SuperIntervals

A fast, memory-efficient data structure for interval intersection queries. SuperIntervals uses a novel superset-index approach that maintains intervals in position-sorted order, enabling cache-friendly searches and SIMD-optimized counting.

Features:

• Linear-time index construction from sorted intervals
• Cache-friendly querying
• SIMD acceleration (AVX2/Neon) for counting operations
• Minimal memory overhead (one size_t per interval)
• Available for C++, Rust, Python, and C
• Optional Eytzinger memory layout for slightly faster queries (C++/Rust only)
• No dependencies, header only

Quick Start

• Intervals are considered end-inclusive
• The index() function must be called before any queries
• Found intervals are returned in reverse position-sorted order

🐍 Python

from superintervals import IntervalSet

iset = IntervalSet()
iset.add(10, 20, 'A')
iset.index()
overlaps = iset.find_overlaps(8, 20)

⚙️ C++

#include "SuperIntervals.hpp"

SuperIntervals<int, std::string> intervals;
intervals.add(1, 5, "A");
intervals.index();
std::vector<std::string> results;
intervals.findOverlaps(4, 9, results);

🦀 Rust

use super_intervals::SuperIntervals;

let mut intervals = SuperIntervals::new();
intervals.add(1, 5, "A");
intervals.index();
let mut results = Vec::new();
intervals.find_overlaps(4, 11, &mut results);

Test programs

Test programs expect plain text BED files and only assess chr1 records - other chromosomes are ignored.

C++ program compares SuperIntervals, ImplicitIntervalTree, IntervalTree and NCLS:

cd test; make
./run-cpp-libs a.bed b.bed

Rust program:

RUSTFLAGS="-Ctarget-cpu=native" cargo run --release --example bed-intersect-si
cargo run --release --example bed-intersect-si a.bed b.bed

Benchmark

Benchmark

SuperIntervals (SI) was compared with:

Coitrees (Rust: https://github.com/dcjones/coitrees)
Implicit Interval Tree (C++: https://github.com/lh3/cgranges)
Interval Tree (C++: https://github.com/ekg/intervaltree)
Nested Containment List (C: https://github.com/pyranges/ncls/tree/master/ncls/src)

Main results:

• Finding interval intersections is roughly ~1.5-3x faster than the next best library (Coitrees for Rust, Implicit Interval Tree for C++), with some exceptions. Coitrees-s was faster for one test (ONT reads, sorted DB53 reads).
• The SIMD counting performance of coitrees and superintervals is similar.

Datasets:

1. `rna / anno` RNA-seq reads and annotations from cgranges repository
2. `ONT reads` nanopore alignments from sample PAO33946 chr1, converted to bed format
3. `DB53 reads` paired-end reads from sample DB53, NCBI BioProject PRJNA417592, chr1, converted to bed format
4. `mito-b, mito-a` paired-end reads from sample DB53 chrM, converted to bed format (mito-b and mito-a are the same)
5. `genes` UCSC genes from hg19

Test programs use internal timers and print data to stdout, measuring the index time, and time to find all intersections. Other steps such as file IO are ignored. Test programs also only assess chr1 bed records - other chromosomes are ignored. For 'chrM' records, the M was replaced with 1 using sed. Data were assessed in position sorted and random order. Datasets can be found on the Releases page, and the test/run_tools.sh script has instructions for how to repeat the benchmark.

Timings were in microseconds using an i9-11900K, 64 GB, 2TB NVMe machine.

Finding interval intersections

Coitrees-s uses the SortedQuerent version of coitrees
SI = superintervals. Eytz refers to the eytzinger layout. -rs is the Rust implementation.

Intervals in sorted order

	Coitrees	Coitrees-s	SuperIntervals-rs	SuperIntervalsEytz-rs	ImplicitITree-C++	IntervalTree-C++	NCLS-C	SuperIntervals-C++	SuperIntervalsEytz-C++
DB53 reads, ONT reads	1668	3179	757	757	3831	44404	10642	1315	1358
DB53 reads, genes	55	84	21	21	122	109	291	42	40
ONT reads, DB53 reads	6504	3354	3859	3854	17949	12280	30772	5290	4462
anno, rna	50	35	18	18	127	90	208	29	22
genes, DB53 reads	1171	1018	301	296	3129	1315	1780	442	323
mito-b, mito-a	34769	34594	16971	16952	93900	107660	251707	33177	32985
rna, anno	31	23	21	20	70	55	233	28	27

Intervals in random order

	Coitrees	Coitrees-s	SuperIntervals-rs	SuperIntervalsEytz-rs	ImplicitITree-C++	IntervalTree-C++	NCLS-C	SuperIntervals-C++	SuperIntervalsEytz-C++
DB53 reads, ONT reads	2943	4663	1356	1355	6505	46743	11947	2491	2169
DB53 reads, genes	78	130	27	26	170	125	305	58	51
ONT reads, DB53 reads	16650	18931	16116	16037	38677	27832	53452	23003	23232
anno, rna	89	105	54	54	188	143	294	58	60
genes, DB53 reads	2222	2424	1693	1684	4490	2701	3605	1251	1749
mito-b, mito-a	38030	86309	18326	18368	125336	118321	256293	42195	41695
rna, anno	53	73	45	45	137	83	311	52	52

