array_range_query

A high-performance, generic Rust implementation of Segment Trees and Lazy Segment Trees for efficient range queries, range updates, and interval operations.

Perfect for competitive programming, algorithm optimization, and solving range query problems with O(log n) time complexity.

Features

• Segment Tree: O(log n) point updates and range queries for any associative operation
• Lazy Segment Tree: O(log n) range updates and range queries with lazy propagation
• Generic Design: Type-safe specifications for custom operations
• Helper Types: Pre-built implementations for sum, min, max operations
• Zero-cost Abstractions: No runtime overhead from generics

What is a Segment Tree?

A Segment Tree is a versatile data structure that allows you to:

• Answer range queries (like sum, min, max, GCD) on an array in O(log n) time
• Perform point updates (modify a single element) in O(log n) time
• Handle any associative operation (operations where (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c))

A Lazy Segment Tree extends this capability to efficiently handle:

• Range updates (modify multiple elements at once) in O(log n) time
• Lazy propagation to defer updates until necessary, significantly improving performance

Why Use Segment Trees?

Segment trees are essential for solving interval query problems efficiently:

Operation	Naive Approach	Segment Tree	Improvement
Range Query	O(n)	O(log n)	Exponential speedup
Point Update	O(1)	O(log n)	Slightly slower
Range Update	O(n)	O(log n) with lazy	Exponential speedup

Common Use Cases

• Competitive Programming: Solving range query problems (Codeforces, LeetCode, AtCoder)
• Database Systems: Interval queries and aggregate functions
• Game Development: Collision detection, spatial queries
• Financial Analysis: Time-series analysis, moving window calculations
• Computer Graphics: Rectangle queries, spatial indexing
• Network Monitoring: Bandwidth tracking, latency analysis

Comparison with Other Data Structures

Data Structure	Range Query	Point Update	Range Update	Best For
Array	O(n)	O(1)	O(n)	Simple access
Prefix Sum	O(1)	O(n)	O(n)	Immutable sum queries
Segment Tree	O(log n)	O(log n)	O(n)	Dynamic range queries
Lazy Segment Tree	O(log n)	O(log n)	O(log n)	Range updates
Fenwick Tree (BIT)	O(log n)	O(log n)	-	Sum/XOR only
Sparse Table	O(1)	-	-	Immutable idempotent ops

Installation

cargo add array_range_query

Quick Start

Basic Segment Tree

use array_range_query::SegTreeSum;

let values = vec![1, 2, 3, 4, 5];
let mut tree = SegTreeSum::<i32>::from_vec(values);

assert_eq!(tree.query(1..4), 9); // Sum of [2, 3, 4]
tree.update(2, 10);              // Change index 2 to 10
assert_eq!(tree.query(..), 22);  // Total sum: 1+2+10+4+5

Lazy Segment Tree

use array_range_query::LazySegTreeAddSum;

let values = vec![1, 2, 3, 4, 5];
let mut tree = LazySegTreeAddSum::<i32>::from_vec(values);

assert_eq!(tree.query(1..4), 9); // Sum of [2, 3, 4]
tree.update(1..4, 10);           // Add 10 to range [1, 4)
assert_eq!(tree.query(..), 45);  // New total: 1+12+13+14+5

Helper Types

Regular Segment Trees

• SegTreeSum<T> — Range sum queries
• SegTreeMin<T> — Range minimum queries
• SegTreeMax<T> — Range maximum queries

Lazy Segment Trees

• LazySegTreeAddSum<T> — Range add updates, sum queries
• LazySegTreeAddMin<T> — Range add updates, min queries
• LazySegTreeAddMax<T> — Range add updates, max queries
• LazySegTreeReplaceSum<T> — Range assignment updates, sum queries

Custom Operations

Define your own operations by implementing the specification traits:

use array_range_query::{SegTree, SegTreeSpec};

struct MaxSpec;
impl SegTreeSpec for MaxSpec {
    type T = i32;
    const ID: Self::T = i32::MIN;
    fn op(a: &mut Self::T, b: &Self::T) { *a = (*a).max(*b); }
}

let tree = SegTree::<MaxSpec>::from_vec(vec![3, 1, 4, 1, 5]);
assert_eq!(tree.query(..), 5); // Maximum element

For lazy segment trees:

use array_range_query::{LazySegTree, LazySegTreeSpec};

struct RangeAddSum;
impl LazySegTreeSpec for RangeAddSum {
    type T = i64; // Data type
    type U = i64; // Update type
    const ID: Self::T = 0;

    fn op_on_data(d1: &mut Self::T, d2: &Self::T) { *d1 += *d2; }
    fn op_on_update(u1: &mut Self::U, u2: &Self::U) { *u1 += *u2; }
    fn op_update_on_data(u: &Self::U, d: &mut Self::T, size: usize) {
        *d += u * size as i64;
    }
}

let mut tree = LazySegTree::<RangeAddSum>::from_vec(vec![1, 2, 3, 4, 5]);
tree.update(1..4, 10); // Add 10 to range [1, 4)
assert_eq!(tree.query(..), 45);

API Reference

SegTree

• new(size) / from_slice(values) / from_vec(values) — Construction
• query(range) — Range query in O(log n)
• update(index, value) — Point update in O(log n)

