LCA+最小生成树 Codeforces609E Minimum spanning tree for each edge

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题意：给一个图，有m条边n个点，如果对于一个最小生成树中要求必须包括第i条边，那么最小生成树的权值总和最小是多少

思路：求出最小生成树，然后对于m条边相当于m次查询，每次查询的时候，相当于求出在最小生成树中(u,v)路径上的边权最大值，那么新添加了一条边，就要把这条最大值的边删掉。所以题目转换成了，求路径上边权最大值。可以用LCA来做，也可以用树链剖分来维护。

LCA维护

#include<map>
#include<set>
#include<cmath>
#include<ctime>
#include<stack>
#include<queue>
#include<cstdio>
#include<cctype>
#include<string>
#include<vector>
#include<cstring>
#include<iostream>
#include<algorithm>
#include<functional>
#define fuck(x) cout<<"["<<x<<"]"
#define FIN freopen("input.txt","r",stdin)
#define FOUT freopen("output.txt","w+",stdout)
using namespace std;
typedef long long LL;
typedef pair<int, int>PII;

const int MX = 2e5 + 5;
const int MS = 4e5 + 5;
const int M = 25;//n的log
const int INF = 0x3f3f3f3f;

struct Edge {
    int u, v, nxt, cost, id;
    bool operator<(const Edge &P) const {
        return cost < P.cost;
    }
} E[MS], A[MX];
int rear, Head[MX];
void edge_init() {
    rear = 0;
    memset(Head, -1, sizeof(Head));
}
void edge_add(int u, int v, int cost) {
    E[rear].u = u;
    E[rear].v = v;
    E[rear].cost = cost;
    E[rear].nxt = Head[u];
    Head[u] = rear++;
}

LL mincost, ans[MX];
int n, m, P[MX];
int find(int x) {
    return P[x] == x ? x : (P[x] = find(P[x]));
}
void MST() {
    mincost = 0;
    for(int i = 1; i <= n; i++) P[i] = i;
    for(int i = 1; i <= m; i++) {
        int p1 = find(A[i].u), p2 = find(A[i].v);
        if(p1 != p2) {
            P[p1] = p2;
            edge_add(A[i].u, A[i].v, A[i].cost);
            edge_add(A[i].v, A[i].u, A[i].cost);
            mincost += A[i].cost;
        }
    }
}
int dep[MX], fa[MX][M], MAX[MX][M];
void DFS(int u, int _dep, int _fa) {
    dep[u] = _dep; fa[u][0] = _fa;
    for(int i = Head[u]; ~i; i = E[i].nxt) {
        int v = E[i].v;
        if(v == _fa) {
            MAX[u][0] = E[i].cost;
            continue;
        }
        DFS(v, _dep + 1, u);
    }
}
void presolve() {
    DFS(1, 0, 1);
    for(int i = 1; i < M; i++) {
        for(int j = 1; j <= n; j++) {
            fa[j][i] = fa[fa[j][i - 1]][i - 1];
            MAX[j][i] = max(MAX[j][i - 1], MAX[fa[j][i - 1]][i - 1]);
        }
    }
}
int LCA(int u, int v) {
    int ret = 0;
    while(dep[u] != dep[v]) {
        if(dep[u] < dep[v]) swap(u, v);
        int d = dep[u] - dep[v];
        for(int i = 0; i < M; i++) {
            if(d >> i & 1) {
                ret = max(ret, MAX[u][i]);
                u = fa[u][i];
            }
        }
    }
    if(u == v) return ret;
    for(int i = M - 1; i >= 0; i--) {
        if(fa[u][i] != fa[v][i]) {
            ret = max(ret, MAX[u][i]);
            ret = max(ret, MAX[v][i]);
            u = fa[u][i];
            v = fa[v][i];
        }
    }
    return max(ret, max(MAX[u][0], MAX[v][0]));
}
int main() {
    //FIN;
    while(~scanf("%d%d", &n, &m)) {
        edge_init();
        for(int i = 1; i <= m; i++) {
            A[i].id = i;
            scanf("%d%d%d", &A[i].u, &A[i].v, &A[i].cost);
        }
        sort(A + 1, A + 1 + m);

        MST();
        presolve();
        for(int i = 1; i <= m; i++) {
            ans[A[i].id] = mincost - LCA(A[i].u, A[i].v) + A[i].cost;
        }
        for(int i = 1; i <= m; i++) {
            printf("%I64d\n", ans[i]);
        }
    }
    return 0;
}

