = head[u]; i != -1; i = E[i].next) {
        int v = E[i].to;

        if (!pre[v]) {
            int lowv = dfs(v, u);
            lowu = min(lowv, lowu);
            if (lowv > pre[u]) {
                bridge[bnum][0] = u;
                bridge[bnum++][1] = v;

                bcc_cnt++;
                while (1) {
                    int x = stack[top--];
                    belong[x] = bcc_cnt;
                    if (x == v) break;
                }
            }
        }else if (pre[v] < pre[u] && v != fa) {
            lowu = min(lowu, pre[v]);
        }   
    }
    return lowu;
}

void solve() {
    memset(degree, 0, sizeof(degree));
    memset(pre, 0, sizeof(pre));
    dfs_clock = bcc_cnt = top = bnum = 0;
    dfs(1, -1);

    if (top) {
        bcc_cnt++;
        while (1) {
            int x = stack[top--];
            belong[x] = bcc_cnt;
            if (x == 1)
                break;
        }
    }

    for (int i = 0; i < bnum; i++) {
        int u = bridge[i][0];
        int v = bridge[i][1];
        degree[belong[u]]++;
        degree[belong[v]]++;
    }

    int leaf = 0;
    for (int i = 1; i <= bcc_cnt; i++)
        if (degree[i] == 1)
            leaf++;
    printf("%d\n", (leaf + 1)/ 2);
}

int main() {
    while (scanf("%d%d", &n, &m) != EOF) {
        init();
        solve();
    }
    return 0;
}