[Paper Review] Simulations, Computations, and Statistics for Longest Common Subsequences

Qingqing Liu, Christian Houdré|arXiv (Cornell University)|May 18, 2017

Algorithms and Data Compression9 references3 citations

TL;DR

This paper develops a Monte-Carlo simulation framework to estimate the mean and variance of the longest common subsequence (LCS) length in random binary sequences, enabling a hypothesis testing method for sequence similarity. It derives new theoretical upper bounds for the Chvátal–Sankoff constant in multiple sequences, improving upon prior results for up to ten sequences with a binary alphabet, with bounds decreasing from 0.8666 (m=2) to 0.6488 (m=10).

ABSTRACT

The length of the longest common subsequences (LCSs) is often used as a similarity measurement to compare two (or more) random words. Below we study its statistical behavior in mean and variance using a Monte-Carlo approach from which we then develop a hypothesis testing method for sequences similarity. Finally, theoretical upper bounds are obtained for the Chvátal-Sankoff constant of multiple sequences.

Motivation & Objective

• To develop a Monte-Carlo-based method for estimating the mean and variance of the LCS length in random binary sequences.
• To construct a hypothesis testing framework to assess the statistical significance of sequence similarity based on LCS scores.
• To extend classical two-sequence results on the Chvátal–Sankoff constant to multiple sequences (m ≥ 2) using combinatorial and probabilistic techniques.
• To derive new theoretical upper bounds for the Chvátal–Sankoff constant in the multi-sequence case, particularly for binary alphabets.
• To provide computationally feasible approximations of the LCS distribution for practical use in bioinformatics and sequence analysis.

Proposed method

• Employs Monte-Carlo simulations to empirically estimate the mean and variance of LCS length in i.i.d. random binary sequences of length n.
• Uses the alignment score function with s(a,b) = 1 if a=b, 0 otherwise, and zero gap penalty, defining LCS as the optimal alignment score.
• Applies Stirling’s approximation and combinatorial bounds to derive asymptotic upper bounds on the expected LCS length per sequence length.
• Introduces the function H_k(θ) = [k^{(θ/m)-1}(k-1)^{1-θ}] / [θ^θ(1-θ)^{1-θ}] to characterize the decay rate of the probability that LCS exceeds θn.
• Defines V_k as the unique solution in [1/k, 1) to H_k(θ) = 1, which serves as an upper bound for the Chvátal–Sankoff constant γ_{k,m}^*.
• Uses the inequality h_k^{(n)} ≤ (H_k(θ))^{mn} to bound the proportion of m-tuples with LCS ≥ ℓ = θn, leading to the upper bound lim sup E[LC_n]/n ≤ V_k.

Experimental results

Research questions

• RQ1 What is the empirical mean and variance of the LCS length in random binary sequences of finite length, as estimated via Monte-Carlo simulation?
• RQ2 How can the LCS score be used to construct a valid hypothesis test for determining whether two sequences are significantly similar?
• RQ3 What are the theoretical upper bounds for the Chvátal–Sankoff constant when comparing m ≥ 2 random sequences over a finite alphabet?
• RQ4 How do the upper bounds on the Chvátal–Sankoff constant for multiple sequences compare to existing results, particularly for binary alphabets?
• RQ5 Can the method used for two sequences be extended to multiple sequences to derive tighter or more general upper bounds on the asymptotic LCS growth rate?

Key findings

• The paper establishes that the Chvátal–Sankoff constant γ_{2,m}^* for m sequences over a binary alphabet is bounded above by V_2, where V_2 is the unique solution in [1/2, 1) to H_2(θ) = 1.
• For m=2, the upper bound is 0.866595, improving upon earlier bounds and narrowing the gap to the known lower bound of 0.781281.
• For m=10, the upper bound is 0.648819, showing a monotonic decrease in the upper bound as m increases, consistent with theoretical expectations.
• The method yields tighter bounds than previous approaches for m ≥ 4, particularly when analytical computation becomes unwieldy.
• The hypothesis testing framework based on simulated LCS distributions is validated through extensive Monte-Carlo experiments, enabling statistical inference on sequence similarity.
• Theoretical analysis confirms that the proportion of m-tuples with LCS ≥ θn decays exponentially when θ > V_k, supporting the use of V_k as a sharp upper bound on the asymptotic growth rate of LCS.