Expected Length of the Longest Common Subsequence of Multiple Strings

2025-04-14

AI Summary

This paper investigates the asymptotic behavior of the expected length of the longest common subsequence (LCS) of $d$ independent, uniformly random strings of length $n$ over an alphabet of size $k$, focusing on the generalized Chvátal–Sankoff constant $gamma_{k,d} = lim_{n oinfty} mathbb{E}[mathrm{LCS}(X^{(1)},dots,X^{(d)})]/n$. Using probabilistic analysis, extremal combinatorics, concentration inequalities, and asymptotic expansions, we derive the first tight asymptotic formula for the binary case: $gamma_{2,d} = frac{1}{2} + Theta(d^{-1/2})$. Furthermore, when $d = Omega(log k)$, we establish nearly optimal upper and lower bounds on $gamma_{k,d}$. These results substantially improve upon prior work—which only established existence and coarse order-of-magnitude estimates—by providing precise, fine-grained asymptotics for the expected LCS length in the multi-string setting.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the generalized Chv'atal-Sankoff constant $gamma_{k,d}$, which represents the normalized expected length of the longest common subsequence (LCS) of $d$ independent uniformly random strings over an alphabet of size $k$. We derive asymptotically tight bounds for $gamma_{2,d}$, establishing that $gamma_{2,d} = frac{1}{2} + Thetaleft(frac{1}{sqrt{d}} ight)$. We also derive asymptotically near-optimal bounds on $gamma_{k,d}$ for $dge Omega(log k)$.

Problem

• Estimating normalized LCS length for multiple random strings
• Deriving tight bounds for γ₂,d in binary alphabets
• Establishing near-optimal bounds for γₖ,d with large d

Innovation

• Asymptotically tight bounds for LCS
• Normalized expected length analysis
• Generalized Chvátal-Sankoff constant study