Segment Tree (Range Query Tree)
Segment Tree is a binary tree data structure for storing intervals/segments that enables efficient range queries (sum, min, max, GCD) and updates in O(log n) time. Each node represents an interval, with leaves representing single elements. Essential for competitive programming, it solves problems like range sum queries with updates, which would be O(n) with arrays but becomes O(log n) with segment trees.
Visualization
Interactive visualization for Segment Tree (Range Query Tree)
Interactive visualization with step-by-step execution
Implementation
1class SegmentTree {
2 private tree: number[];
3 private n: number;
4
5 constructor(arr: number[]) {
6 this.n = arr.length;
7 this.tree = new Array(4 * this.n).fill(0);
8 this.build(arr, 0, 0, this.n - 1);
9 }
10
11 private build(arr: number[], node: number, start: number, end: number): void {
12 if (start === end) {
13 this.tree[node] = arr[start];
14 return;
15 }
16
17 const mid = Math.floor((start + end) / 2);
18 const leftChild = 2 * node + 1;
19 const rightChild = 2 * node + 2;
20
21 this.build(arr, leftChild, start, mid);
22 this.build(arr, rightChild, mid + 1, end);
23
24 this.tree[node] = this.tree[leftChild] + this.tree[rightChild];
25 }
26
27 query(L: number, R: number): number {
28 return this.queryRange(0, 0, this.n - 1, L, R);
29 }
30
31 private queryRange(node: number, start: number, end: number, L: number, R: number): number {
32 // No overlap
33 if (R < start || L > end) return 0;
34
35 // Complete overlap
36 if (L <= start && end <= R) return this.tree[node];
37
38 // Partial overlap
39 const mid = Math.floor((start + end) / 2);
40 const leftSum = this.queryRange(2 * node + 1, start, mid, L, R);
41 const rightSum = this.queryRange(2 * node + 2, mid + 1, end, L, R);
42
43 return leftSum + rightSum;
44 }
45
46 update(index: number, value: number): void {
47 this.updateValue(0, 0, this.n - 1, index, value);
48 }
49
50 private updateValue(node: number, start: number, end: number, index: number, value: number): void {
51 if (start === end) {
52 this.tree[node] = value;
53 return;
54 }
55
56 const mid = Math.floor((start + end) / 2);
57 if (index <= mid) {
58 this.updateValue(2 * node + 1, start, mid, index, value);
59 } else {
60 this.updateValue(2 * node + 2, mid + 1, end, index, value);
61 }
62
63 this.tree[node] = this.tree[2 * node + 1] + this.tree[2 * node + 2];
64 }
65}Deep Dive
Theoretical Foundation
Segment Tree divides array into segments. Each internal node stores aggregate information (sum/min/max) for its segment. Root represents entire array [0,n-1], each child represents half of parent's range. Leaves represent single elements. For range [l,r] query, recursively query relevant segments, combining at most O(log n) nodes. For update, modify path from leaf to root (O(log n) nodes). Space: O(4n). Lazy propagation optimizes range updates from O(n log n) to O(log n).
Complexity
Time
O(log n)
O(log n)
O(log n)
Space
O(n)
Applications
Industry Use
Competitive programming contests
Database range queries
Computational geometry
Graphics rendering (range min/max queries)
Stock market analysis (range statistics)
Network monitoring (aggregate statistics)
Use Cases
Related Algorithms
AVL Tree (Self-Balancing BST)
A self-balancing binary search tree where the heights of the two child subtrees of any node differ by at most one. Named after inventors Adelson-Velsky and Landis (1962), AVL trees guarantee O(log n) time for search, insert, and delete operations by maintaining balance through rotations. It was the first self-balancing binary search tree to be invented.
Fenwick Tree (Binary Indexed Tree - BIT)
A data structure that can efficiently update elements and calculate prefix sums in O(log n) time. Also known as Binary Indexed Tree (BIT), invented by Peter Fenwick in 1994. Fenwick trees are more space-efficient and simpler than segment trees for range sum queries, using only O(n) space compared to segment tree's O(4n).
Lowest Common Ancestor (LCA)
Lowest Common Ancestor (LCA) finds the deepest node that is an ancestor of two given nodes in a tree. This fundamental tree query operation has applications in computational biology (phylogenetic trees), version control systems (finding merge base), network routing, and range queries. Multiple algorithms exist: simple recursive O(n), binary lifting O(log n) per query after O(n log n) preprocessing, and Tarjan's offline algorithm for batch queries.
Morris Traversal (Threaded Binary Tree)
Morris Traversal is a brilliant tree traversal technique that achieves O(1) space complexity without using recursion or an explicit stack. Invented by Joseph H. Morris in 1979, it temporarily modifies the tree structure by creating 'threads' (temporary links) that allow the algorithm to traverse back to ancestors. After traversal, the original tree structure is restored. This makes it ideal for memory-constrained environments and demonstrates creative problem-solving in algorithm design.