线段树 2016.4.13

一、定义

线段树（Segment Tree ）是一种二叉搜索树

不是真正的完全二叉树，但十分近似

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

由于二叉树结构的特性，它基本能保持每个操作的时间复杂度为O(logN)

线段树在数组存储的方式下需要占用的空间约为4N

二、创建线段树

如果当前结点记录的区间只有一个值，则直接赋值
否则递归构造左右子树
回溯的时候给当前结点赋值

#include <iostream>#include <cstdio>#include <cstring>using namespace std;struct SegTree {    int left, right;    int val;};const int maxn = 5 + 5;int num[maxn];SegTree Tree[maxn * 4];void Build(int low, int high, int index);int main(){    for (int i=1; i<=5; ++i) {        num[i] = i;    }    Build(1, 5, 1);    for (int i=1; i<=9; ++i) {        printf("%d %d %d\n", Tree[i].left, Tree[i].right, Tree[i].val);    }    return 0;}void Build(int low, int high, int index){    Tree[index].left = low;    Tree[index].right = high;    if (low == high) {        Tree[index].val = num[low];        return;    } else {        int mid = low + (high-low)/2;        Build(low, mid, index*2);        Build(mid+1, high, index*2+1);    }}

#include <iostream>#include <cstdio>#include <cstring>using namespace std;struct SegTree {    int left, right;    int Max;};const int maxn = 5 + 5;int num[maxn];SegTree Tree[maxn * 4];void Build(int low, int high, int index);int main(){    for (int i=1; i<=5; ++i) {        num[i] = i;    }    Build(1, 5, 1);    for (int i=1; i<=9; ++i) {        printf("%d %d %d\n", Tree[i].left, Tree[i].right, Tree[i].Max);    }    return 0;}void Build(int low, int high, int index){    Tree[index].left = low;    Tree[index].right = high;    if (low == high) {        Tree[index].Max = num[low];        return;    } else {        int mid = low + (high-low)/2;        Build(low, mid, index*2);        Build(mid+1, high, index*2+1);        Tree[index].Max = max(Tree[index*2].Max, Tree[index*2+1].Max);    }}

#include <iostream>#include <cstdio>#include <cstring>using namespace std;struct SegTree {    int left, right;    int sum;};const int maxn = 5 + 5;int num[maxn];SegTree Tree[maxn * 4];void Build(int low, int high, int index);int main(){    for (int i=1; i<=5; ++i) {        num[i] = i;    }    Build(1, 5, 1);    for (int i=1; i<=9; ++i) {        printf("%d %d %d\n", Tree[i].left, Tree[i].right, Tree[i].sum);    }    return 0;}void Build(int low, int high, int index){    Tree[index].left = low;    Tree[index].right = high;    if (low == high) {        Tree[index].sum = num[low];        return;    } else {        int mid = low + (high-low)/2;        Build(low, mid, index*2);        Build(mid+1, high, index*2+1);        Tree[index].sum = Tree[index*2].sum + Tree[index*2+1].sum;    }}

三、单结点更新 + 区间查询

#include <iostream>#include <cstdio>#include <cstring>using namespace std;struct SegTree {    int left, right;    int sum;};const int maxn = 5 + 5;int num[maxn];SegTree Tree[maxn * 4];void Build(int low, int high, int index);void Update(int target, int add, int index);int Query(int low, int high, int index);int main(){    for (int i=1; i<=5; ++i) {        num[i] = i;    }    Build(1, 5, 1);    Update(2, 2, 1);    printf("%d\n", Query(1, 4, 1));    printf("%d\n", Query(4, 5, 1));    for (int i=1; i<=9; ++i) {        printf("%d %d %d\n", Tree[i].left, Tree[i].right, Tree[i].sum);    }    return 0;}void Build(int low, int high, int index){    Tree[index].left = low;    Tree[index].right = high;    if (low == high) {        Tree[index].sum = num[low];        return;    } else {        int mid = low + (high-low)/2;        Build(low, mid, index*2);        Build(mid+1, high, index*2+1);        Tree[index].sum = Tree[index*2].sum + Tree[index*2+1].sum;    }}void Update(int target, int add, int index){    if (Tree[index].left == Tree[index].right && Tree[index].left == target) {        Tree[index].sum += add;        return;    } else {        int mid = Tree[index].left + (Tree[index].right - Tree[index].left)/2;        if (target <= mid) {            Update(target, add, index*2);        } else {            Update(target, add, index*2+1);        }        Tree[index].sum = Tree[index*2].sum + Tree[index*2+1].sum;    }}int Query(int low, int high, int index){    if (Tree[index].left == low && Tree[index].right == high) {        return Tree[index].sum;    } else {        int mid = Tree[index].left + (Tree[index].right-Tree[index].left)/2;        if (high <= mid) {            return Query(low, high, index*2);        } else if (low > mid) {            return Query(low, high, index*2+1);        } else {            return Query(low, mid, index*2) + Query(mid+1, high, index*2+1);        }    }}

