Approximating the maximum consecutive subsums of a sequence Dedicated to Professor Gad M. Landau, on the occasion of his 60th birthday

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 of Parikh vectors in a bit string. Recently, several attempts have been made to build indexes for all Parikh vectors of a binary string in subquadratic time. However, no algorithm is known to date which can beat by more than a polylogarithmic factor the naive Θ(n²) procedure. We show how to construct a (1+ε)-approximate index for all Parikh vectors of a binary string in O(n^log2n/log(1+ε)) time, for any constant ε>0. Such index is approximate, in the sense that it leaves a small chance for false positives (no false negatives are 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 subquadratic running time.