Segment Tree

Segment Tree is a data structure that allows efficient querying and updating of intervals or segments of an array.

• It is particularly useful for problems involving range queries, such as finding the sum, minimum, maximum, or any other operation over a specific range of elements in an array.
• The tree is built recursively by dividing the array into segments until each segment represents a single element.
• This structure enables fast query and update operations with a time complexity of O(log n), making it a powerful tool in algorithm design and optimization .

The following diagram shows a segment tree built for an array [1, 3, 5, 7, 9, 11] of size 5. The example tree is built for range sum queries. Every node stores sum of a range. The root nodes stores sum of the whole array and leaf nodes store sums of single elements in the array. Please refer Segment Tree Introduction article for details about construction and query.

Types of Operations:

The operations that the segment tree can perform must be binary and associative. Some of the examples of operations are:

• Finding Range Sum Queries
• Searching index with given prefix sum
• Finding Range Maximum/Minimum
• Counting frequency of Range Maximum/Minimum
• Finding Range GCD/LCM
• Finding Range AND/OR/XOR
• Finding number of zeros in the given range or finding index of Kth zero

Basics of Segment Tree:

• Introduction
• Persistent Segment Tree
• Efficient implementation
• Iterative Segment Tree
• Range Sum and Update in Array
• Dynamic Segment Trees
• Applications, Advantages and Disadvantages

Lazy Propagation:

• Lazy Propagation
• Lazy Propagation | Set 2
• Flipping Sign Problem

Range Queries:

• Queries for any non-repeating element
• Range Minimum Query
• Querying maximum number of divisors
• Min-Max Range Queries in Array
• Range LCM Queries
• Number of primes in a subarray
• Largest Sum Contiguous Subarray
• Longest Correct Bracket Subsequence
• Maximum Occurrence in a Given Range
• Maximum product pair in range with updates
• Chessboard Pieces
• Number of subsets equal to a given String
• Minimum distance between two Zeros
• Queries to evaluate the given equation in a range [L, R]
• Maximum weight in given price range for Q queries

Some interesting problem on Segment Tree:

• Longest Increasing Subsequences (LIS)
• Longest subarray consisting of same elements by at most K decrements
• Generate original permutation from given array of inversions
• Maximum of all subarrays of size K
• Build a segment tree for N-ary rooted tree
• Count number of increasing sub-sequences : O(NlogN)
• Calculate the Sum of GCD over all subarrays
• Cartesian tree from inorder traversal
• LIS using Segment Tree
• Reconstructing Segment Tree

Applications of Segment Tree:

• Interval scheduling: Segment trees can be used to efficiently schedule non-overlapping intervals, such as scheduling appointments or allocating resources.
• Range-based statistics: Segment trees can be used to compute range-based statistics such as variance, standard deviation, and percentiles.
• Image processing: Segment trees are used in image processing algorithms to divide an image into segments based on color, texture, or other attributes.