[Paper Review] Systematic assessment of the expected length, variance and distribution of Longest Common Subsequences

Kang Ning, Kwok Pui Choi|arXiv (Cornell University)|Jun 18, 2013

Algorithms and Data Compression10 references3 citations

TL;DR

This paper presents a systematic Monte Carlo simulation study on the expected length, variance, and distribution of the Longest Common Subsequence (LCS) across multiple sequences, non-uniform alphabet distributions, and large alphabets. It confirms theoretical bounds on the Chvátal-Sankoff constant γ₂, reveals variance grows linearly with sequence length, and benchmarks heuristic LCS algorithms, showing the Deposition and Extension algorithm achieves the best performance ratio, with results 5–10% shorter than optimal LCS lengths.

ABSTRACT

The Longest Common Subsequence (LCS) problem is a very important problem in math- ematics, which has a broad application in scheduling problems, physics and bioinformatics. It is known that the given two random sequences of infinite lengths, the expected length of LCS will be a constant. however, the value of this constant is not yet known. Moreover, the variance distribution of LCS length is also not fully understood. The problem becomes more difficult when there are (a) multiple sequences, (b) sequences with non-even distribution of alphabets and (c) large alphabets. This work focus on these more complicated issues. We have systematically analyze the expected length, variance and distribution of LCS based on extensive Monte Carlo simulation. The results on expected length are consistent with currently proved theoretical results, and the analysis on variance and distribution provide further insights into the problem.

Motivation & Objective

• Address the lack of systematic empirical analysis on LCS expected length, variance, and distribution under complex settings such as multiple sequences, non-uniform alphabet distributions, and large alphabets.
• Provide simulation-based estimates and bounds for the Chvátal-Sankoff constant γᵢ for i > 2 and q > 2, extending beyond the well-known binary case.
• Assess the growth rate of LCS variance to resolve open conjectures about whether it scales linearly with sequence length n.
• Establish a benchmark for heuristic LCS algorithms using systematic simulation results to evaluate performance ratios.
• Validate and compare the performance of state-of-the-art heuristic algorithms, including Long Run, Greedy, Tournament, and Deposition and Extension, on simulated and real datasets.

Proposed method

• Employ extensive Monte Carlo simulations to estimate the expected LCS length, variance, and distribution across diverse sequence configurations.
• Use finite-state automata and Markov chain modeling to compute theoretical lower bounds for γ₂, based on 931 states in the automaton.
• Apply dynamic programming on selected representative sequences to compute upper bounds for LCS length when exact computation is infeasible.
• Implement and evaluate four heuristic algorithms: Long Run, Greedy, Tournament, and Deposition and Extension, with time and space complexity analysis.
• Use performance ratio (heuristic LCS length / optimal LCS length) as the key metric to compare algorithm efficiency and accuracy.
• Focus on sequences with non-uniform alphabet probabilities and large alphabets, applying subset selection of sequences based on alphabet frequency to reduce computational complexity.

Experimental results

Research questions

• RQ1 What is the expected length of the LCS for multiple sequences (i > 2) over alphabets of size q > 2, and how does it compare to theoretical bounds?
• RQ2 How does the variance of LCS length grow with sequence length n, and does it follow a linear scaling pattern?
• RQ3 What is the distribution of LCS length under non-uniform alphabet distributions and large alphabet sizes?
• RQ4 How do heuristic LCS algorithms perform relative to optimal LCS lengths under various sequence configurations?
• RQ5 What is the performance ratio of the Deposition and Extension algorithm compared to other heuristics across simulated and real datasets?

Key findings

• The expected LCS length for two binary sequences converges to a constant γ₂, with simulation results consistent with theoretical lower and upper bounds, supporting γ₂ ∈ [0.773911, 0.837623].
• Variance of LCS length grows linearly with sequence length n, contradicting earlier conjectures that it was o(n²/³), and supporting the existence of a positive constant c such that Var(|LCS|) > cn.
• For multiple sequences and non-uniform alphabet distributions, the expected LCS length varies significantly, with no universal trend, indicating strong dependence on sequence and alphabet configuration.
• The Deposition and Extension algorithm achieves the best performance ratio among heuristics, generating LCS lengths within 5–10% of the optimal length on average.
• The Long Run algorithm performs second best, while Greedy and Tournament algorithms show the worst performance, especially as the number of sequences increases.
• Upper bounds for LCS length are computed via dynamic programming on frequency-selected representative sequences, enabling scalable estimation for large alphabets or long sequences.