POJ 3237：Tree（树链剖分）

http://poj.org/problem?id=3237

题意：树链剖分。操作有三种：改变一条边的边权，将 a 到 b 的每条边的边权都翻转（即 w[i] = -w[i]），询问 a 到 b 的最大边权。

思路：一开始没有用区间更新，每次翻转的时候都更新到叶子节点，居然也能过，后来看别人的发现也是可以区间更新的。

第一种：无区间更新水过

 #include <cstdio>
 #include <algorithm>
 #include <iostream>
 #include <cstring>
 #include <string>
 #include <cmath>
 #include <queue>
 #include <vector>
 using namespace std;
 #define N 100010
 #define INF -0x7fffffff
 #define lson rt<<1, l, m
 #define rson rt<<1|1, m + 1, r
 struct node
 {
     int v, w, next;
 }edge[N*];
 int top[N], size[N], dep[N], son[N], fa[N], tid[N], time;
 int tot, head[N], e[N][];
 int tree[N<<];

 void init()
 {
     memset(head, -, sizeof(head));
     memset(son, -, sizeof(son));
     time = tot = ;
 }

 void add(int u, int v, int w)
 {
     edge[tot].v = v; edge[tot].next = head[u]; edge[tot].w = w; head[u] = tot++;
 }

 void dfs1(int u, int f, int d)
 {
     size[u] = ;
     dep[u] = d;
     fa[u] = f;
     for(int i = head[u]; ~i; i = edge[i].next) {
         int v = edge[i].v;
         if(v == f) continue;
         dfs1(v, u, d + );
         size[u] += size[v];
         if(son[u] == - || size[son[u]] < size[v]) son[u] = v;
     }
 }

 void dfs2(int u, int tp)
 {
     top[u] = tp;
     tid[u] = ++time;
     if(son[u] == -) return ;
     dfs2(son[u], tp);
     for(int i = head[u]; ~i; i = edge[i].next) {
         int v = edge[i].v;
         if(v != son[u] && v != fa[u])
             dfs2(v, v);
     }
 }

 void pushup(int rt)
 {
     tree[rt] = max(tree[rt<<], tree[rt<<|]);
 }

 void build(int rt, int l, int r)
 {
     tree[rt] = ;
     if(l == r) return ;
     int m = (l + r) >> ;
     build(lson); build(rson);
 }

 void update(int rt, int l, int r, int id, int w) // 更新边
 {
     if(l == r && l == id) {
         tree[rt] = w;
         return ;
     }
     if(l == r) return ;
     int m = (l + r) >> ;
     if(id <= m) update(lson, id, w);
     else update(rson, id, w);
     pushup(rt);
 }

 void Update(int rt, int l, int r, int L, int R) // 翻转值
 {
     if(l == r && L <= l && r <= R) {
         tree[rt] = -tree[rt];
         return ;
     }
     if(l == r) return ;
     int m = (l + r) >> ;
     if(L <= m) Update(lson, L, R);
     if(R > m) Update(rson, L, R);
     pushup(rt);
 }

 int query(int rt, int l, int r, int L, int R)
 {
     int ans = INF;
     if(L <= l && r <= R) {
         ans = max(ans, tree[rt]);
         return ans;
     }
     int m = (l + r) >> ;
     if(L <= m) ans = max(ans, query(lson, L, R));
     if(m < R) ans = max(ans, query(rson, L, R));
     return ans;
 }

 int change(int u, int v) // 查询 (u, v) 最长距离
 {
     int ans = INF;
     while(top[u] != top[v]) {
         if(dep[top[u]] < dep[top[v]]) swap(u, v);
         ans = max(ans, query(, , time, tid[top[u]], tid[u]));
         u = fa[top[u]];
     }
     if(u == v) return ans;
     if(dep[u] > dep[v]) swap(u, v);
     ans = max(ans, query(, , time, tid[son[u]], tid[v]));
     return ans;
 }

