Near-Linear Time Edit Distance for Indel Channels

TL;DR

This paper presents an efficient algorithm to compute the edit distance between two related strings in near-linear time, providing theoretical support for practical alignment heuristics like BLAST and FASTA.

Contribution

The authors introduce a simple $O(n \, \ln n)$ time algorithm for edit distance under a probabilistic model, with a novel analysis partitioning alignments to justify heuristic methods.

Findings

Algorithm runs in $O(n \ln n)$ time with high probability.

Provides theoretical justification for practical alignment heuristics.

Techniques may apply to average-case analysis of dynamic programming problems.

Abstract

We consider the following model for sampling pairs of strings: $s_1$ is a uniformly random bitstring of length $n$, and $s_2$ is the bitstring arrived at by applying substitutions, insertions, and deletions to each bit of $s_1$ with some probability. We show that the edit distance between $s_1$ and $s_2$ can be computed in $O(n \ln n)$ time with high probability, as long as each bit of $s_1$ has a mutation applied to it with probability at most a small constant. The algorithm is simple and only uses the textbook dynamic programming algorithm as a primitive, first computing an approximate alignment between the two strings, and then running the dynamic programming algorithm restricted to entries close to the approximate alignment. The analysis of our algorithm provides theoretical justification for alignment heuristics used in practice such as BLAST, FASTA, and MAFFT, which also start by computing approximate alignments quickly and then find the best alignment near the approximate alignment. Our main technical contribution is a partitioning of alignments such that the number of the subsets in the partition is not too large and every alignment in one subset is worse than an alignment considered by our algorithm with high probability. Similar techniques may be of interest in the average-case analysis of other problems commonly solved via dynamic programming.