UESTC 1717 Journey（DFS+LCA）（Sichuan State Programming Contest 2012）

Description

Bob has traveled to byteland, he find the N cities in byteland formed a tree structure, a tree structure is very special structure, there is exactly one path connecting each pair of nodes, and a tree with N nodes has N - 1 edges.

As a traveler, Bob wants to journey between those N cities, and he know the time each road will cost. he advises the king of byteland building a new road to save time, and then, a new road was built. Now Bob has Q journey plan, give you the start city and destination city, please tell Bob how many time is saved by add a road if he always choose the shortest path. Note that if it's better not journey from the new roads,
the answer is 0.

Input

First line of the input is a single integer T(1 <= T <= 20), indicating there are T test cases.

For each test case, the first will line contain two integers N(2 <= N <= 10^5) and Q(1 <= Q <= 10^5), indicating the number of cities in byteland and the journey plans. Then N line followed, each line will contain three integer x, y(1 <= x,y <= N) and z(1 <= z <= 1000) indicating there is a road cost z time connect the x-th city and the y-th city, the first N - 1 roads will form a tree structure, indicating the original roads, and the N-th line is the road built after Bob advised the king. Then Q line followed, each line will contain two integer x and y(1 <= x,y <= N), indicating there is a journey plan from the x-th city to y-th city.

Output

For each case, you should first output "Case #t:" in a single line, where t indicating the case number between 1 and T, then Q lines followed, the i-th line contains one integer indicating the time could saved in i-th journey plan.

题目大意：给一棵树T，每条边都有一个权值，然后又一条新增边，多次询问：从点x到点y在T上走的最短距离，在加上那条新增边之后，最短距离可以减少多少。

思路：任意确定一个根root，DFS计算每个点到根的距离dis[]，然后每两点间的最短距离为dis[x]+dis[y]-2*dis[LCA(x,y)]。若新加入一条边u--v，那么如果我们必须经过u--v，那么从x到y的最短距离就为dis(x,u)+dis(u,v)+dis(v,y)或dis(x,v)+dis(v,u)+dis(u,y)。这样在线处理答案就行。

PS：至于求LCA的方法可以参考2007年郭华阳的论文《RMQ&LCA问题》，RMQ可以用ST算法，至于那个O(n)的±1RMQ有空再写把……

#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;

const int MAXN = 100010;
const int MAXM = MAXN * 2;

int head[MAXN];
int next[MAXM], to[MAXM], cost[MAXM];
int ecnt, root;

void init() {
    ecnt = 1;
    memset(head, 0, sizeof(head));
}

void addEdge(int u, int v, int c) {
    to[ecnt] = v; cost[ecnt] = c; next[ecnt] = head[u]; head[u] = ecnt++;
    to[ecnt] = u; cost[ecnt] = c; next[ecnt] = head[v]; head[v] = ecnt++;
}

int dis[MAXN];

void dfs(int f, int u, int di) {
    dis[u] = di;
    for(int p = head[u]; p; p = next[p]) {
        if(to[p] == f) continue;
        dfs(u, to[p], di + cost[p]);
    }
}

int RMQ[2*MAXN], mm[2*MAXN], best[20][2*MAXN];

void initMM() {
    mm[0] = -1;
    for(int i = 1; i <= MAXN * 2 - 1; ++i)
       mm[i] = ((i&(i-1)) == 0) ? mm[i-1] + 1 : mm[i-1];
}

void initRMQ(int n) {
    int i, j, a, b;
    for(i = 1; i <= n; ++i) best[0][i] = i;
    for(i = 1; i <= mm[n]; ++i) {
        for(j = 1; j <= n + 1 - (1 << i); ++j) {
          a = best[i - 1][j];
          b = best[i - 1][j + (1 << (i - 1))];
          if(RMQ[a] < RMQ[b]) best[i][j] = a;
          else best[i][j] = b;
        }
    }
}

int askRMQ(int a,int b) {
    int t;
    t = mm[b - a + 1]; b -= (1 << t)-1;
    a = best[t][a]; b = best[t][b];
    return RMQ[a] < RMQ[b] ? a : b;
}

int dfs_clock, num[2*MAXN], pos[MAXN];//LCA

void dfs_LCA(int f, int u, int dep) {
    pos[u] = ++dfs_clock;
    RMQ[dfs_clock] = dep; num[dfs_clock] = u;
    for(int p = head[u]; p; p = next[p]) {
        if(to[p] == f) continue;
        dfs_LCA(u, to[p], dep + 1);
        ++dfs_clock;
        RMQ[dfs_clock] = dep; num[dfs_clock] = u;
    }
}

int LCA(int u, int v) {
    if(pos[u] > pos[v]) swap(u, v);
    return num[askRMQ(pos[u], pos[v])];
}

void initLCA(int n) {
    dfs_clock = 0;
    dfs_LCA(0, root, 0);
    initRMQ(dfs_clock);
}

int mindis(int x, int y) {
    return dis[x] + dis[y] - 2 * dis[LCA(x, y)];
}

int main() {
    int T, n, Q;
    int x, y, z, p;
    int u, v;
    initMM();
    scanf("%d", &T);
    for(int t = 1; t <= T; ++t) {
        printf("Case #%d:
", t);
        scanf("%d%d", &n, &Q);
        init();
        for(int i = 0; i < n - 1; ++i) {
            scanf("%d%d%d", &x, &y, &z);
            addEdge(x, y, z);
        }
        scanf("%d%d%d", &x, &y, &z);
        root = x;
        dis[root] = 0;
        for(p = head[root]; p; p = next[p]) dfs(root, to[p], cost[p]);
        initLCA(n);
        while(Q--) {
            scanf("%d%d", &u, &v);
            int ans1 = mindis(u, v);
            int ans2 = min(mindis(u, x) + mindis(y, v), mindis(u, y) + mindis(x, v)) + z;
            if(ans1 > ans2) printf("%d
", ans1 - ans2);
            else printf("0
");
        }
    }
}