Substring Query Complexity of String Reconstruction
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Suppose an oracle knows a string S that is unknown to us and we want to determine. The oracle can answer queries of the form "Is s a substring of S?". The Substring Query Complexity of a string S, denoted χ(S), is the minimum number of adaptive substring queries that are needed to exactly reconstruct (or learn) S. It has been introduced in 1995 by Skiena and Sundaram, who showed that χ(S) ≥σ n/4 -O(n) in the worst case, where σ is the size of the alphabet of S and n its length, and gave an algorithm that spends (σ-1)n+O(σ√(n)) queries to reconstruct S. We show that for any binary string S, χ(S) is asymptotically equal to the Kolmogorov complexity of S and therefore lower bounds any other measure of compressibility. However, since this result does not yield an efficient algorithm for the reconstruction, we present new algorithms to compute a set of substring queries whose size grows as a function of other known measures of complexity, e.g., the number rle of runs in S, the size g of the smallest grammar producing (only) S or the size z_no of the non-overlapping LZ77 factorization of S. We first show that any string of length n over an integer alphabet of size σ with rle runs can be reconstructed with q=O( rle (σ + logn/ rle)) substring queries in linear time and space. We then present an algorithm that spends q ∈ O(σ glog n) ⊆ O(σ z_nolog (n/z_no)log n) substring queries and runs in O(n(log n + logσ)+ q) time using linear space. This algorithm actually reconstructs the suffix tree of the string using a dynamic approach based on the centroid decomposition.
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