Segment Tree

Applications:

http://poj.org/summerschool/gw_interval_tree.pdf

1. find range minimum/maximum

The structure of Segment Tree is a binary tree which each node has two attributes start and end denote an segment / interval.

start and end are both integers, they should be assigned in following rules:

• The root's start and end is given by build method.
• The left child of node A has start=A.left, end=(A.left + A.right) / 2.
• The right child of node A has start=(A.left + A.right) / 2 + 1, end=A.right.
• if start equals to end, there will be no children for this node.

Segment Tree (a.k.a Interval Tree) is an advanced data structure which can support queries like:

• which of these intervals contain a given point
• which of these points are in a given interval

1. Build segment tree

/ Definition of SegmentTreeNode: public class SegmentTreeNode { public int start, end; public SegmentTreeNode left, right; public SegmentTreeNode(int start, int end) { this.start = start, this.end = end; this.left = this.right = null; } } / public class Solution { / @param start, end: Denote an segment / interval @return: The root of Segment Tree */ public SegmentTreeNode build(int start, int end) { // write your code here if (start > end) { return null; } if (start == end) { return new SegmentTreeNode(start, end); } int mid = start + (end-start)/2; SegmentTreeNode root = new SegmentTreeNode(start, end); root.left = build(start, mid); root.right = build(mid+1, end); return root; } }

2. Build segment tree with max value of the interval

/ Definition of SegmentTreeNode: public class SegmentTreeNode { public int start, end, max; public SegmentTreeNode left, right; public SegmentTreeNode(int start, int end, int max) { this.start = start; this.end = end; this.max = max this.left = this.right = null; } } / public class Solution { / @param A: a list of integer @return: The root of Segment Tree */ public SegmentTreeNode build(int[] A) { // write your code here if (A == null || A.length == 0) { return null; } return build(A, 0, A.length-1); } public SegmentTreeNode build(int[]A, int start, int end) { // write your code here if (start > end) { return null; } if (start == end) { return new SegmentTreeNode(start, end, A[start]); } int mid = start + (end-start)/2; SegmentTreeNode root = new SegmentTreeNode(start, end, 0); root.left = build(A, start, mid); root.right = build(A, mid+1, end); root.max = Math.max(root.left.max, root.right.max); return root; } }

3. Query the max value in a given interval

Runtime 10s / Definition of SegmentTreeNode: public class SegmentTreeNode { public int start, end, max; public SegmentTreeNode left, right; public SegmentTreeNode(int start, int end, int max) { this.start = start; this.end = end; this.max = max this.left = this.right = null; } } / public class Solution { / @param root, start, end: The root of segment tree and an segment / interval @return: The maximum number in the interval [start, end] / public int query(SegmentTreeNode root, int start, int end) { // write your code here if (root == null || start > root.end || end < root.start) { return 0; } if (root.start == root.end) { return root.max; } int mid = ((root.start + root.end) / 2); return Math.max(query(root.left, start, end), query(root.right, start, end)); } } Runtime 5s public class Solution { /* @param root, start, end: The root of segment tree and an segment / interval @return: The maximum number in the interval [start, end] */ public int query(SegmentTreeNode root, int start, int end) { // write your code here if (root == null || start > root.end || end < root.start) { return 0; } if (root.start >= start && root.end <= end) { return root.max; } int mid = ((root.start + root.end) / 2); return Math.max(query(root.left, start, Math.min(mid, end)), query(root.right, Math.max(mid, start), end)); } }

4. Query the count of numbers in a given interval

/ Definition of SegmentTreeNode: public class SegmentTreeNode { public int start, end, count; public SegmentTreeNode left, right; public SegmentTreeNode(int start, int end, int count) { this.start = start; this.end = end; this.count = count; this.left = this.right = null; } } / public class Solution { / @param root, start, end: The root of segment tree and an segment / interval @return: The count number in the interval [start, end] / public int query(SegmentTreeNode root, int start, int end) { // write your code here if (root == null || start > end) { return 0; } if (start <= root.start && end >= root.end) { return root.count; } int mid = root.start + (root.end-root.start)/2; return query(root.left, start, Math.min(mid, end)) + query(root.right, Math.max(mid, start), end); } }