【BZOJ-4127】Abs 树链剖分 + 线段树 (有趣的姿势)

4127: Abs

Time Limit: 40 Sec Memory Limit: 256 MB
Submit: 381 Solved: 132
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Description

给定一棵树,设计数据结构支持以下操作

1 u v　d　表示将路径 (u,v) 加d

2 u　v 表示询问路径 (u,v) 上点权绝对值的和

Input

第一行两个整数n和m，表示结点个数和操作数

接下来一行n个整数a_i,表示点i的权值

接下来n-1行,每行两个整数u,v表示存在一条(u,v)的边

接下来m行,每行一个操作,输入格式见题目描述

Output

对于每个询问输出答案

Sample Input

4 4
-4 1 5 -2
1 2
2 3
3 4
2 1 3
1 1 4 3
2 1 3
2 3 4

Sample Output

10
13
9

HINT

对于100%的数据，n,m <= 10^5 且 0<= d,|a_i|<= 10^8

Source

Solution

树链剖分显然,把树上路径问题转化为序列问题

然后线段树维护区间权值绝对值和,支持区间加

注意Delta>=0这个条件,即1操作保证加数不为负,即实际值不发生减小

于是线段树维护一些东西:

l,r左右端点;maxf区间最大负数;num区间正数个数-负数个数;tag区间加的标记;sum区间绝对值和

maxf的意义在于,对于区间加Delta,那么如果maxf+Delta<0很显然1操作后会出现变号的情况,对于维护绝对值和必然会做出影响,所以用来进行判断

num的意义在于计算sum的变化,这里同样可以考虑维护正数个数和负数个数,但Code起来比较不方便

tag的意义在于,如果区间+Delta不发生变号情况(即maxf+Delta<0||maxf>=0)的时候,可以直接打上标记,否则则需要把标记下放至叶节点,在向上更新答案