void Update(int target, int add, int index);Update(2, 2, 1);Query(1, 4, 1) = 12;Query(4, 5, 1) = 9;

#include <iostream>#include <cstdio>#include <cstring>using namespace std;struct SegTree {    int left, right;    int Max;};const int maxn = 5 + 5;int num[maxn];SegTree Tree[maxn * 4];void Build(int low, int high, int index);void Update(int target, int val, int index);int Query(int low, int high, int index);int main(){    for (int i=1; i<=5; ++i) {        num[i] = i;    }    Build(1, 5, 1);    Update(3, 6, 1);    printf("%d\n", Query(2, 5, 1));    printf("%d\n", Query(3, 4, 1));    for (int i=1; i<=9; ++i) {        printf("%d %d %d\n", Tree[i].left, Tree[i].right, Tree[i].Max);    }    return 0;}void Build(int low, int high, int index){    Tree[index].left = low;    Tree[index].right = high;    if (low == high) {        Tree[index].Max = num[low];        return;    } else {        int mid = low + (high-low)/2;        Build(low, mid, index*2);        Build(mid+1, high, index*2+1);        Tree[index].Max = max(Tree[index*2].Max, Tree[index*2+1].Max);    }}void Update(int target, int val, int index){    if (Tree[index].left == Tree[index].right && Tree[index].left == target) {        Tree[index].Max = val;        return;    } else {        int mid = Tree[index].left + (Tree[index].right - Tree[index].left)/2;        if (target <= mid) {            Update(target, val, index*2);        } else {            Update(target, val, index*2+1);        }        Tree[index].Max = max(Tree[index*2].Max, Tree[index*2+1].Max);    }}int Query(int low, int high, int index){    if (Tree[index].left == low && Tree[index].right == high) {        return Tree[index].Max;    } else {        int mid = Tree[index].left + (Tree[index].right-Tree[index].left)/2;        if (high <= mid) {            return Query(low, high, index*2);        } else if (low > mid) {            return Query(low, high, index*2+1);        } else {            return max(Query(low, mid, index*2), Query(mid+1, high, index*2+1));        }    }}

void Update(int target, int val, int index);Update(3, 6, 1);Query(2, 5, 1) = 6;Query(3, 4, 1) = 6;