树链剖分维护

#include<map>
#include<set>
#include<cmath>
#include<ctime>
#include<stack>
#include<queue>
#include<cstdio>
#include<cctype>
#include<string>
#include<vector>
#include<cstring>
#include<iostream>
#include<algorithm>
#include<functional>
#define fuck(x) cout<<"["<<x<<"]"
#define FIN freopen("input.txt","r",stdin)
#define FOUT freopen("output.txt","w+",stdout)
using namespace std;
typedef long long LL;
typedef pair<int, int>PII;

const int MX = 2e5 + 5;
const int MS = 4e5 + 5;
const int INF = 0x3f3f3f3f;
#define lson l,m,rt<<1
#define rson m+1,r,rt<<1|1

struct Edge {
    int u, v, nxt, cost, id;
    bool operator<(const Edge &P) const {
        return cost < P.cost;
    }
} E[MS], A[MS];
int _rear, Head[MX];
void edge_init() {
    _rear = 0;
    memset(Head, -1, sizeof(Head));
}
void edge_add(int u, int v, int cost) {
    E[_rear].u = u;
    E[_rear].v = v;
    E[_rear].cost = cost;
    E[_rear].nxt = Head[u];
    Head[u] = _rear++;
}
bool cmp(Edge a, Edge b) {
    return a.id < b.id;
}

int n, m, P[MX];
LL mincost;
int find(int x) {
    return P[x] == x ? x : (P[x] = find(P[x]));
}
void MST_solve() {
    mincost = 0;
    edge_init();
    sort(A + 1, A + 1 + m);
    for(int i = 1; i <= n; i++) P[i] = i;
    for(int i = 1; i <= m; i++) {
        int p1 = find(A[i].u), p2 = find(A[i].v);
        if(p1 != p2) {
            P[p1] = p2;
            mincost += A[i].cost;
            edge_add(A[i].u, A[i].v, A[i].cost);
            edge_add(A[i].v, A[i].u, A[i].cost);
            //printf("[%d,%d,%d]",A[i].u,A[i].v,A[i].cost);
        }
    }
}

int MAX[MX << 2], TA[MX];
void push_up(int rt) {
    MAX[rt] = max(MAX[rt << 1], MAX[rt << 1 | 1]);
}
void build(int l, int r, int rt) {
    if(l == r) {
        MAX[rt] = TA[l];
        return;
    }
    int m = (l + r) >> 1;
    build(lson);
    build(rson);
    push_up(rt);
}
int query(int L, int R, int l, int r, int rt) {
    if(L <= l && r <= R) {
        return MAX[rt];
    }
    int m = (l + r) >> 1, ret = -INF;
    if(L <= m) ret = max(ret, query(L, R, lson));
    if(R > m) ret = max(ret, query(L, R, rson));
    return ret;
}

int fa[MX], top[MX], siz[MX], son[MX], dep[MX], id[MX], rear;
void DFS1(int u, int f, int d) {
    fa[u] = f; dep[u] = d;
    son[u] = 0; siz[u] = 1;
    for(int i = Head[u]; ~i; i = E[i].nxt) {
        int v = E[i].v;
        if(v == f) continue;
        DFS1(v, u, d + 1);
        siz[u] += siz[v];
        if(siz[son[u]] < siz[v]) {
            son[u] = v;
        }
    }
}
void DFS2(int u, int tp) {
    top[u] = tp;
    id[u] = ++rear;
    if(son[u]) DFS2(son[u], tp);
    for(int i = Head[u]; ~i; i = E[i].nxt) {
        int v = E[i].v;
        if(v == fa[u] || v == son[u]) continue;
        DFS2(v, v);
    }
}
void HLD_presolve() {
    rear = 0;
    DFS1(1, 0, 1);
    DFS2(1, 1);
    for(int i = 0; i < 2 * (rear - 1); i += 2) {
        int u = E[i].u, v = E[i].v;
        if(dep[u] < dep[v]) swap(u, v);
        TA[id[u]] = E[i].cost;
    }
    TA[1] = -INF;
    build(1, rear, 1);
}
int HLD_query(int u, int v) {
    int tp1 = top[u], tp2 = top[v], ans = -INF;
    while(tp1 != tp2) {
        if(dep[tp1] < dep[tp2]) {
            swap(u, v);
            swap(tp1, tp2);
        }
        ans = max(ans, query(id[tp1], id[u], 1, rear, 1));
        u = fa[tp1]; tp1 = top[u];
    }
    if(u == v) return ans;
    if(dep[u] > dep[v]) swap(u, v);
    ans = max(ans, query(id[son[u]], id[v], 1, rear, 1));
    return ans;
}

int main() {
    //FIN;
    while(~scanf("%d%d", &n, &m)) {
        for(int i = 1; i <= m; i++) {
            A[i].id = i;
            scanf("%d%d%d", &A[i].u, &A[i].v, &A[i].cost);
        }
        MST_solve();
        HLD_presolve();
        sort(A + 1, A + 1 + m, cmp);
        for(int i = 1; i <= m; i++) {
            int u = A[i].u, v = A[i].v, cost = A[i].cost;
            printf("%I64d\n", mincost + cost - HLD_query(u, v));
        }
    }
    return 0;
}