The Bounded Search Tree Algorithm for the Closest String Problem Has Quadratic Smoothed Complexity

Abstract

Given a set S of n strings, each of length ℓ, and a non-negative value d, we define a center string as a string of length ℓ that has Hamming distance at most d from each string in S. The Closest String problem aims to determine whether there exists a center string for a given set of strings S and input parameters n, ℓ, and d. When n is relatively large with respect to ℓ then the basic majority algorithm solves the Closest String problem efficiently, and the problem can also be solved efficiently when either n, ℓ or d is reasonably small [12]. Hence, the only case for which there is no known efficient algorithm is when n is between logℓ/ loglogℓ and logℓ. Using smoothed analysis, we prove that such Closest String instances can be solved efficiently by the O(nℓ + nd ·d ^d)-time algorithm by Gramm et al. [13]. In particular, we show that for any given Closest String instance I, the expected running time of this algorithm on a small perturbation of I is $O\left(n\ell + nd \cdot d^{2 + o(1)} \right)$.