算法笔记--线段树

算法笔记

模板：

#include<bits/stdc++.h>
using namespace std;
#define ll long long 
#define ls rt<<1,l,m
#define rs rt<<1|1,m+1,r
#define pb push_back
const int INF=0x3f3f3f3f;
const int N=1e6+5;
int tree[4*N];
int lazy[4*N]; 
//将子节点值更新到父亲节点
/*对于区间求和而言*/ 
void push_up(int rt)
{
    tree[rt]=tree[rt<<1]+tree[rt<<1|1];
}
//节点懒惰标记下推 
/*对于区间求和而言*/
void push_down(int rt,int len)
{
    tree[rt<<1]+=lazy[rt]*(len-(len>>1));
    lazy[rt<<1]+=lazy[rt];
    tree[rt<<1|1]+=lazy[rt]*(len>>1);
    lazy[rt<<1|1]+=lazy[rt];
    lazy[rt]=0;
} 
//建树
void build(int rt,int l,int r)
{
    if(l==r)
    {
        scanf("%d",&tree[rt]);//最好用scanf,防止超时
        return ;
    }
    int m=(l+r)>>1;
    build(ls);
    build(rs);
    push_up(rt);
} 
//单节点更新 
void update(int p,int delta,int rt,int l,int r)
{
    if(l==r)
    {
        tree[rt]+=delta;
        return ;
    }
    int m=(l+r)>>1;
    if(p<=m)update(p,delta,ls);
    else update(p,delta,rs);
    push_up(rt);
}
//区间更新
void Update(int L,int R,int delta,int rt,int l,int r)
{
    if(L<=l&&r<=R)
    {
        tree[rt]+=(r-l+1)*delta;
        lazy[rt]+=delta;
        return ;
    }
    if(lazy[rt])push_down(rt,r-l+1);
    int m=(l+r)>>1;
    if(L<=m)Update(L,R,delta,ls);
    if(R>m)Update(L,R,delta,rs);
    push_up(rt); 
} 
//区间查询
/*对于区间求和而言*/
int query(int L,int R,int rt,int l,int r)
{
    if(L<=l&&r<=R)return tree[rt];
    if(lazy[rt])push_down(rt,r-l+1);
    int m=(l+r)>>1,res=0;
    if(L<=m)res+=query(L,R,ls);
    if(R>m)res+=query(L,R,rs);
    return res;
} 

POJ - 3468A Simple Problem with Integers

代码：

#include<iostream>
#include<cstdio>
using namespace std;
#define ll long long
#define ls rt<<1,l,m
#define rs rt<<1|1,m+1,r
#define pb push_back
const int INF=0x3f3f3f3f;
const int N=1e5+5;
ll tree[4*N];
ll lazy[4*N]; 
void push_up(int rt)
{
    tree[rt]=tree[rt<<1]+tree[rt<<1|1];
}
void push_down(int rt,int len)
{
    tree[rt<<1]+=lazy[rt]*(len-(len>>1));
    lazy[rt<<1]+=lazy[rt];
    tree[rt<<1|1]+=lazy[rt]*(len>>1);
    lazy[rt<<1|1]+=lazy[rt];
    lazy[rt]=0;
}
void build(int rt,int l,int r)
{
    if(l==r)
    {
        scanf("%lld",&tree[rt]);
        return ;
    }
    int m=(l+r)>>1;
    build(ls);
    build(rs);
    push_up(rt);
}
void Update(int L,int R,int delta,int rt,int l,int r)
{
    if(L<=l&&r<=R)
    {
        tree[rt]+=delta*(r-l+1);
        lazy[rt]+=delta;
        return;
    }
    if(lazy[rt])push_down(rt,r-l+1);
    int m=(l+r)>>1;
    if(L<=m)Update(L,R,delta,ls);
    if(R>m)Update(L,R,delta,rs);
    push_up(rt);
}
ll query(int L,int R,int rt,int l,int r)
{
    if(L<=l&&r<=R)return tree[rt];
    if(lazy[rt])push_down(rt,r-l+1);
    int m=(l+r)>>1;
    ll res=0;
    if(L<=m)res+=query(L,R,ls);
    if(R>m)res+=query(L,R,rs);
    return res;
} 
int main()
{
    /*ios::sync_with_stdio(false);
    cin.tie(0);*/
    int n,q;
    scanf("%d%d",&n,&q);
    build(1,1,n);
    while(q--)
    {
        getchar();
        char s;
        int a,b,c;
        scanf("%c",&s);
        if(s=='Q')
        {
            scanf("%d%d",&a,&b);
            printf("%lld
",query(a,b,1,1,n));
        }
        else
        {
            scanf("%d%d%d",&a,&b,&c);
            Update(a,b,c,1,1,n);
        }
    }
    return 0;
} 