 void uuu(int u, int v) // 将值翻转
 {
     while(top[u] != top[v]) {
         if(dep[top[u]] < dep[top[v]]) swap(u, v);
         Update(, , time, tid[top[u]], tid[u]);
         u = fa[top[u]];
     }
     if(u == v) return ;
     if(dep[u] > dep[v]) swap(u, v);
     Update(, , time, tid[son[u]], tid[v]);
 }

 void debug()
 {
     printf("debug *************\n");
     printf("%d\n", time);
     for(int i = ; i <= time; i++) {
         printf("%d, ", top[i]);
     }
     puts("");
 }

 int main()
 {
     int t;
     scanf("%d", &t);
     while(t--) {
         int n;
         scanf("%d", &n);
         init();
         for(int i = ; i < n; i++) {
             scanf("%d%d%d", &e[i][], &e[i][], &e[i][]);
             add(e[i][], e[i][], e[i][]);
             add(e[i][], e[i][], e[i][]);
         }
         dfs1(, , );
         dfs2(, );
         build(, , time);
         for(int i = ; i < n; i++) {
             if(dep[e[i][]] > dep[e[i][]]) swap(e[i][], e[i][]);
             update(, , time, tid[e[i][]], e[i][]);
         }
 //        for(int i = 1; i <= n; i++) printf("tid[%d] :%d\n", i, tid[i]);
         char s[];
         int a, b;
         while(scanf("%s", s) == ) {
             if(s[] == 'D') break;
             scanf("%d%d", &a, &b);
             if(s[] == 'Q') {
 //                puts("");
                 printf("%d\n", change(a, b));
 //                puts("");
             } else if(s[] == 'N') {
                 uuu(a, b);
             } else {
                 update(, , time, tid[e[a][]], b);
             }
         }
     }
     return ;
 }

第二种：区间更新

比上一种快了一倍的时间。就是记录一个最大值和最小值，在翻转的时候，最大值 = -最小值， 最小值 = -最大值，每次标记取反。这样可以快速求出来。

 #include <cstdio>
 #include <algorithm>
 #include <iostream>
 #include <cstring>
 #include <string>
 #include <cmath>
 #include <queue>
 #include <vector>
 using namespace std;
 #define N 100010
 #define INF 0x7fffffff
 #define lson rt<<1, l, m
 #define rson rt<<1|1, m + 1, r
 struct node
 {
     int v, w, next;
 }edge[N*];
 int top[N], size[N], dep[N], son[N], fa[N], tid[N], time;
 int tot, head[N], e[N][];
 int col[N<<], Min[N<<], Max[N<<];

 void init()
 {
     memset(head, -, sizeof(head));
     memset(son, -, sizeof(son));
     time = tot = ;
 }

 void add(int u, int v, int w)
 {
     edge[tot].v = v; edge[tot].next = head[u]; edge[tot].w = w; head[u] = tot++;
 }

 void dfs1(int u, int f, int d)
 {
     size[u] = ;
     dep[u] = d;
     fa[u] = f;
     for(int i = head[u]; ~i; i = edge[i].next) {
         int v = edge[i].v;
         if(v == f) continue;
         dfs1(v, u, d + );
         size[u] += size[v];
         if(son[u] == - || size[son[u]] < size[v]) son[u] = v;
     }
 }

 void dfs2(int u, int tp)
 {
     top[u] = tp;
     tid[u] = ++time;
     if(son[u] == -) return ;
     dfs2(son[u], tp);
     for(int i = head[u]; ~i; i = edge[i].next) {
         int v = edge[i].v;
         if(v != son[u] && v != fa[u])
             dfs2(v, v);
     }
 }

 void pushup(int rt)
 {
     Min[rt] = min(Min[rt<<], Min[rt<<|]);
     Max[rt] = max(Max[rt<<], Max[rt<<|]);
 }

 void pushdown(int rt)
 {
     int tmp;
     if(col[rt]) {
         tmp = -Max[rt<<];
         Max[rt<<] = -Min[rt<<];
         Min[rt<<] = tmp;

