Tight bounds for string reconstruction using substring queries

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Abstract

We resolve two open problems presented in [8]. First, we consider the problem of reconstructing an unknown string T over a fixed alphabet using queries of the form "does the string S appear in T?" for some query string S. We show that Ω(ε-1/2n2) queries are needed in order to reconstruct a 1 - ε fraction of the strings of length n. This lower bound is asymptotically optimal since it is known that O(ε1/2n2) queries are sufficient. The second problem is reconstructing a string using queries of the form "does a string from S appear in T?", where S is a set of strings. We show that a 1 - ε fraction of the strings of length n can be reconstructed using O(n) queries, where the maximum length of a string in the queries is 2 logσ n + logσ 1/ε + O(1). This construction is optimal up to constants.

Original languageEnglish
Pages (from-to)448-459
Number of pages12
JournalLecture Notes in Computer Science
Volume3624
DOIs
StatePublished - 1 Jan 2005
Externally publishedYes
Event8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005 and 9th International Workshop on Randomization and Computation, RANDOM 2005 - Berkeley, CA, United States
Duration: 22 Aug 200524 Aug 2005

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