Counting interval intersections

Intervals in sorted order

	Coitrees	SuperIntervals-rs	SuperIntervalsEytz-rs	SuperIntervals-C++	SuperIntervalsEytz-C++
DB53 reads, ONT reads	551	370	371	241	263
DB53 reads, genes	28	12	12	8	7
ONT reads, DB53 reads	2478	1909	1890	2209	1312
anno, rna	26	14	14	22	11
genes, DB53 reads	747	321	336	446	290
mito-b, mito-a	6894	6727	6746	3088	2966
rna, anno	9	13	13	12	10

Intervals in random order

	Coitrees	SuperIntervals-rs	SuperIntervalsEytz-rs	SuperIntervals-C++	SuperIntervalsEytz-C++
DB53 reads, ONT reads	1988	972	969	1016	778
DB53 reads, genes	53	20	20	16	13
ONT reads, DB53 reads	6692	8864	8733	8182	9523
anno, rna	52	49	48	47	50
genes, DB53 reads	1503	1628	1592	1120	1623
mito-b, mito-a	14354	7579	7600	4442	4383
rna, anno	22	30	29	25	25

Python

Install using pip install superintervals

from superintervals import IntervalSet

iset = IntervalSet()

# Add interval start, end, identifier. Integer values are supported
iset.add(10, 20, 0)
iset.add(19, 18, 1)
iset.add(8, 11, 2)

# Index method must be called before queries
iset.index()

iset.any_overlaps(8, 20)
# >>> True

iset.count_overlaps(8, 20)
# >>> 3

iset.find_overlaps(8, 20)
# >>> [1, 0, 2]

iset.set_search_interval(8, 20)
for itv in iset:
    print(itv)

# >>> (19, 18, 1) 
# >>> (10, 20, 0) 
# >>> (8, 11, 2)

Cpp

#include <iostream>
#include <vector>
#include "SuperIntervals.hpp"

int main() {
    // Create a SuperIntervals instance for integer intervals with string data
    // Specify with S, T template types
    SuperIntervals<int, std::string> intervals;

    // Add some intervals
    intervals.add(1, 5, "Interval A");
    intervals.add(3, 7, "Interval B");
    intervals.add(6, 10, "Interval C");
    intervals.add(8, 12, "Interval D");

    // Index the intervals (must be called before querying)
    intervals.index();

    // Find overlaps for the range [4, 9]
    std::vector<std::string> overlaps;
    intervals.findOverlaps(4, 9, overlaps);

    // Print the overlapping intervals
    for (const auto& interval : overlaps) {
        std::cout << interval << std::endl;
    }
    
    // Count the intervals instead
    std::cout << "Count: " << intervals.countOverlaps(4, 9) << std::endl;
    
    // Count stabbed intervals at point 7
    std::cout << "Number of intervals containing point 7: " << intervals.countStabbed(7) << std::endl;

    return 0;
}

There is also a SuperIntervalsEytz subclasses that can be used. SuperIntervalsEytz uses an Eytzinger memory layout that can sometimes offer faster query times at the cost of higher memory usage and slower indexing time.

Rust

Fetch using cargo add.

use super_intervals::SuperIntervals;

fn main() {
    // Create a new instance of SuperIntervals
    let mut intervals = SuperIntervals::new();

    // Add some intervals with associated data of type T
    intervals.add(1, 5, "Interval A");
    intervals.add(10, 15, "Interval B");
    intervals.add(7, 12, "Interval C");

    // Call index() to prepare the intervals for queries
    intervals.index();

    // Query for overlapping intervals with a range (4, 11)
    let mut found_intervals = Vec::new();
    intervals.find_overlaps(4, 11, &mut found_intervals);
    
    // Display found intervals
    for interval in found_intervals {
        println!("Found overlapping interval: {}", interval);
    }

    // Count overlaps with a range (4, 11)
    let overlap_count = intervals.count_overlaps(4, 11);
    println!("Number of overlapping intervals: {}", overlap_count);
}

There is also SuperIntervalsEytz implementation. SuperIntervalsEytz uses an Eytzinger memory layout that can sometimes offer faster query times at the cost of higher memory usage and slower indexing time.

Acknowledgements

• The rust test program borrows heavily from the coitrees package
• The superset-index implemented here exploits a similar interval ordering as described in Schmidt 2009 "Interval Stabbing Problems in Small Integer Ranges". However, the superset-index has several advantages including
  1. An implicit memory layout
  2. General purpose implementation (not just small integer ranges)
  3. SIMD counting algorithm
• The Eytzinger layout was adapted from Sergey Slotin, Algorithmica