Extreme Witnesses and Their Applications

We study the problem of computing the so called minimum and maximum witnesses for Boolean vector convolution. We also consider a generalization of the problem which is to determine for each positive value at a coordinate of the convolution vector, q smallest (largest) witnesses, where q is the minimum of a parameter k and the number of witnesses for this coordinate. We term this problem the smallest k-witness problem or the largest k-witness problem, respectively. We also study the corresponding smallest and largest k-witness problems for Boolean matrix product. First, we present an O~ (n^1.5k^0.5) -time algorithm for the smallest or largest k-witness problem for the Boolean convolution of two n-dimensional vectors, where the notation O~() suppresses polylogarithmic in n factors. In consequence, we obtain new upper time bounds on reporting positions of mismatches in potential string alignments and on computing restricted cases of the (min , + ) vector convolution. Next, we present a fast (substantially subcubic in n and linear in k) algorithm for the smallest or largest k-witness problem for the Boolean matrix product of two n× n Boolean matrices. It yields fast algorithms for reporting k lightest (heaviest) triangles in a vertex-weighted graph.