数据结构--树--线段树（Segment Tree）

文章目录

- 1. 概念
- 2. 建树
- 3. 查询
- 4. 修改
- 5. 完整代码及测试

在这里插入图片描述

上图 from 熊掌搜索

类似数据结构：树状数组

1. 概念

线段树是一种二叉树，是用来表示一个区间的树：

常常用来查询区间的：和、最小值、最大值
树结点中存放不是普通二叉树的值，其结点结构如下

class TreeNode
{
public:int sum;//区间和int MAX;//区间最大的int MIN;//区间最小的int start, end;//区间左右端点TreeNode *left, *right;//左右节点TreeNode(int s, int e, int v):start(s),end(e),sum(v){left = right = NULL;MAX = v;MIN = v;}
};

2. 建树

传入数组，及其左右极限端点
自底向上建树

	TreeNode* build(vector<int>& A, int L, int R){if(L > R)return NULL;TreeNode* rt = new TreeNode(L,R,A[L]);if(L == R)return rt;int mid = L+((R-L)>>1);//对半分开rt->left = build(A,L,mid);rt->right = build(A,mid+1,R);rt->sum = 0;if(rt->left){rt->sum += rt->left->sum;rt->MAX = max(rt->MAX, rt->left->MAX);rt->MIN = min(rt->MIN, rt->left->MIN);}if(rt->right){rt->sum += rt->right->sum;rt->MAX = max(rt->MAX, rt->right->MAX);rt->MIN = min(rt->MIN, rt->right->MIN);}return rt;}

3. 查询

时间复杂度： $O(log⁡n)O(\log n)$

	vector<int> query(TreeNode *rt, int s, int e)//查询区间的sum，min，max{if(s > rt->end || e < rt->start)return {0, INT_MAX, INT_MIN};//没有交集if(s <= rt->start && rt->end <= e)return {rt->sum, rt->MIN, rt->MAX};//完全包含区间，取其值//不完全包含，左右查找vector<int> l = query(rt->left, s, e);vector<int> r = query(rt->right,s, e);//汇总信息vector<int> summary(3);summary[0] = l[0] + r[0];summary[1] = min(l[1], r[1]);summary[2] = max(l[2], r[2]);return summary;}

4. 修改

时间复杂度： $O(log⁡n)O(\log n)$

	void modify(TreeNode *rt, int id, int val){if(rt->start == rt->end){	//叶子节点rt->sum = val;//和为自身rt->MAX = val;rt->MIN = val;data[id] = val;return;}int mid = (rt->start + rt->end)/2;if(id > mid)modify(rt->right, id, val);elsemodify(rt->left, id, val);root->sum = 0;if(rt->left){rt->sum += rt->left->sum;rt->MAX = max(rt->MAX, rt->left->MAX);rt->MIN = min(rt->MIN, rt->left->MIN);}if(rt->right){rt->sum += rt->right->sum;rt->MAX = max(rt->MAX, rt->right->MAX);rt->MIN = min(rt->MIN, rt->right->MIN);}}

5. 完整代码及测试

/*** @description: 线段树* @author: michael ming* @date: 2020/3/13 0:21* @modified by:* @Website: https://michael.blog.csdn.net/*/
#include<vector>
#include<iostream>
#include<climits>
using namespace std;
class TreeNode
{
public:int sum;//区间和int MAX;//区间最大的int MIN;//区间最小的int start, end;//区间左右端点TreeNode *left, *right;//左右节点TreeNode(int s, int e, int v):start(s),end(e),sum(v){left = right = NULL;MAX = v;MIN = v;}
};
class SegmentTree
{
public:TreeNode* root;vector<int> data;SegmentTree(vector<int>& A){root = build(A, 0, A.size()-1);data = A;}~SegmentTree(){destroy(root);}void destroy(TreeNode* rt){if(!rt) return;destroy(rt->left);destroy(rt->right);delete rt;}TreeNode* build(vector<int>& A, int L, int R){if(L > R)return NULL;TreeNode* rt = new TreeNode(L,R,A[L]);if(L == R)return rt;int mid = L+((R-L)>>1);//对半分开rt->left = build(A,L,mid);rt->right = build(A,mid+1,R);rt->sum = 0;if(rt->left){rt->sum += rt->left->sum;rt->MAX = max(rt->MAX, rt->left->MAX);rt->MIN = min(rt->MIN, rt->left->MIN);}if(rt->right){rt->sum += rt->right->sum;rt->MAX = max(rt->MAX, rt->right->MAX);rt->MIN = min(rt->MIN, rt->right->MIN);}return rt;}vector<int> query(TreeNode *rt, int s, int e)//查询区间的sum，min，max{if(s > rt->end || e < rt->start)return {0, INT_MAX, INT_MIN};//没有交集if(s <= rt->start && rt->end <= e)return {rt->sum, rt->MIN, rt->MAX};//完全包含区间，取其值//不完全包含，左右查找vector<int> l = query(rt->left, s, e);vector<int> r = query(rt->right,s, e);//汇总信息vector<int> summary(3);summary[0] = l[0] + r[0];summary[1] = min(l[1], r[1]);summary[2] = max(l[2], r[2]);return summary;}void modify(TreeNode *rt, int id, int val){if(rt->start == rt->end){	//叶子节点rt->sum = val;//和为自身rt->MAX = val;rt->MIN = val;data[id] = val;return;}int mid = (rt->start + rt->end)/2;if(id > mid)modify(rt->right, id, val);elsemodify(rt->left, id, val);root->sum = 0;if(rt->left){rt->sum += rt->left->sum;rt->MAX = max(rt->MAX, rt->left->MAX);rt->MIN = min(rt->MIN, rt->left->MIN);}if(rt->right){rt->sum += rt->right->sum;rt->MAX = max(rt->MAX, rt->right->MAX);rt->MIN = min(rt->MIN, rt->right->MIN);}}
};
//-------------test---------------------
void printVec(vector<int> &a)
{for(auto& ai : a)cout << ai << " ";cout << endl;
}int main()
{vector<int> v = {1,2,7,8,5};printVec(v);cout << "建立线段树" << endl;SegmentTree sgtree(v);printVec(sgtree.data);cout << "查询区间的sum，MIN，MAX" << endl;vector<int> qy_res = sgtree.query(sgtree.root,1,3);printVec(qy_res);cout << "修改某位置的值" << endl;sgtree.modify(sgtree.root,1,100);printVec(sgtree.data);cout << "查询区间的sum，MIN，MAX" << endl;qy_res = sgtree.query(sgtree.root,1,3);printVec(qy_res);return 0;
}

运行结果：valgrind ./a.out

==16895== Memcheck, a memory error detector
==16895== Copyright (C) 2002-2017, and GNU GPL'd, by Julian Seward et al.
==16895== Using Valgrind-3.14.0 and LibVEX; rerun with -h for copyright info
==16895== Command: ./a.out
==16895== 
1 2 7 8 5 
建立线段树
1 2 7 8 5 
查询区间的sum，MIN，MAX
17 2 8 
修改某位置的值
1 100 7 8 5 
查询区间的sum，MIN，MAX
115 7 100 
==16895== 
==16895== HEAP SUMMARY:
==16895==     in use at exit: 0 bytes in 0 blocks
==16895==   total heap usage: 29 allocs, 29 frees, 616 bytes allocated
==16895== 
==16895== All heap blocks were freed -- no leaks are possible
==16895== 
==16895== For counts of detected and suppressed errors, rerun with: -v
==16895== ERROR SUMMARY: 0 errors from 0 contexts (suppressed: 0 from 0)