2017 ACM-ICPC 亚洲区（西安赛区）网络赛 xor （根号分治）

xor

There is a tree with nn nodes. For each node, there is an integer value a_iai, (1 \le a_i \le 1,000,000,0001≤ai≤1,000,000,000 for 1 \le i \le n1≤i≤n). There is qq queries which are described as follow: Assume the value on the path from node aa to node bb is t_0, t_1, \cdots t_mt0,t1,⋯tm. You are supposed to calculate t_0t0 xor t_ktk xor t_{2k}t2k xor ... xor t_{pk}tpk (pk \le m)(pk≤m).

Input Format

There are multi datasets. (\sum n \le 50,000, \sum q \le 500,000)(∑n≤50,000,∑q≤500,000).

For each dataset: In the first n-1n−1 lines, there are two integers u,vu,v, indicates there is an edge connect node uuand node vv.

In the next nn lines, There is an integer a_iai (1 \le a_i \le 1,000,000,0001≤ai≤1,000,000,000).

In the next qq lines, There is three integers a,ba,b and kk. (1 \le a,b,k \le n1≤a,b,k≤n).

Output Format

For each query, output an integer in one line, without any additional space.

样例输入

样例输出

题目来源

2017 ACM-ICPC 亚洲区（西安赛区）网络赛

【题意】给您一棵树，每个节点有一个权值，q次询问，每次给出u,v,k，求从u到v的路径中，从u开始，每隔k个节点亦或一下的结果。

【分析】根号分治，大于根号n的暴力跳（倍增跳），小于根号n的用数组存起来，复杂度最高为O(N)*sqrt(N)*log(N).

#include <bits/stdc++.h>

#define inf 0x3f3f3f3f

#define met(a,b) memset(a,b,sizeof a)

#define pb push_back

#define mp make_pair

#define rep(i,l,r) for(int i=(l);i<=(r);++i)

#define inf 0x3f3f3f3f

using namespace std;

typedef long long ll;

const int N = 5e4+;;

const int M = ;

const int mod = ;

const int mo=;

const double pi= acos(-1.0);

typedef pair<int,int>pii;

int n,q,sz;

int a[N],fa[N][],up[N][M+],dep[N];

vector<int>edg[N],vec;

int find(int u,int k){

    for(int i=;i>=;i--){

        if(k>>i&){

            u=fa[u][i];

            if(u==)return ;

        }

    }

    return u;

}

void dfs(int u,int f){

    fa[u][]=f;

    for(int i=;i<;i++){

        fa[u][i]=fa[fa[u][i-]][i-];

    }

    for(int i=;i<=sz;i++){

        up[u][i]=a[u];

        int v=find(u,i);

        up[u][i]^=up[v][i];

    }

    for(int i=;i<edg[u].size();i++){

        int v=edg[u][i];

        if(v==f)continue;

        dep[v]=dep[u]+;

        dfs(v,u);

    }

}

int LCA(int u,int v){

    int U=u,V=v;

    if(dep[u]<dep[v])swap(u,v);

    for(int i=;i>=;i--){

        if(dep[fa[u][i]]>=dep[v]){

            u=fa[u][i];

        }

    }

    if(u==v)return (u);

    for(int i=;i>=;i--){

        if(fa[u][i]!=fa[v][i]){

            u=fa[u][i];v=fa[v][i];

        }

    }

    return (fa[u][]);

}

int solve(int u,int v,int k,int lca){

    int res=(dep[u]+dep[v]-*dep[lca])%k;

    int U=u,V=v;

    v=find(v,res);

    int ans=;

    while(dep[u]>=dep[lca]){

        ans^=a[u];

        u=find(u,k);

        if(!u)break;

    }

    if(V==lca||dep[v]<=dep[lca]||v==)return ans;

    V=v;

    while(dep[v]>=dep[lca]){

        ans^=a[v];

        v=find(v,k);

        if(!v)break;

    }

    if((dep[U]-dep[lca])%k==&&(dep[V]-dep[lca])%k==)ans^=a[lca];

    return ans;

}

void init(){

    met(fa,);met(up,);

    for(int i=;i<N;i++){

        edg[i].clear();

    }

}

int main(){

    while(~scanf("%d%d",&n,&q)){

        init();

        sz=round(sqrt(n));

        for(int i=,u,v;i<n;i++){

            scanf("%d%d",&u,&v);

            edg[u].pb(v);edg[v].pb(u);

        }

        for(int i=;i<=n;i++)scanf("%d",&a[i]);

        dep[]=;

        dfs(,);

        while(q--){

            int u,v,k;

            scanf("%d%d%d",&u,&v,&k);

            int lca=LCA(u,v),ans=;;

            if(k>sz){

                ans=solve(u,v,k,lca);

            }

            else {

                int dis=dep[u]-dep[lca];

                int s=(dis/k+)*k;

                int x=find(u,s);

                ans=up[u][k]^up[x][k];

                int res=(dep[u]+dep[v]-*dep[lca])%k;

                if(lca!=v&&dep[v]-dep[lca]>res){

                    v=find(v,res);

                    dis=dep[v]-dep[lca];

                    s=(dis/k+)*k;

                    x=find(v,s);

                    ans^=up[v][k]^up[x][k];

                    if((dep[u]-dep[lca])%k==&&(dep[v]-dep[lca])%k==)ans^=a[lca];

                }

            }

            printf("%d\n",ans);

        }

    }

    return ;

}