Longest Common Subsequence: Tabular vs. Closed-Form Equation Computation of Subsequence Probability

The Longest Common Subsequence Problem (LCS) deals with finding the longest subsequence among a given set of strings. The LCS problem is an NP-hard problem which makes it a target for lots of effort to find a better solution with heuristics methods. The baseline for most famous heuristics functions is a tabular random, probabilistic approach. This approach approximates the length of the LCS in each iteration. The combination of beam search and tabular probabilistic-based heuristics has led to a large number of proposals and achievements in algorithms for solving the LCS problem. In this work, we introduce a closed-form equation of the probabilistic table calculation for the first time. Moreover, we present other corresponding forms of the closed-form equation and prove all of them. The closed-form equation opens new ways for analysis and further approximations. Using the theorems and beam search, we propose an analytic method for estimating the length of the LCS of the remaining subsequence. Furthermore, we present another heuristic function based on the Coefficient of Variation. The results show that our proposed methods outperform the state-of-the-art methods on the LCS problem.