 HDU 1698

#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define pb push_back
#define ls rt<<1,l,m
#define rs rt<<1|1,m+1,r
#define mem(a,b) memset(a,b,sizeof(a))

const int N=1e5+5;
int lazy[4*N];
int tree[4*N];
void push_up(int rt)
{
    tree[rt]=tree[rt<<1]+tree[rt<<1|1];
}
void push_down(int rt,int len)
{
    lazy[rt<<1]=lazy[rt];
    lazy[rt<<1|1]=lazy[rt];
    tree[rt<<1]=lazy[rt]*(len-(len>>1));
    tree[rt<<1|1]=lazy[rt]*(len>>1);
    lazy[rt]=0;
}
void build(int rt,int l,int r)
{
    lazy[rt]=0;
    if(l==r)
    {
        tree[rt]=1;
        return  ;
    }
    int m=(l+r)>>1;
    build(ls);
    build(rs);
    push_up(rt);
}
/*void update(int p,int delta,int rt,int l,int r)
{
    if(l==r)
    {
        tree[rt]=delta;
        return ;
    }
    int m=(l+r)>>1;
    if(p<=m)update(p,delta,ls);
    else update(p,delta,rs);
    push_up(rt);
}*/
void Update(int L,int R,int delta,int rt,int l,int r)
{
    if(L<=l&&r<=R)
    {
        tree[rt]=delta*(r-l+1);
        lazy[rt]=delta;
        return ;
    }
    if(lazy[rt])push_down(rt,r-l+1);
    int m=(l+r)>>1;
    if(L<=m)Update(L,R,delta,ls);
    if(R>m)Update(L,R,delta,rs);
    push_up(rt);
}
int query(int L,int R,int rt,int l,int r)
{
    if(L<=l&&r<=R)return tree[rt];
    if(lazy[rt])push_down(rt,r-l+1);
    int m=(l+r)>>1,ans=0;
    if(L<=m)ans+=query(L,R,ls);
    if(R>m)ans+=query(L,R,rs);
    return ans;
}
int main()
{
    /*ios::sync_with_stdio(false);
    cin.tie(0);*/
    int T,n,q,l,r,a;
    scanf("%d",&T);
    int cs=0;
    while(T--)
    {
        scanf("%d",&n);
        build(1,1,n);
        scanf("%d",&q);
        while(q--)
        {
            scanf("%d%d%d",&l,&r,&a);
            Update(l,r,a,1,1,n);
        }
        printf("Case %d: The total value of the hook is %d.
",++cs,query(1,n,1,1,n));
        //for(int i=1;i<=2*n;i++)cout<<tree[i]<<endl;
    }
    return 0;
}

ZOJ 1610

#include<bits/stdc++.h>
using namespace std;
#define long long
#define pb push_back
#define ls rt<<1,l,m
#define rs rt<<1|1,m+1,r
#define mem(a,b) memset(a,b,sizeof(a))

const int N=8e3+5;
int vis[N];
int ans[N];
int lazy[N<<2];
void push_down(int rt)
{
    lazy[rt<<1]=lazy[rt<<1|1]=lazy[rt];
    lazy[rt]=-1;
}
void update(int L,int R,int c,int rt,int l,int r)
{
    if(L<=l&&r<=R)
    {
        lazy[rt]=c;
        return ;
    }
    if(lazy[rt]==c)return ;
    if(lazy[rt]!=-1)push_down(rt);
    int m=(l+r)>>1;
    if(L<=m)update(L,R,c,ls);
    if(R>m)update(L,R,c,rs);
}
void query(int L,int R,int rt,int l,int r)
{
    if(lazy[rt]!=-1)
    {
        for(int i=l;i<=r;i++)
        vis[i]=lazy[rt];
        return ;
    }
    if(l!=r&&lazy[rt]==-1)
    {
        int m=(l+r)>>1;
        if(L<=m)query(L,R,ls);
        if(R>m)query(L,R,rs);
    }
}
int main()
{
    ios::sync_with_stdio(false);
    cin.tie(0);
    int n,l,r,c;
    while(cin>>n)
    {
        mem(lazy,-1);
        mem(vis,-1);
        mem(ans,0);
        for(int i=0;i<n;i++)cin>>l>>r>>c,update(l+1,r,c,1,1,8000);
        //cout<<1<<endl;
        query(1,8000,1,1,8000);
        int i=1;
        while(i<N)
        {
            if(vis[i]==-1){
                i++;
                continue;
            }
            int c=vis[i];
            while(i<N&&vis[i]==c)i++;
            ans[c]++;
        }
        for(int i=0;i<N;i++)if(ans[i])cout<<i<<' '<<ans[i]<<endl;
        cout<<endl;
    }
    return 0;
}

