Longest Sorted Bucket Sequence: An Algorithmic Bridge Between LCS and 2D LIS

Official implementation and research repository for the Longest Sorted Bucket Sequence (LSBS) algorithm, a unique algorithmic approach that bridges the Longest Common Subsequence (LCS) and 2D Longest Increasing Subsequence (LIS) problems.

See online benchmark demo here.

Overview

LSBS is a unique algorithm that solves the Longest Common Subsequence problem by transforming it into a 2D Longest Increasing Subsequence problem through the use of a Common Element Table (CET). The approach provides both theoretical insights and practical implementations for sequence analysis.

Key Innovation

• LSBS Algorithm: Transforms LCS computation into 2D LIS finding
• Common Element Table (CET): Efficiently maps character occurrences between sequences
• Multiple LIS Backends: Supports both Dynamic Programming and Patience Sorting approaches
• Performance Analysis: Comprehensive comparison with classical LCS algorithms

Quick Start

Interactive Mode

python __main__.py

Programmatic Usage

from lcs_lsbs import lcs_lsbs

# Basic usage
sequence1 = "ABCBDAB"
sequence2 = "BDCAB"
length, lcs = lcs_lsbs(sequence1, sequence2)
print(f"LCS length: {length}, LCS: '{lcs}'")  # Output: LCS length: 4, LCS: 'BDAB'

# Using different LIS algorithms
from lis_funcs import lis_ps_2d
length, lcs = lcs_lsbs(sequence1, sequence2, f=lis_ps_2d)

Algorithm Implementations

This repository includes multiple algorithm implementations for comparative analysis:

LCS Algorithms

• LCS-LSBS (lcs_lsbs.py): Our novel LSBS-based approach
• Classic DP (lcs_dp.py): Traditional dynamic programming solution
• Recursive variants (lcs_rec.py): Naive, memoized, and cached implementations

LIS Algorithms

• 2D DP LIS (lis_funcs.py): Classic O(n²) dynamic programming
• 2D Patience Sorting (lis_funcs.py): Optimized O(n log n) approach
• 1D Patience Sorting (lis_funcs.py): Traditional LIS with patience sorting

Supporting Components

• Common Element Table (cet.py): Core data structure for LSBS
• LSBS Core (lsbs.py): Main LSBS algorithm implementation

Project Structure

.
├── __main__.py           # Interactive CLI interface
├── lcs_lsbs.py          # Main LSBS-based LCS implementation
├── lsbs.py              # Core LSBS algorithm
├── cet.py               # Common Element Table implementation
├── lis_funcs.py         # Collection of LIS algorithm variants
├── lcs_dp.py            # Classic DP LCS implementation
├── lcs_rec.py           # Recursive LCS implementations
├── tests/
│   └── lcs_lsbs_test.py # Unit tests and benchmarks
├── LICENSE              # GPL v3 License
└── README.md            # This file

Testing

Run the comprehensive test suite:

python -m pytest tests/

Or run individual algorithm tests:

# Test LSBS implementation
python lsbs.py

# Test classic DP LCS
python lcs_dp.py

# Test recursive LCS variants
python lcs_rec.py

# Test LIS algorithms
python lis_funcs.py

Performance Characteristics

Algorithm	Time Complexity	Space Complexity	Notes
LSBS (DP backend)	O(n + m + k²)	O(nm)	k = number of matches
LSBS (PS backend)	O(n + m + k log k)	O(nm)	Optimal for sparse matches
Classic DP LCS	O(nm)	O(nm)	Traditional approach
Recursive LCS	O(2^(n+m))	O(n+m)	Exponential without memoization

Algorithm Details

Common Element Table (CET)

The CET maps each character in sequence A to all its occurrence positions in sequence B:

• Row 1: Indices of characters in sequence A
• Row 2: Queues of matching positions in sequence B

LSBS Process

1. Build CET: Create mapping of character occurrences
2. Flatten to pairs: Convert to (position_in_B, bucket_index) pairs
3. Apply 2D LIS: Find longest increasing subsequence on both dimensions
4. Reconstruct LCS: Map result back to original characters

Research Context

This implementation accompanies the IEEE manuscript accepted for publication in ISCI2026:

"Bucket-Constrained LCS Algorithm from 2D-LIS using Patience Sorting and Dynamic Programming"

The work demonstrates how classical sequence problems can be viewed through different algorithmic lenses, potentially offering new insights for:

• Bioinformatics sequence alignment
• Text similarity analysis
• Pattern matching applications

Requirements

• Python 3.8+
• Core: No external dependencies (uses only standard library)
• Benchmarks: Refer to requirements.txt

License

This project is dual-licensed to provide maximum flexibility:

Choose Your License:

MIT License (More Permissive)

• Commercial use without restrictions
• Can be included in proprietary software
• Minimal requirements (just attribution)
• Ideal for: Companies, proprietary projects, maximum adoption

GPL 3.0 License (Copyleft)

• Commercial use allowed
• Derivative works must also be open source
• Must provide source code when distributing
• Ideal for: Open source projects, research collaboration, ensuring code stays open

How to Choose:

• For commercial/proprietary use: Choose MIT License
• For open source projects: Choose GPL 3.0 License
• For research: Either license works, GPL 3.0 ensures derivatives stay open

See the LICENSE file for complete license texts.

Contributing

Contributions are welcome! Please feel free to submit a Pull Request. For major changes, please open an issue first to discuss what you would like to change.

Citation

If you use this work in your research, please cite:

@software{lsbs2026,
  title={Longest Sorted Bucket Sequence: An Algorithmic Bridge Between LCS and 2D LIS},
  author={[Awwal Mohammed; Caroline Sumathi Selvarajah]},
  year={2026},
  url={https://github.com/awwalm/lsbs-pub}
}

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Note: This is a research implementation. For production use, consider performance optimizations and additional testing for your specific use case.