線段樹

線段樹

        線段樹是一棵樹，而且是二叉搜索樹。它將一個區間劃分成一些單元區間，每個單元區間對應線段樹中的一個葉結點。
        主要應用：適用於和區間統計有關的問題，例如大數據的動態修改
及多次查詢就比較適合使用這種樹，效果比較好。

        性質：

         1. 樹中的每一個結點表示了一個區間[a,b]。a,b通常是整數。
         2. 對於每一個非葉結點所表示的結點[a,b]，其左兒子表示的區間爲
         [a,(a+b)/2]，右兒子表示的區間爲[(a+b)/2,b]。
        3. 同一層的節點所代表的區間，相互不會重疊。
         4. 葉子節點的區間是單位長度，不可再分。

        構造過程：
         1. 構造節點node,對應的區間爲[start,end]
         2. 構造node的左孩子，對應的區間爲[start,start + (end - start) / 2]
         3. 構造node右孩子，對應的區間爲[start + (end - start) / 2 + 1,end]

        線段樹的構造過程就是二叉樹遞歸的創建的過程，代碼如下：

SegmentTreeNode* SegmentTree::GenerateTree(vector<int> &nums, int start, int end) {
    if(start > end) return nullptr;
    SegmentTreeNode *node = new SegmentTreeNode(start,end);
    if(start == end) {
        node->SetSum(nums[start]);
        return node;
    }
    int mid = start + (end - start) / 2;
    SegmentTreeNode *left = GenerateTree(nums,start,mid);
    SegmentTreeNode *right = GenerateTree(nums,mid+1,end);
    node->SetLeft(left);
    node->SetRight(right);
    node->SetSum(left->GetSum() + right->GetSum());
    return node;
}

        其中SegmentTreeNode 以及 SegmentTree爲自己定義的類，最後附上整個的代碼。

#include <vector>
using namespace std;

class SegmentTreeNode {
public:
    SegmentTreeNode(int a, int b);
    int GetStart();
    int GetEnd();
    void SetSum(int sum);
    int GetSum();
    void SetLeft(SegmentTreeNode* left);
    SegmentTreeNode* GetLeft();
    void SetRight(SegmentTreeNode* right);
    SegmentTreeNode* GetRight();

private:
    int start, end, sum;
    SegmentTreeNode* left;
    SegmentTreeNode* right;

};


class SegmentTree {
public:
    void BuildTree(vector<int> &nums, int start, int end);
    SegmentTreeNode *GetRoot();
    int ModifyTree(int i, int val);
    int QueryTree(int i, int j);

private:
    SegmentTreeNode *GenerateTree(vector<int> &nums, int start, int end);
    int _ModifyTree(int i, int val,SegmentTreeNode *node);
    int _QueryTree(int i, int j,SegmentTreeNode *node);
    SegmentTreeNode *root;
};


SegmentTreeNode::SegmentTreeNode(int a, int b):start(a),
                                               end(b),
                                               sum(0),
                                               left(nullptr),
                                               right(nullptr) {



};

int SegmentTreeNode::GetStart() {
    return start;
}
int SegmentTreeNode::GetEnd() {
    return end;
}
void SegmentTreeNode::SetSum(int sum) {
    this->sum = sum;
}
int SegmentTreeNode::GetSum() {
    return sum;
}

void SegmentTreeNode::SetLeft(SegmentTreeNode* left) {
    this->left = left;
}
SegmentTreeNode* SegmentTreeNode::GetLeft() {
    return left;
}
void SegmentTreeNode::SetRight(SegmentTreeNode* right) {
    this->right = right;
}

SegmentTreeNode* SegmentTreeNode::GetRight() {
    return right;
}

void SegmentTree::BuildTree(vector<int> &nums, int start, int end) {
    root = GenerateTree(nums, start, end);
}

SegmentTreeNode* SegmentTree::GenerateTree(vector<int> &nums, int start, int end) {
    if(start > end) return nullptr;
    SegmentTreeNode *node = new SegmentTreeNode(start,end);
    if(start == end) {
        node->SetSum(nums[start]);
        return node;
    }
    int mid = start + (end - start) / 2;
    SegmentTreeNode *left = GenerateTree(nums,start,mid);
    SegmentTreeNode *right = GenerateTree(nums,mid+1,end);
    node->SetLeft(left);
    node->SetRight(right);
    node->SetSum(left->GetSum() + right->GetSum());
    return node;
}

int SegmentTree::ModifyTree(int i, int val) {
    return _ModifyTree(i, val, root);
}

int SegmentTree::QueryTree(int i, int j) {
    return _QueryTree(i, j, root);
}

int SegmentTree::_ModifyTree(int i, int val,SegmentTreeNode *node) {
    if(node == nullptr) return 0;
    int diff;
    if(node->GetStart() == i && node->GetEnd() == i) {
        diff = val - node->GetSum();
        node->SetSum(val);
        return diff;
    }
    int mid = (node->GetStart() + node->GetEnd()) / 2;
    if(i > mid) {
        diff = _ModifyTree(i,val,node->GetRight());
    } else {
        diff = _ModifyTree(i,val,node->GetLeft());
    }
    node->SetSum(node->GetSum() + diff);
    return diff;
}

int SegmentTree::_QueryTree(int i, int j,SegmentTreeNode *node) {
    if(node == nullptr) return 0;
    if(node->GetStart() == i && node->GetEnd() == j) return node->GetSum();
    int mid = (node->GetStart() + node->GetEnd()) / 2;
    if(i > mid) return _QueryTree(i,j,node->GetRight());
    if(j <= mid) return _QueryTree(i,j,node->GetLeft());
    return _QueryTree(i,mid,node->GetLeft()) + _QueryTree(mid+1,j,node->GetRight());
}

        利用此代碼在https://leetcode.com上AC過了兩道題目。

Question 1: Given an integer array nums, find the sum of the elements
between indices i and j (i ≤ j), inclusive.

Example: Given nums = [-2, 0, 3, -5, 2, -1]

sumRange(0, 2) -> 1 sumRange(2, 5) -> -1 sumRange(0, 5) -> -3 Note:
You may assume that the array does not change. There are many calls to
sumRange function.

Question 2: Given an integer array nums, find the sum of the elements
between indices i and j (i ≤ j), inclusive.

The update(i, val) function modifies nums by updating the element at
index i to val. Example: Given nums = [1, 3, 5]

sumRange(0, 2) -> 9 update(1, 2) sumRange(0, 2) -> 8 Note: The array
is only modifiable by the update function. You may assume the number
of calls to update and sumRange function is distributed evenly.

        源碼的下載地址：https://github.com/FyhSky/SegmentTree

        轉載請註明出處：http://blog.csdn.net/skynullcode