Code

#include<iostream>
#include<cstdio>
#include<cmath>
#include<cstring>
#include<algorithm>
using namespace std;
int read()
{
    int x=0,f=1; char ch=getchar();
    while (ch<'0' || ch>'9') {if (ch=='-') f=-1; ch=getchar();}
    while (ch>='0' && ch<='9') {x=x*10+ch-'0'; ch=getchar();}
    return x*f;
}
#define maxn 110000
int n,m,a[maxn];
struct Edgenode{int to,next;}edge[maxn<<1];
int head[maxn],cnt=1;
void add(int u,int v){cnt++; edge[cnt].to=v; edge[cnt].next=head[u]; head[u]=cnt;}
void insert(int u,int v){add(u,v); add(v,u);}
//----------------------------------------------------------------------------------
int size[maxn],fa[maxn],deep[maxn],son[maxn],pl[maxn],sz,pre[maxn],top[maxn],pr[maxn];
void dfs_1(int now)
{
    size[now]=1;
    for (int i=head[now]; i; i=edge[i].next)
        if (edge[i].to!=fa[now])
            {
                fa[edge[i].to]=now;
                deep[edge[i].to]=deep[now]+1;
                dfs_1(edge[i].to);
                if (size[son[now]]<size[edge[i].to]) son[now]=edge[i].to;
                size[now]+=size[edge[i].to];
            }
}
void dfs_2(int now,int chain)
{
    pl[now]=++sz; pre[sz]=a[now]; top[now]=chain;
    if (son[now]) dfs_2(son[now],chain);
    for (int i=head[now]; i; i=edge[i].next)
        if (edge[i].to!=son[now] && edge[i].to!=fa[now])
            dfs_2(edge[i].to,edge[i].to);
    pr[now]=sz;
}
//----------------------------------------------------------------------------------
struct TreeNode
{
    int l,r;long long maxf,tag,sum,num;
    void Add(int k) 
        {
            if (k<0) maxf=k,sum=-k,num=-1;
                else maxf=0,sum=k,num=1;
            tag=0;
        }
}tree[maxn<<2];
long long Maxf(long long x,long long y)
{
    if (x>=0 && y>=0) return 0;
    if (x>=0 || y>=0) return min(x,y); 
    return max(x,y);
}
void Update(int now)
{
    tree[now].maxf=Maxf(tree[now<<1].maxf,tree[now<<1|1].maxf);
    tree[now].sum=tree[now<<1].sum+tree[now<<1|1].sum;
    tree[now].num=tree[now<<1].num+tree[now<<1|1].num;
}
void BuildTree(int now,int l,int r)
{
    tree[now].l=l,tree[now].r=r;
    if (l==r) {tree[now].Add(pre[l]); return;}
    int mid=(l+r)>>1;
    BuildTree(now<<1,l,mid); BuildTree(now<<1|1,mid+1,r);
    Update(now);
}
void Pushdown(int now)
{
    if (!tree[now].tag || tree[now].l==tree[now].r) return;
    int tag=tree[now].tag; tree[now].tag=0; 
    tree[now<<1].maxf+=tag; tree[now<<1].sum+=tree[now<<1].num*tag; tree[now<<1].tag+=tag;
    tree[now<<1|1].maxf+=tag; tree[now<<1|1].sum+=tree[now<<1|1].num*tag; tree[now<<1|1].tag+=tag;
}
void Change(int now,int L,int R,int D)
{
    int l=tree[now].l,r=tree[now].r;
    if (L<=l && R>=r && (tree[now].maxf>=0 || tree[now].maxf+D<0)) 
        {tree[now].maxf+=D; tree[now].sum+=(long long)tree[now].num*D; tree[now].tag+=D; return;}
    if (l==r) {tree[now].Add(tree[now].maxf+D); return;}
    Pushdown(now);
    int mid=(l+r)>>1;
    if (L<=mid) Change(now<<1,L,R,D);
    if (R>mid) Change(now<<1|1,L,R,D);
    Update(now);
}
long long Query(int now,int L,int R)
{    
    Pushdown(now);
    int l=tree[now].l,r=tree[now].r;
    if (L<=l && R>=r) return tree[now].sum;    
    int mid=(l+r)>>1; long long re=0;
    if (L<=mid) re+=Query(now<<1,L,R);
    if (R>mid) re+=Query(now<<1|1,L,R);
    return re;
}
void DeBug(int now)
{
    int l=tree[now].l,r=tree[now].r;
    if (l==r) {printf("l==r=%d   Val=%d  maxf=%lld   tag=%lld   sum=%lld  num=%lld\n",l,a[l],tree[now].maxf,tree[now].tag,tree[now].sum,tree[now].num);return;}
    int mid=(l+r)>>1;
    DeBug(now<<1); DeBug(now<<1|1); 
}
//----------------------------------------------------------------------------------
void Solve_1(int x,int y,int D)
{
    while (top[x]!=top[y])
        {
            if (deep[top[x]]<deep[top[y]]) swap(x,y);
            Change(1,pl[top[x]],pl[x],D);
            x=fa[top[x]];
        }
    if (deep[x]>deep[y]) swap(x,y);
    Change(1,pl[x],pl[y],D);
}
long long Solve_2(int x,int y)
{
    long long re=0;
    while (top[x]!=top[y])
        {
            if (deep[top[x]]<deep[top[y]]) swap(x,y);
            re+=Query(1,pl[top[x]],pl[x]);
            x=fa[top[x]]; 
        }
    if (deep[x]>deep[y]) swap(x,y);
    re+=Query(1,pl[x],pl[y]);
    return re;
}
//----------------------------------------------------------------------------------
int main()
{
//    freopen("4127.in","r",stdin);
//    freopen("4127.out","w",stdout);
    n=read(),m=read();
    for (int i=1; i<=n; i++) a[i]=read();
    for (int u,v,i=1; i<=n-1; i++) u=read(),v=read(),insert(u,v);
    dfs_1(1); dfs_2(1,1); BuildTree(1,1,n);
    int opt,u,v,w;
    while (m--)
        {
            opt=read();
        //    DeBug(1); 
            if (opt==1) u=read(),v=read(),w=read(),Solve_1(u,v,w);
            else u=read(),v=read(),printf("%lld\n",Solve_2(u,v));
        }
    return 0;
}

友情附送数据生成器:(Designed by YveH)

#include<ctime>
#include<cstdio>
#include<cstdlib>
using namespace std;
int main()
{
    freopen("4127.in","w",stdout);
    srand(time(0));
    int n=1000,q=10000;
    printf("%d %d\n",n,q);
    for (int i=1;i<=n;i++)
        printf("%d ",rand()%10000-5000);
    printf("\n");
    for (int i=2;i<=n;i++)
        printf("%d %d\n",i,rand()%(i-1)+1);
    for (int i=1;i<=q;i++)
    {
        int opt=rand()%2+1;
        printf("%d ",opt);
        if (opt==1)
            printf("%d %d %d\n",rand()%n+1,rand()%n+1,rand()%10000);
        if (opt==2)
            printf("%d %d\n",rand()%n+1,rand()%n+1);
    }
    return 0;
}

数据生成器

友情附送对拍:(Designed by YveH)

#include<iostream>
#include<cstdio>
#include<windows.h>
using namespace std;
int main()
{
    while (1)
    {
        system("4127data.exe");
        system("4127.exe");
        system("4127STD.exe");
        if (system("fc 4127.out 4127std.out"))
            break;
    }
    return 0;
}