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

  • Christina Boucher
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6907)

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 \(O\left(n\ell + nd \cdot d^{2 + o(1)} \right)\).

Keywords

Close String Binary Search Tree Recursive Step Search Tree Algorithm Halting State 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2011

Authors and Affiliations

  • Christina Boucher
    • 1
  1. 1.Department of Computer Science and EngineeringUniversity of CaliforniaSan DiegoUSA

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