Near linear time construction of an approximate index for all maximum consecutive sub-sums of a sequence

Authors:
Ferdinando Cicalese;Eduardo Laber;Oren Weimann;Raphael Yuster
Affiliations:
Department of Computer Science, University of Salerno, Italy;Department of Informatics, PUC-Rio, Rio de Janeiro, Brazil;Department of Computer Science, University of Haifa, Israel;Department of Mathematics, University of Haifa, Israel
Venue:
CPM'12 Proceedings of the 23rd Annual conference on Combinatorial Pattern Matching
Year:
2012

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Abstract

We present a novel approach for computing all maximum consecutive subsums in a sequence of positive integers in near linear time. Solutions for this problem over binary sequences can be used for reporting existence (and possibly one occurrence) of Parikh vectors in a bit string. Recently, several attempts have been tried to build indexes for all Parikh vectors of a binary string in subquadratic time. However, to the best of our knowledge, no algorithm is know to date which can beat by more than a polylogarithmic factor the natural Θ(n2) exhaustive procedure. Our result implies an approximate construction of an index for all Parikh vectors of a binary string in O(n1+η) time, for any constant η0. Such index is approximate, in the sense that it leaves a small chance for false positives, i.e., Parikh vectors might be reported which are not actually present in the string. No false negative is possible. However, we can tune the parameters of the algorithm so that we can strictly control such a chance of error while still guaranteeing strong sub-quadratic running time.