线段树模板+详解

线段树

是一种二叉搜索树，与区间树相似，它将一个区间划分成一些单元区
间，每个单元区间对应线段树中的一个叶结点。
对于线段树中的每一个非叶子节点[a,b]，它的左儿子表示的区间为[a,
(a+b)/2]，右儿子表示的区间为[(a+b)/2+1,b]。因此线段树是平衡二叉树，最后的子
节点数目为N，即整个线段区间的长度。
使用线段树可以快速的查找某一个节点在若干条线段中出现的次数，时间复杂
度为O(logN） 。而未优化的空间复杂度为2N，因此有时需要离散化让空间压缩。

模板一：

//单点替换、单点增减、区间求和、区间最值#include <cstdio>#include <algorithm>using namespace std;#define lson l , m , rt << 1#define rson m + 1 , r , rt << 1 | 1const int maxn = 222222;int MAX[maxn<<2];int MIN[maxn<<2];int SUM[maxn<<2];int max(int a,int b){if(a>b)return a;else return b;}int min(int a,int b){if(a<b)return a;else return b;}void PushUP(int rt){MAX[rt] = max(MAX[rt<<1] , MAX[rt<<1|1]);MIN[rt] = min(MIN[rt<<1] , MIN[rt<<1|1]);SUM[rt] = SUM[rt<<1] + SUM[rt<<1|1];}void build(int l,int r,int rt) {if (l == r){ scanf("%d",&MAX[rt]);MIN[rt] = MAX[rt];SUM[rt] = MAX[rt];//printf("mi = %d\n",MIN[rt]);// printf("ma = %d\n",MAX[rt]);return ;} int m = (l + r) >> 1;build(lson);build(rson);PushUP(rt);} void update(int p,int tihuan,int l,int r,int rt){if (l == r) {MAX[rt] = tihuan;MIN[rt] = tihuan;SUM[rt] = tihuan;return ;} int m = (l + r) >> 1;if (p <= m) update(p , tihuan ,lson);else update(p , tihuan , rson);PushUP(rt);} void update1(int p,int add,int l,int r,int rt){if (l == r) {SUM[rt] = SUM[rt] + add;return ;} int m = (l + r) >> 1;if (p <= m) update1(p , add ,lson);else update1(p , add , rson);PushUP(rt);} int query(int L,int R,int l,int r,int rt){if (L <= l && r <= R){return MAX[rt];} int m = (l + r) >> 1;int ret = -1;if (L <= m) ret = max(ret , query(L , R , lson));if (R > m) ret = max(ret , query(L , R , rson));return ret;} int query1(int L,int R,int l,int r,int rt){if (L <= l && r <= R){return MIN[rt];} int m = (l + r) >> 1;int ret = 99999;if (L <= m) ret = min(ret , query1(L , R , lson));if (R > m) ret = min(ret , query1(L , R , rson));return ret;} int queryhe(int L,int R,int l,int r,int rt){if (L <= l && r <= R){return SUM[rt];} int m = (l + r) >> 1;int ret = 0;if (L <= m) ret += queryhe(L , R , lson);if (R > m) ret += queryhe(L , R , rson);return ret;} int main(){int n , m;while (~scanf("%d%d",&n,&m)){build(1 , n , 1);while (m --) {char op[2];int a , b;scanf("%s%d%d",op,&a,&b);if (op[0] == 'Q') //区间求最大{/* for(int i = 1;i<=10;i++)printf("%d ",MAX[i]);puts("");*/printf("%d\n",query(a , b , 1 , n , 1));} else if(op[0]=='U') //单点替换update(a , b , 1 , n , 1);else if(op[0]=='M')//区间求最小{/*for(int i = 1;i<=10;i++)printf("%d ",MIN[i]);puts("");*/printf("%d\n",query1(a , b , 1 , n , 1));} else if(op[0]=='H')//区间求和{printf("%d\n",queryhe(a , b , 1 , n , 1));} else if(op[0]=='S')//单点增加{scanf("%d%d",&a,&b);update1(a , b , 1 , n , 1);}else if(op[0]=='E')//单点减少{scanf("%d%d",&a,&b);update1(a , -b , 1 , n , 1);}}} return 0;}

