The Smoothed Complexity of Edit Distance

We initiate the study of the smoothed complexity of sequence alignment, by proposing a semi-random model of edit distance between two input strings, generated as follows. First, an adversary chooses two binary strings of length

d

and a longest common subsequence

A

of them. Then, every character is perturbed independently with probability

p

, except that

A

is perturbed in exactly the same way inside the two strings.

We design two efficient algorithms that compute the edit distance on smoothed instances up to a constant factor approximation. The first algorithm runs in near-linear time, namely

d

1 + 

ε

for any fixed

ε

> 0. The second one runs in time sublinear in

d

, assuming the edit distance is not too small. These approximation and runtime guarantees are significantly better then the bounds known for worst-case inputs, e.g. near-linear time algorithm achieving approximation roughly

d

1/3

, due to Batu, Ergün, and Sahinalp [SODA 2006].

Our technical contribution is twofold. First, we rely on finding matches between substrings in the two strings, where two substrings are considered a match if their edit distance is relatively small, a prevailing technique in commonly used heuristics, such as PatternHunter of Ma, Tromp and Li [Bioinformatics, 2002]. Second, we effectively reduce the smoothed edit distance to a simpler variant of (worst-case) edit distance, namely, edit distance on permutations (a.k.a. Ulam’s metric). We are thus able to build on algorithms developed for the Ulam metric, whose much better algorithmic guarantees usually do not carry over to general edit distance.