POJ 3264

#include<iostream>
#include<cstdio>
#include<cstring>
#include<cmath>
#include<algorithm>
#include<vector>
#include<queue>
#include<map>
#include<set>
using namespace std;
#define ll long long
#define pb push_back
#define ls rt<<1,l,m
#define rs rt<<1|1,m+1,r
#define mem(a,b) memset(a,b,sizeof(a))
const int N=5e4+5;
const int INF=0x3f3f3f3f;
struct node
{
    int mx,mn;    
}tree[N<<2];
void push_up(int rt)
{
    tree[rt].mx=max(tree[rt<<1].mx,tree[rt<<1|1].mx);
    tree[rt].mn=min(tree[rt<<1].mn,tree[rt<<1|1].mn);
}
void build(int rt,int l,int r)
{
    if(l==r)
    {
        scanf("%d",&tree[rt].mx);
        tree[rt].mn=tree[rt].mx;
        return ;
    }
    int m=(l+r)>>1;
    build(ls);
    build(rs);
    push_up(rt);
}
int querymx(int L,int R,int rt,int l,int r)
{
    if(L<=l&&r<=R)return tree[rt].mx;
    int m=(l+r)>>1,ans=0;
    if(L<=m)ans=max(ans,querymx(L,R,ls));
    if(R>m)ans=max(ans,querymx(L,R,rs));
    return ans;
}
int querymn(int L,int R,int rt,int l,int r)
{
    if(L<=l&&r<=R)return tree[rt].mn;
    int m=(l+r)>>1,ans=INF;
    if(L<=m)ans=min(ans,querymn(L,R,ls));
    if(R>m)ans=min(ans,querymn(L,R,rs));
    return ans;
} 
int main()
{
    /*ios::sync_with_stdio(false);
    cin.tie(0);*/
    int n,q,l,r;
    scanf("%d%d",&n,&q);
    build(1,1,n);
    while(q--)
    {
        scanf("%d%d",&l,&r);
        printf("%d
",querymx(l,r,1,1,n)-querymn(l,r,1,1,n));
    }
    return 0;    
} 

HDU 4027

#include<bits/stdc++.h>
using namespace std;
#define ll long long 
#define pb push_back
#define ls rt<<1,l,m
#define rs rt<<1|1,m+1,r
#define mem(a,b) memset(a,b,sizeof(a))

const int N=1e5+5;
ll tree[N<<2];
void push_up(int rt)
{
    tree[rt]=tree[rt<<1]+tree[rt<<1|1];
} 
void build(int rt,int l,int r)
{
    if(l==r)
    {
        scanf("%lld",&tree[rt]);
        return ;
    }
    int m=(l+r)>>1;
    build(ls);
    build(rs);
    push_up(rt);
}
void update(int L,int R,int rt,int l,int r)
{
    if(l==r)
    {
        tree[rt]=(int)sqrt(tree[rt]);
        return ;
    }
    int m=(l+r)>>1;
    if(L<=m)update(L,R,ls);
    if(R>m)update(L,R,rs);
    push_up(rt); 
}
ll query(int L,int R,int rt,int l,int r)
{
    if(L<=l&&r<=R)return tree[rt];
    int m=(l+r)>>1;
    ll ans=0;
    if(L<=m)ans+=query(L,R,ls);
    if(R>m)ans+=query(L,R,rs);
    return ans;
}
int main()
{
    int n,m,t,l,r,cs=0;
    while(~scanf("%d",&n))
    {
        build(1,1,n);
        scanf("%d",&m);
        printf("Case #%d:
",++cs);
        while(m--)
        {
            scanf("%d%d%d",&t,&l,&r);
            if(l>r)swap(l,r);
            if(t==0)
            {
                if(query(l,r,1,1,n)!=r-l+1)update(l,r,1,1,n);
            }
            else
            {
                printf("%lld
",query(l,r,1,1,n));
            }
        }
        puts("");
    }
    return 0;    
} 

HDU 1540

线段树区间合并

#include<bits/stdc++.h>
using namespace std;
#define pb push_back
#define ls rt<<1,l,m
#define rs rt<<1|1,m+1,r
#define mem(a,b) memset(a,b,sizeof(a))