LazySegTree

• new(size) / from_slice(values) / from_vec(values) — Construction
• query(range) — Range query in O(log n)
• update(range, value) — Range update in O(log n)

Range Types

All query and update methods accept any range type:

• 2..5 (half-open)
• 2..=4 (inclusive)
• ..3 (from start)
• 2.. (to end)
• .. (entire range)

Solving Classic Problems

Problem 1: Range Sum Queries with Point Updates

Example: Given an array, answer queries for the sum of elements in any range and support updating individual elements.

use array_range_query::SegTreeSum;

let mut tree = SegTreeSum::<i32>::from_vec(vec![1, 3, 5, 7, 9, 11]);
// What's the sum from index 1 to 4?
assert_eq!(tree.query(1..4), 15); // 3 + 5 + 7
// Update index 2 to 10
tree.update(2, 10);
assert_eq!(tree.query(1..4), 20); // 3 + 10 + 7

Problem 2: Range Minimum Queries (RMQ)

Example: Find the minimum element in any subarray efficiently.

use array_range_query::SegTreeMin;

let tree = SegTreeMin::<i32>::from_vec(vec![4, 2, 8, 1, 9, 3, 6]);
assert_eq!(tree.query(1..5), 1); // min(2, 8, 1, 9) = 1
assert_eq!(tree.query(0..3), 2); // min(4, 2, 8) = 2

Problem 3: Range Updates with Range Queries

Example: Add a value to all elements in a range, then query range sums.

use array_range_query::LazySegTreeAddSum;

let mut tree = LazySegTreeAddSum::<i64>::from_vec(vec![1, 2, 3, 4, 5]);
// Add 10 to elements from index 1 to 3
tree.update(1..4, 10);
// Elements are now [1, 12, 13, 14, 5]
assert_eq!(tree.query(0..5), 45); // sum = 1+12+13+14+5

Problem 4: Range Assignment

Example: Set all elements in a range to a specific value.

use array_range_query::LazySegTreeReplaceSum;

let mut tree = LazySegTreeReplaceSum::<i32>::from_vec(vec![1, 2, 3, 4, 5]);
// Set all elements from index 1 to 4 to value 10
tree.update(1..4, 10);
// Elements are now [1, 10, 10, 10, 5]
assert_eq!(tree.query(..), 36); // sum = 1+10+10+10+5

Performance

• Construction: O(n)
• Point update: O(log n)
• Range query: O(log n)
• Range update (lazy): O(log n)
• Space: O(n)

Advanced Topics

Implementing Custom Operations

This library supports any monoid operation (associative operation with an identity element). Common examples:

• Sum (identity: 0)
• Product (identity: 1)
• Min (identity: ∞)
• Max (identity: -∞)
• GCD (identity: 0)
• Bitwise OR (identity: 0)
• Bitwise AND (identity: all 1s)

Example of implementing GCD:

use array_range_query::{SegTree, SegTreeSpec};

struct GcdSpec;
impl SegTreeSpec for GcdSpec {
    type T = u32;
    const ID: Self::T = 0;
    fn op(a: &mut Self::T, b: &Self::T) {
        *a = gcd(*a, *b);
    }
}

fn gcd(a: u32, b: u32) -> u32 {
    if b == 0 { a } else { gcd(b, a % b) }
}

let tree = SegTree::<GcdSpec>::from_vec(vec![12, 18, 24, 30]);
assert_eq!(tree.query(..), 6); // GCD of all elements

When to Use Lazy Propagation

Use Lazy Segment Trees when you need:

1. Range updates: Modifying multiple elements efficiently
2. Batch operations: Applying the same operation to intervals
3. Deferred computation: Delaying updates until queries require them

Examples: Range increment, range assignment, range multiplication.

Segment Tree vs. Other Interval Structures

Fenwick Tree (Binary Indexed Tree):

• Simpler implementation, less memory
• Only works for invertible operations (sum, XOR)
• Can't handle min/max queries
• No range updates without tricks

Sparse Table:

• O(1) query time for idempotent operations
• Immutable (no updates)
• Higher space complexity O(n log n)
• Best for static range minimum/maximum queries

Segment Tree (this library):

• Supports any associative operation
• Handles both queries and updates
• Works with lazy propagation for range updates
• Most versatile for competitive programming

Requirements

• Rust 2021 edition
• Dependencies (for helpers): num-traits, min_max_traits

Related Topics & Keywords

This library is relevant for:

• Data Structures: Segment Tree, Interval Tree, Binary Indexed Tree (BIT), Fenwick Tree, Range Tree
• Algorithms: Divide and Conquer, Lazy Propagation, Range Queries, Interval Operations
• Problem Types: Range Minimum Query (RMQ), Range Sum Query (RSQ), Range Maximum Query, Range GCD Query
• Applications: Competitive Programming, Algorithm Contests, Interval Scheduling, Computational Geometry
• Concepts: Monoid Operations, Associative Operations, Binary Trees, Tree-based Data Structures
• Performance: O(log n) queries, O(log n) updates, Efficient range operations

Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

Learn More

• Segment Tree Tutorial (CP-Algorithms)
• Lazy Propagation Explained
• Range Query Problems on LeetCode

License

MIT License. See LICENSE for details.