segment树(线段树)

线段树(segment tree)是一种Binary Search Tree或者叫做ordered binary tree。对于线段树中的每一个非叶子节点[a,b]，它的左子树表示的区间为[a,(a+b)/2]，右子树表示的区间为[(a+b)/2+1,b]。如下图：

[0-2]

/ \

[0-1] [2-2]

/ \

[0-0] [1-1]

下面看一道leetcode上的题，求动态区间的和(Range Sum Query - Mutable)，题目如下：

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

分析如下：

一、构造线段树节点：

class SegmentTreeNode {
int start, end;
int sum;
SegmentTreeNode ltree, rtree;
public SegmentTreeNode(int s, int e) {
start = s;
end = e;
}
}

二、建立线段树(根据数组nums，建立一个动态区间求和的线段树)：

public SegmentTreeNode buildTree(int[] nums, int left, int right) {
SegmentTreeNode root = new SegmentTreeNode(left, right);
if (left == right) {
root.sum = nums[left];
} else {
int mid = left + (right - left)/2;
root.ltree = buildTree(nums, left, mid);
root.rtree = buildTree(nums, mid+1, right);
root.sum = root.ltree.sum + root.rtree.sum;
}
return root;
}

三、线段树的更新(更新int数组下标i的值为val)：

private void update(SegmentTreeNode root, int i, int val) {
if (root.start == root.end) {
root.sum = val;
} else {
int mid = root.start + (root.end-root.start)/2;
if (i <= mid) {
update(root.ltree, i, val);
} else {
update(root.rtree, i, val);
}
root.sum = root.ltree.sum + root.rtree.sum;
}
}

四、线段树的查询(查询int数组下标 i 到 j 的元素之和)：

private int sumRange(SegmentTreeNode root, int i, int j) {
if (root.start == i && root.end == j) {
return root.sum;
} else {
int mid = root.start + (root.end - root.start)/2;
if (j <= mid) {
return sumRange(root.ltree, i, j);
} else if (i > mid) {
return sumRange(root.rtree, i, j);
} else {
return sumRange(root.ltree, i, root.ltree.end) + sumRange(root.rtree, root.rtree.start, j);
}
}
}

综上所述，上面Range Sum Query - Mutable的AC代码如下：

class SegmentTreeNode {
int start, end;
int sum;
SegmentTreeNode ltree, rtree;
public SegmentTreeNode(int s, int e) {
start = s;
end = e;
}
}

public class NumArray {
SegmentTreeNode root = null;

public NumArray(int[] nums) {
if(nums == null || nums.length == 0) {
return;
}
root = buildTree(nums, 0, nums.length-1);
}

public SegmentTreeNode buildTree(int[] nums, int left, int right) {
SegmentTreeNode root = new SegmentTreeNode(left, right);
if (left == right) {
root.sum = nums[left];
} else {
int mid = left + (right - left)/2;
root.ltree = buildTree(nums, left, mid);
root.rtree = buildTree(nums, mid+1, right);
root.sum = root.ltree.sum + root.rtree.sum;
}
return root;
}

void update(int i, int val) {
update(root, i, val);
}

private void update(SegmentTreeNode root, int i, int val) {
if (root.start == root.end) {
root.sum = val;
} else {
int mid = root.start + (root.end-root.start)/2;
if (i <= mid) {
update(root.ltree, i, val);
} else {
update(root.rtree, i, val);
}
root.sum = root.ltree.sum + root.rtree.sum;
}
}

public int sumRange(int i, int j) {
return sumRange(root, i, j);
}

private int sumRange(SegmentTreeNode root, int i, int j) {
if (root.start == i && root.end == j) {
return root.sum;
} else {
int mid = root.start + (root.end - root.start)/2;
if (j <= mid) {
return sumRange(root.ltree, i, j);
} else if (i > mid) {
return sumRange(root.rtree, i, j);
} else {
return sumRange(root.ltree, i, root.ltree.end) + sumRange(root.rtree, root.rtree.start, j);
}
}
}
}