const int N=5e4+5;
struct tree{
    int ll,rr,len;//ll表示以当前区间左端点为左端点最长连续区间的长度,rr表示以当前区间右端点为右端点最长连续区间的长度，len表示当前区间的长度
}tree[N<<2];
void push_up(int rt){
    if(tree[rt<<1].ll==tree[rt<<1].len)tree[rt].ll=tree[rt<<1].ll+tree[rt<<1|1].ll;
    else tree[rt].ll=tree[rt<<1].ll;
    if(tree[rt<<1|1].rr==tree[rt<<1|1].len)tree[rt].rr=tree[rt<<1|1].rr+tree[rt<<1].rr;
    else tree[rt].rr=tree[rt<<1|1].rr;
}
void build(int rt,int l,int r){
    tree[rt].ll=tree[rt].rr=tree[rt].len=r-l+1;
    if(l==r)return ;
    int m=(l+r)>>1;
    build(ls);
    build(rs);
}
void update(int p,int v,int rt,int l,int r){
    if(l==r){
        tree[rt].ll=tree[rt].rr=v;
        return ;
    }
    int m=(l+r)>>1;
    if(p<=m)update(p,v,ls);
    else update(p,v,rs);
    push_up(rt);
}
int query(int p,int rt,int l,int r){
    if(rt==1){
        if(l+tree[rt].ll-1>=p)return tree[rt].ll;
        if(r-tree[rt].rr+1<=p)return tree[rt].rr;
    }
    else{
        if(l+tree[rt].ll-1>=p)return tree[rt].ll+tree[rt-1].rr;
        if(r-tree[rt].rr+1<=p)return tree[rt].rr+tree[rt+1].ll;
    }
    if(l==r)return 0;
    int m=(l+r)>>1;
    if(p<=m)return query(p,ls);
    else return query(p,rs);
}
int main(){
    ios::sync_with_stdio(false);
    cin.tie(0);
    int n,m,x;
    char c;
    while(cin>>n>>m){
        stack<int>s;
        build(1,1,n);
        while(m--){
            cin>>c;
            if(c=='D'){
                cin>>x;
                s.push(x);
                update(x,0,1,1,n);
            }
            else if(c=='Q'){
                cin>>x;
                cout<<query(x,1,1,n)<<endl;
            }
            else{
                x=s.top();
                s.pop();
                update(x,1,1,1,n);
            }
        }
    }
    return 0;
}

HDU 3974

dfs序+线段树

先将每个节点按dfs序的in[u]映射到线段上，然后将线段映射成线段树

#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define pb push_back
#define ls rt<<1,l,m
#define rs rt<<1|1,m+1,r
#define mem(a,b) memset(a,b,sizeof(a))

const int N=5e4+5;
int head[N];
int in[N];
int ind[N];
int out[N];
int tree[N<<2];
int lazy[N<<2];
struct edge{
    int to,next;
}edge[N];
int cnt=0;
void add_edge(int u,int v){
    edge[cnt].to=v;
    edge[cnt].next=head[u];
    head[u]=cnt++;
}
void dfs(int u,int &x){
    in[u]=x;
    for(int i=head[u];~i;i=edge[i].next)dfs(edge[i].to,++x);
    out[u]=x;
}
void push_down(int rt){
    if(lazy[rt]){
        tree[rt<<1]=lazy[rt];
        tree[rt<<1|1]=lazy[rt];
        lazy[rt<<1]=lazy[rt];
        lazy[rt<<1|1]=lazy[rt];
        lazy[rt]=0;
    }
}
void update(int L,int R,int v,int rt,int l,int r){
    if(L<=l&&r<=R){
        tree[rt]=v;
        lazy[rt]=v;
        return ;
    }
    push_down(rt);
    int m=(l+r)>>1;
    if(L<=m)update(L,R,v,ls);
    if(R>m)update(L,R,v,rs);
}
int query(int p,int rt,int l,int r){
    if(l==r)return tree[rt];
    push_down(rt);
    int m=(l+r)>>1;
    if(p<=m)return query(p,ls);
    else return query(p,rs);
}
int main(){
    ios::sync_with_stdio(false);
    cin.tie(0);
    int T,cs=0;
    cin>>T;
    while(T--){
        mem(head,-1);
        mem(ind,0);
        cnt=0;
        int n,rt,u,v,l,r,m;
        char c;
        cin>>n;
        for(int i=1;i<n;i++){
            cin>>u>>v;
            add_edge(v,u);
            ind[u]++;
        }
        for(int i=1;i<=n;i++)if(ind[i]==0)rt=i;
        int x=1;
        dfs(rt,x);
        mem(tree,-1);
        mem(lazy,0);
        cout<<"Case #"<<++cs<<":"<<endl;
        cin>>m;
        while(m--){
            cin>>c;
            if(c=='C'){
                cin>>l;
                cout<<query(in[l],1,1,n)<<endl;
            }
            else {
                cin>>l>>r;
                update(in[l],out[l],r,1,1,n);
            }
        }
    }
    return 0;
}

线段树易错点：

1.push_down后记得push_up

2.区间修改或者查询应该有两个if判断if(L <= m); if(R > m);很容易写成if(L <= m); else ;这是不对的

3.lazy标记的修改很容易错