四、区间更新 + 区间查询

#include <iostream>#include <cstdio>#include <cstring>using namespace std;struct SegTree {    int left, right;    int val;};const int maxn = 5 + 5;SegTree Tree[maxn * 4];void Build(int low, int high, int index);void Update(int low, int high, int val, int index);int Query(int low, int high, int index);int main(){    Build(1, 5, 1);//    Update(2, 3, 2, 1);//    Update(4, 5, 3, 1);//    printf("%d\n", Query(1, 3, 1));    Update(1, 3, 2, 1);    Update(4, 4, 3, 1);    printf("%d\n", Query(1, 4, 1));    printf("%d\n", Query(1, 2, 1));    for (int i=1; i<=9; ++i) {        printf("%d %d %d\n", Tree[i].left, Tree[i].right, Tree[i].val);    }    return 0;}void Build(int low, int high, int index){    Tree[index].left = low;    Tree[index].right = high;    Tree[index].val = 1;    if (low == high) {        return;    } else {        int mid = low + (high-low)/2;        Build(low, mid, index*2);        Build(mid+1, high, index*2+1);    }}void Update(int low, int high, int val, int index){    if (Tree[index].val == val || (Tree[index].left == low && Tree[index].right == high)) {        Tree[index].val = val;        return;    } else {        if (Tree[index].val != 0) {            Tree[index*2].val = Tree[index].val;            Tree[index*2+1].val = Tree[index].val;            Tree[index].val = 0;        }        int mid = Tree[index].left + (Tree[index].right - Tree[index].left)/2;        if (high <= mid) {            Update(low, high, val, index*2);        } else if (low > mid) {            Update(low, high, val, index*2+1);        } else {            Update(low, mid, val, index*2);            Update(mid+1, high, val, index*2+1);        }    }}int Query(int low, int high, int index){    int mid = Tree[index].left + (Tree[index].right - Tree[index].left)/2;    if (Tree[index].left == low && Tree[index].right == high) {        if (Tree[index].val != 0) {            return Tree[index].val * (high-low+1);        } else {            return Query(low, mid, index*2) + Query(mid+1, high, index*2+1);        }    } else {        if (high <= mid) {            return Query(low, high, index*2);        } else if (low > mid) {            return Query(low, high, index*2+1);        } else {            return Query(low, mid, index*2) + Query(mid+1, high, index*2+1);        }    }}

void Update(int low, int high, int val, int index);Update(2, 3, 2, 1);Update(4, 5, 3, 1);Query(1, 3, 1) = 5;

Update(1, 3, 2, 1);Update(4, 4, 3, 1);Query(1, 4, 1) = 9;Query(1, 2, 1) = 2;

#include <iostream>#include <cstdio>#include <cstring>using namespace std;struct SegTree {    int left, right;    int sum;    int val;};const int maxn = 5 + 5;int num[maxn];SegTree Tree[maxn * 4];void Build(int low, int high, int index);void Update(int low, int high, int add, int index);int Query(int low, int high, int index);int main(){    for (int i=1; i<=5; ++i) {        num[i] = i;    }    Build(1, 5, 1);    Update(2, 3, 2, 1);    printf("%d\n", Query(2, 2, 1));    printf("%d\n", Query(3, 4, 1));    Update(4, 5, 3, 1);    printf("%d\n", Query(5, 5, 1));    for (int i=1; i<=9; ++i) {        printf("%d %d %d\n", Tree[i].left, Tree[i].right, Tree[i].sum);    }    return 0;}void Build(int low, int high, int index){    Tree[index].left = low;    Tree[index].right = high;    Tree[index].val = 0;    if (low == high) {        Tree[index].sum = num[low];        return;    } else {        int mid = low + (high-low)/2;        Build(low, mid, index*2);        Build(mid+1, high, index*2+1);        Tree[index].sum = Tree[index*2].sum + Tree[index*2+1].sum;    }}void Update(int low, int high, int add, int index){    if (Tree[index].left == low && Tree[index].right == high) {        Tree[index].val += add;        return;    } else {        Tree[index].sum += add * (high-low+1);        int mid = Tree[index].left + (Tree[index].right - Tree[index].left)/2;        if (high <= mid) {            Update(low, high, add, index*2);        } else if (low > mid) {            Update(low, high, add, index*2+1);        } else {            Update(low, mid, add, index*2);            Update(mid+1, high, add, index*2+1);        }    }}int Query(int low, int high, int index){    if (Tree[index].left == low && Tree[index].right == high) {        return Tree[index].sum + (high-low + 1) * Tree[index].val;    } else {        Tree[index].sum += (Tree[index].right-Tree[index].left + 1) * Tree[index].val;        int mid = Tree[index].left + (Tree[index].right - Tree[index].left)/2;        Update(Tree[index].left, mid, Tree[index].val, index*2);        Update(mid+1, Tree[index].right, Tree[index].val, index*2+1);        Tree[index].val = 0;        if (high <= mid) {            return Query(low, high, index*2);        } else if (low > mid) {            return Query(low, high, index*2+1);        } else {            return Query(low, mid, index*2) + Query(mid+1, high, index*2+1);        }    }}

void Update(int low, int high, int add, int index);Update(2, 3, 2, 1);

Query(2, 2, 1) = 4;Query(3, 4, 1) = 9;

Update(4, 5, 3, 1);

Query(5, 5, 1)

Query(5, 5, 1) = 8;