bzoj 1468

 

大概思路：树点分治，重心树中每个重心维护一个总的平衡树，树中保存属于该重心的点到该重心的距离，然后对于去掉该重心后形成的子树分别再保存一份。

用这种方式实现的话，还可以支持修改与多次查询，每次操作都是O(logn*logn)

感悟：

点分治正如一个前辈说的，是“借助”重心的力量，来维护与查询与某点距离相关的问题，我们知道点u到它各个重心的距离关系，各个重心又保存了所有属于它的点的信息，可以证明，一个点至多属于O(logn)个重心，所有重心拥有的点总和为O(nlogn)。

本质上来说，树的点分治是通过找重心来对树分治，然后建立了一种层次关系（想象一下，找出一棵树的重心，然后在3维空间向上提升，幷让重心向剩下的各块的重心连边，然后再将各块分层，这样就形成了这种层次关系，表现为树结构就是所谓的重心树）。

 

/**************************************************************
    Problem: 1468
    User: idy002
    Language: C++
    Result: Accepted
    Time:6536 ms
    Memory:54384 kb
****************************************************************/
 
#include <cstdio>
#include <vector>
#include <map>
#define fprintf(...)
#define N 40010
#define S 2000000
using namespace std;
 
struct Stat {
    int c, d, s;
    Stat( int c, int d, int s ):c(c),d(d),s(s){}
};
struct Splay {
    static int son[S][2], pre[S], key[S], siz[S], ntot;
    int root;
    void init() { root=0; }
    inline void update( int nd ) {
        siz[nd] = siz[son[nd][0]]+siz[son[nd][1]]+1;
    }
    void rotate( int nd, int d ) {
        int p=pre[nd];
        int s=son[nd][!d];
        int ss=son[s][d];
         
        son[nd][!d] = ss;
        son[s][d] = nd;
        if( p ) son[p][ nd==son[p][1] ] = s;
        else root = s;
 
        pre[nd] = s;
        pre[s] = p;
        if( ss ) pre[ss] = nd;
         
        update( nd );
        update( s );
    }
    void splay( int nd, int top=0 ) {
        while( pre[nd]!=top ) {
            int p = pre[nd];
            int nl = nd==son[p][0];
            if( pre[p]==top ) {
                rotate( p, nl );
            } else {
                int pp = pre[p];
                int pl = p==son[pp][0];
                if( nl==pl ) {
                    rotate( pp, pl );
                    rotate( p, nl );
                } else {
                    rotate( p, nl );
                    rotate( pp, pl );
                }
            }
        }
    }
    int newnode( int p, int k ) {
        int nd = ++ntot;
        son[nd][0] = son[nd][1] = 0;
        pre[nd] = p;
        key[nd] = k;
        siz[nd] = 1;
        return nd;
    }
    void insert( int k ) {
        if( !root ) {
            root = newnode( 0, k );
            return;
        }
        int nd = root;
        while( son[nd][ k>key[nd] ] ) nd=son[nd][ k>key[nd] ];
        son[nd][ k>key[nd] ] = newnode( nd, k );
        splay( son[nd][ k>key[nd] ] );
    }
    int query( int k ) {
        int nd=root;
        int lnd;
        int rt = 0;
        while(nd) {
            lnd = nd;
            if( key[nd]>k ) {
                nd = son[nd][0];
            } else {
                rt += siz[son[nd][0]]+1;
                nd = son[nd][1];
            }
        }
        splay( lnd );
        return rt;
    }
};
int Splay::son[S][2], Splay::pre[S], Splay::key[S], Splay::siz[S], Splay::ntot;
 
 
int n, qdis;
vector<int> g[N], ww[N];
vector<Stat> st[N];
map<int,Splay > spy[N];
int vis[N], anc[N], siz[N], bac[N], dis[N];
int qu[N], bg, ed;
 
void build( int rt ) {
    /* find the core of the block containing rt */
    qu[bg=ed=1] = rt;
    anc[rt] = rt;
    siz[rt] = bac[rt] = 0;
    while( ed>=bg ) {
        int u=qu[bg++];
        for( int t=0; t<g[u].size(); t++ ) {
            int v=g[u][t];
            if( vis[v] || v==anc[u] ) continue;
            qu[++ed] = v;
            anc[v] = u;
            siz[v] = bac[v] = 0;
        }
    }
    for( int i=ed; i>=1; i-- ) {
        int u=qu[i];
        int p=anc[u];
        siz[u]++;
        siz[p] += siz[u];
        if( bac[p]<siz[u] ) bac[p]=siz[u];
    }
    for( int i=ed; i>=1; i-- ) {
        int u=qu[i];
        if( bac[u]<siz[rt]-siz[u] ) bac[u]=siz[rt]-siz[u];
    }
    for( int i=ed; i>=1; i-- ) {
        int u=qu[i];
        if( bac[u]<bac[rt] ) rt=u;
    }
    /* statistics the info of the block */
    spy[rt][0].init();
    spy[rt][0].insert( 0 );
    st[rt].push_back( Stat(rt,0,0) );
    vis[rt] = true;
    fprintf( stderr, "%d as core\n", rt );
    for( int tm=0; tm<g[rt].size(); tm++ ) {
        int r=g[rt][tm], w=ww[rt][tm];
        if( vis[r] ) continue;
        qu[bg=ed=1] = r;
        anc[r] = rt;
        dis[r] = w;
        while( ed>=bg ) {
            int u=qu[bg++];
            for( int t=0; t<g[u].size(); t++ ) {
                int v=g[u][t], w=ww[u][t];
                if( vis[v] || v==anc[u] ) continue;
                qu[++ed] = v;
                anc[v] = u;
                dis[v] = dis[u]+w;
            }
        }
        spy[rt][r].init();
        for( int i=ed; i>=1; i-- ) {
            int u=qu[i];
            spy[rt][0].insert( dis[u] );
            spy[rt][r].insert( dis[u] );
            st[u].push_back( Stat(rt,dis[u],r) );
        }
        build( r );
    }
}
 
int main() {
    scanf( "%d", &n );
    for( int i=1,u,v,w; i<n; i++ ) {
        scanf( "%d%d%d", &u, &v, &w );
        g[u].push_back( v );
        g[v].push_back( u );
        ww[u].push_back( w );
        ww[v].push_back( w );
    }
    scanf( "%d", &qdis );
    build(1);
    for( int u=1; u<=n; u++ ) {
        fprintf( stderr, "Stat of %d:\n", u );
        for( int t=0; t<st[u].size(); t++ ) {
            fprintf( stderr, "core=%d dis=%d subcore:%d\n", st[u][t].c, st[u][t].d, st[u][t].s );
        }
        fprintf( stderr, "\n" );
    }
     
    int ans = 0;
    for( int u=1; u<=n; u++ ) {
        int tans=0;
        for( int t=0; t<st[u].size(); t++ ) {
            Stat &s = st[u][t];
            tans += spy[s.c][0].query( qdis-s.d )- (s.s?spy[s.c][s.s].query( qdis-s.d ):0);
        }
        fprintf( stderr, "%d added %d\n", u, tans-1 );
        ans += tans-1;
    }
    printf( "%d\n", ans/2 );
}