模板二：

//区间替换#include <iostream>#include <cstdio>#include <cstdlib>#include <cstring>#include <algorithm>#define max(a,b) (a>b)?a:b#define min(a,b) (a>b)?b:a#define lson l , m , rt << 1#define rson m + 1 , r , rt << 1 | 1#define LL long longconst int maxn = 100100;using namespace std;int lazy[maxn<<2];int sum[maxn<<2];void PushUp(int rt)//由左孩子、右孩子向上更新父节点{sum[rt] = sum[rt<<1] + sum[rt<<1|1];} void PushDown(int rt,int m) //向下更新{if (lazy[rt]) //懒惰标记{lazy[rt<<1] = lazy[rt<<1|1] = lazy[rt];sum[rt<<1] = (m - (m >> 1)) * lazy[rt];sum[rt<<1|1] = ((m >> 1)) * lazy[rt];lazy[rt] = 0;}} void build(int l,int r,int rt)//建树{lazy[rt] = 0;if (l== r){scanf("%d",&sum[rt]);return ;} int m = (l + r) >> 1;build(lson);build(rson);PushUp(rt);} void update(int L,int R,int c,int l,int r,int rt)//更新{//if(L>l||R>r) return;if (L <= l && r <= R){lazy[rt] = c;sum[rt] = c * (r - l + 1);//printf("%d %d %d %d %d\n", rt, sum[rt], c, l, r);return ;} PushDown(rt , r - l + 1);int m = (l + r) >> 1;if (L <= m) update(L , R , c , lson);if (R > m) update(L , R , c , rson);PushUp(rt);} LL query(int L,int R,int l,int r,int rt){if (L <= l && r <= R){//printf("%d\n", sum[rt]);return sum[rt];} PushDown(rt , r - l + 1);int m = (l + r) >> 1;LL ret = 0;if (L <= m) ret += query(L , R , lson);if (m < R) ret += query(L , R , rson);return ret;} int main(){int n , m;char str[5];while(scanf("%d%d",&n,&m)){build(1 , n , 1);while (m--){scanf("%s",str);int a , b , c;if(str[0]=='T'){scanf("%d%d%d",&a,&b,&c);update(a , b , c , 1 , n , 1);} else if(str[0]=='Q'){scanf("%d%d",&a,&b);cout<<query(a,b,1,n,1)<<endl;}}} return 0;}

模板三:

//区间增减#include <iostream>#include <cstdio>#include <cstdlib>#include <cstring>#include <algorithm>#define max(a,b) (a>b)?a:b#define min(a,b) (a>b)?b:a#define lson l , m , rt << 1#define rson m + 1 , r , rt << 1 | 1#define LL __int64const int maxn = 100100;using namespace std;LL lazy[maxn<<2];LL sum[maxn<<2];void putup(int rt){sum[rt] = sum[rt<<1] + sum[rt<<1|1];}void putdown(int rt,int m){if (lazy[rt]){lazy[rt<<1] += lazy[rt];lazy[rt<<1|1] += lazy[rt];sum[rt<<1] += lazy[rt] * (m - (m >> 1));sum[rt<<1|1] += lazy[rt] * (m >> 1);lazy[rt] = 0;}} void build(int l,int r,int rt) {lazy[rt] = 0;if (l == r){scanf("%I64d",&sum[rt]);return ;} int m = (l + r) >> 1;build(lson);build(rson);putup(rt);} void update(int L,int R,int c,int l,int r,int rt){if (L <= l && r <= R){lazy[rt] += c;sum[rt] += (LL)c * (r - l + 1);return ;} putdown(rt , r - l + 1);int m = (l + r) >> 1;if (L <= m) update(L , R , c , lson);if (m < R) update(L , R , c , rson);putup(rt);} LL query(int L,int R,int l,int r,int rt){if (L <= l && r <= R){return sum[rt];} putdown(rt , r - l + 1);int m = (l + r) >> 1;LL ret = 0;if (L <= m) ret += query(L , R , lson);if (m < R) ret += query(L , R , rson);return ret;} int main(){int n , m;int a , b , c;char str[5];scanf("%d%d",&n,&m);build(1 , n , 1);while (m--){scanf("%s",str);if (str[0] == 'Q'){scanf("%d%d",&a,&b);printf("%I64d\n",query(a , b , 1 , n , 1));} else if(str[0]=='C'){scanf("%d%d%d",&a,&b,&c);update(a , b , c , 1 , n , 1);}} return 0;}

(PS:什么都不会的同学看看这个QAQ==）