【刷题笔记】线段树

· 区间查询

query函数：

当线段树当前结点范围被要查询范围包围时直接返回。

否则判断左右子树的遍历方向。测试数据不会出现越界的情况，不需考虑。

配套题目：click

#include<bits/stdc++.h>
using namespace std;
int a[100010],treea[400010],treeb[400010],ma,mi;
void build(int p,int l,int r)
{
    if(l==r)
    {
        treea[p]=treeb[p]=a[l];
        return;
    }
    int mid=(l+r)/2;
    build(2*p,l,mid);
    build(2*p+1,mid+1,r);
    treea[p]=max(treea[2*p],treea[2*p+1]);
    treeb[p]=min(treeb[2*p],treeb[2*p+1]);
}
void search(int p,int l,int r,int x,int y)
{
    if(x<=l&&r<=y)
    {
        ma=max(ma,treea[p]);
        mi=min(mi,treeb[p]);
        return;
    }
    int mid=(l+r)/2;
    if(y<=mid) search(2*p,l,mid,x,y);
    else if(x>mid) search(2*p+1,mid+1,r,x,y);
    else{
        search(2*p,l,mid,x,y);
        search(2*p+1,mid+1,r,x,y);
    }
}
int main()
{
    int n,q; cin>>n>>q;
    for(int i=1;i<=n;i++) cin>>a[i];
    build(1,1,n);
    while(q--)
    {
        ma=0, mi=0x3f3f3f3f;
        int x,y; cin>>x>>y;
        search(1,1,n,x,y);
        cout<<ma-mi<<endl;
    }
    return 0;
}

· 分步建树

将数组长度作为分界线进行左右子树建立，当l==r时，就是最新元素的位置

配套题目：click

#include<bits/stdc++.h>
using namespace std;
typedef long long ll;
const int MAXN=2e5+5;
ll tree[4*MAXN],t;
void update(int node,int nodeleft,int noderight,int len,ll val){
    if(nodeleft==noderight){
        tree[node]=val; return;
    }
    int mid=(nodeleft+noderight)>>1;
    if(mid>=len) update(node*2,nodeleft,mid,len,val);
    else update(node*2+1,mid+1,noderight,len,val);
    tree[node]=max(tree[node*2],tree[node*2+1]);
}
ll find(int node,int nodeleft,int noderight,int l,int r){
    if(l<=nodeleft&&noderight<=r){
        return tree[node];
    }
    int mid=(nodeleft+noderight)>>1;
    if(r<=mid) return find(node*2,nodeleft,mid,l,r);
    else if(l>mid) return find(node*2+1,mid+1,noderight,l,r);
    else{
        return max(find(node*2,nodeleft,mid,l,r),find(node*2+1,mid+1,noderight,l,r));
    }
}
int main(){
    int n,d; cin>>n>>d;
    int len=0;
    for(int i=0;i<n;i++){
        char ch; int a;
        cin>>ch>>a;
        if(ch=='A'){
            len++;
            update(1,1,MAXN-1,len,(a+t)%d);
        }
        else{
            t=find(1,1,MAXN-1,len-a+1,MAXN-1);
            cout<<t<<endl;
        }
    }
    return 0;
}

· 区间查询与更改模板

题目：click

#include<iostream>
#define int long long
using namespace std;
typedef long long ll;
const int N = 100005;
int w[N];
struct node {
	int l, r;
	ll sum;
	ll add;
}tree[N<<2];
void build(int node,int nodeleft,int noderight) {
	if (nodeleft == noderight) tree[node] = { nodeleft,noderight,w[nodeleft],0 };
	else {
		tree[node] = { nodeleft,noderight };
		int mid = (nodeleft + noderight) / 2;
		build(node * 2, nodeleft, mid);
		build(node * 2 + 1, mid + 1, noderight);
		tree[node].sum = tree[node * 2].sum + tree[node * 2 + 1].sum;
	}
}
void pushdown(int node) {
	struct node& root = tree[node], & left = tree[node * 2], & right = tree[node * 2 + 1];
	if (root.add) {
		left.add += root.add; left.sum += (left.r - left.l + 1) * root.add;
		right.add += root.add; right.sum += (right.r - right.l + 1) * root.add;
		root.add = 0;
	}
}
void update(int node, int left, int right, int val) {
	if (left <= tree[node].l && tree[node].r <= right) {
		tree[node].sum += (tree[node].r - tree[node].l + 1) * val;
		tree[node].add += val;
	}
	else {
		pushdown(node);
		int mid = (tree[node].l + tree[node].r) / 2;
		if (left <= mid) update(node * 2, left, right, val);
		if (right > mid) update(node * 2 + 1, left, right, val);
		tree[node].sum = tree[node * 2].sum + tree[node * 2 + 1].sum;
	}
}
int query(int node, int left, int right) {
	if (left <= tree[node].l && tree[node].r <= right) return tree[node].sum;
	pushdown(node);
	int mid = (tree[node].l + tree[node].r) / 2;
	if (right <= mid) return query(node * 2, left, right);
	else if (left > mid) return query(node * 2 + 1, left, right);
	else {
		return query(node * 2, left, right) + query(node * 2 + 1, left, right);
	}
}
signed main() {
	int n, m; cin >> n >> m;
	for (int i = 1; i <= n; i++) cin >> w[i];
	build(1, 1, n);
	while (m--) {
		char c; cin >> c;
		if (c == 'C') {
			int l, r, val; cin >> l >> r >> val;
			update(1, l, r, val);
		}
		else {
			int l, r; cin >> l >> r;
			cout << query(1, l, r) << endl;
		}
	}
	return 0;
}

· 单点修改区间查询模板

题目：click

#include<bits/stdc++.h>
using namespace std;
const int N = 2e5 + 5;
int num[N];
struct node {
	int l, r;
	int MAX;
}tree[N<<2];
void build(int node, int nodeleft, int noderight) {
	if (nodeleft == noderight) tree[node] = { nodeleft,noderight,num[nodeleft] };
	else {
		tree[node] = { nodeleft,noderight };
		int mid = (nodeleft + noderight) / 2;
		build(node * 2, nodeleft, mid);
		build(node * 2 + 1, mid + 1, noderight);
		tree[node].MAX = max(tree[node * 2].MAX, tree[node * 2 + 1].MAX);
	}
}
void update(int node, int id, int val) {
	if (tree[node].l == id && tree[node].r == id) {
		tree[node].MAX = val; 
	}
	else {
		int mid = (tree[node].l + tree[node].r) / 2;
		if (id <= mid) update(node * 2, id, val);
		else update(node * 2 + 1, id, val);
		tree[node].MAX = max(tree[node * 2].MAX, tree[node * 2 + 1].MAX);
	}
}
int query(int node, int left, int right) {
	if (left <= tree[node].l && tree[node].r <= right) return tree[node].MAX;
	else {
		int mid = (tree[node].l + tree[node].r) / 2;
		if (right <= mid) return query(node * 2, left, right);
		else if (left > mid) return query(node * 2 + 1, left, right);
		else {
			return max(query(node * 2, left, right), query(node * 2 + 1, left, right));
		}
	}
}
int main() {
	int n, m; 
	while (cin >> n >> m) {
		for (int i = 1; i <= n; i++) cin >> num[i];
		build(1, 1, n);
		while (m--) {
			char c; cin >> c;
			int a, b; cin >> a >> b;
			if (c == 'Q') {
				cout << query(1, a, b) << endl;
			}
			else {
				update(1, a, b);
			}
		}
	}
	return 0;
}