         tmp = -Max[rt<<|];
         Max[rt<<|] = -Min[rt<<|];
         Min[rt<<|] = tmp;

         col[rt<<] ^= ;
         col[rt<<|] ^= ;
         col[rt] ^= ;
     }
 }

 void build(int rt, int l, int r)
 {
     Min[rt] = INF;
     Max[rt] = -INF;
     col[rt] = ;
     if(l == r) return ;
     int m = (l + r) >> ;
     build(lson); build(rson);
 }

 void update(int rt, int l, int r, int id, int w) // 更新边
 {
     if(l == r && l == id) {
         Min[rt] = Max[rt] = w;
         return ;
     }
     pushdown(rt);
     int m = (l + r) >> ;
     if(id <= m) update(lson, id, w);
     else update(rson, id, w);
     pushup(rt);
 }

 void Update(int rt, int l, int r, int L, int R) // 翻转值
 {
     int tmp;
     if(L <= l && r <= R) {
         tmp = -Max[rt];
         Max[rt] = -Min[rt];
         Min[rt] = tmp;
         col[rt] ^= ;
         return ;
     }
     pushdown(rt);
     int m = (l + r) >> ;
     if(L <= m) Update(lson, L, R);
     if(R > m) Update(rson, L, R);
     pushup(rt);
 }

 int query(int rt, int l, int r, int L, int R)
 {
     int ans = -INF;
     if(L <= l && r <= R) {
         ans = max(ans, Max[rt]);
         return ans;
     }
     pushdown(rt);
     int m = (l + r) >> ;
     if(L <= m) ans = max(ans, query(lson, L, R));
     if(m < R) ans = max(ans, query(rson, L, R));
     return ans;
 }

 int change(int u, int v) // 查询 (u, v) 最长距离
 {
     int ans = -INF;
     while(top[u] != top[v]) {
         if(dep[top[u]] < dep[top[v]]) swap(u, v);
         ans = max(ans, query(, , time, tid[top[u]], tid[u]));
         u = fa[top[u]];
     }
     if(u == v) return ans;
     if(dep[u] > dep[v]) swap(u, v);
     ans = max(ans, query(, , time, tid[son[u]], tid[v]));
     return ans;
 }

 void uuu(int u, int v) // 将值翻转
 {
     while(top[u] != top[v]) {
         if(dep[top[u]] < dep[top[v]]) swap(u, v);
         Update(, , time, tid[top[u]], tid[u]);
         u = fa[top[u]];
     }
     if(u == v) return ;
     if(dep[u] > dep[v]) swap(u, v);
     Update(, , time, tid[son[u]], tid[v]);
 }

 int main()
 {
     int t;
     scanf("%d", &t);
     while(t--) {
         int n;
         scanf("%d", &n);
         init();
         for(int i = ; i < n; i++) {
             scanf("%d%d%d", &e[i][], &e[i][], &e[i][]);
             add(e[i][], e[i][], e[i][]);
             add(e[i][], e[i][], e[i][]);
         }
         dfs1(, , );
         dfs2(, );
         build(, , time);
         for(int i = ; i < n; i++) {
             if(dep[e[i][]] > dep[e[i][]]) swap(e[i][], e[i][]);
             update(, , time, tid[e[i][]], e[i][]);
         }
         char s[];
         int a, b;
         while(scanf("%s", s) == ) {
             if(s[] == 'D') break;
             scanf("%d%d", &a, &b);
             if(s[] == 'Q') {
                 printf("%d\n", change(a, b));
             } else if(s[] == 'N') {
                 uuu(a, b);
             } else {
                 update(, , time, tid[e[a][]], b);
             }
         }
     }
     return ;
 }

 /*
 1

 3
 1 2 1
 2 3 2
 QUERY 1 2
 CHANGE 1 3
 QUERY 1 2
 NEGATE 1 3
 QUERY 1 3
 DONE
 */