Indexed Dynamic Programming to boost Edit Distance and LCSS Computation

TL;DR

This paper introduces an indexing technique that refines dynamic programming algorithms, leading to faster computation of Edit Distance and LCSS on various string classes by analyzing intermediate difficulty instances.

Contribution

It presents a novel indexing-based approach to improve dynamic programming algorithms for Edit Distance and LCSS, refining complexity analysis beyond worst-case scenarios.

Findings

Faster algorithms for Edit Distance and LCSS on intermediate difficulty instances.

Parameterization of complexity based on relevant dynamic programming matrix areas.

Application of indexing techniques to refine existing dynamic programming solutions.

Abstract

There are efficient dynamic programming solutions to the computation of the Edit Distance from $S\in[1..\sigma]^n$ to $T\in[1..\sigma]^m$, for many natural subsets of edit operations, typically in time within $O(nm)$ in the worst-case over strings of respective lengths $n$ and $m$ (which is likely to be optimal), and in time within $O(n{+}m)$ in some special cases (e.g. disjoint alphabets). We describe how indexing the strings (in linear time), and using such an index to refine the recurrence formulas underlying the dynamic programs, yield faster algorithms in a variety of models, on a continuum of classes of instances of intermediate difficulty between the worst and the best case, thus refining the analysis beyond the worst case analysis. As a side result, we describe similar properties for the computation of the Longest Common Sub Sequence $LCSS(S,T)$ between $S$ and $T$, since it is a particular case of Edit Distance, and we discuss the application of similar algorithmic and analysis techniques for other dynamic programming solutions. More formally, we propose a parameterized analysis of the computational complexity of the Edit Distance for various set of operators and of the Longest Common Sub Sequence in function of the area of the dynamic program matrix relevant to the computation.