A Las Vegas probabilistic algorithm is presented that finds the Smith normal form S ∈ Q[x] of a nonsingular input matrix A ∈ Z [x]. The algorithm requires an expected number of O (̃nd(d + n log ||A||)) bit operations (where ||A|| bounds the magnitude of all integer coefficients appearing in A and d bounds the degrees of entries of A). In practice, the main cost of the computation is obtaining a ...

We prove several new results regarding the relationship between probabilistic time, BPTime (t), and alternating time, ΣO(1)Time (t). Our main results are the following: 1. We prove that BPTime (t) ⊆ Σ3Time (t · poly log t). Previous results show that BPTime (t) ⊆ Σ2Time ( t2 · log t ) (Sipser and Gács, STOC ’83; Lautemann, IPL ’83) and BPTime (t) ⊆ ΣcTime (t) for a large constant c > 3 (Ajtai, ...

The Burrows Wheeler transform has applications in data compression as well as full text indexing. Despite its important applications and various existing algorithmic approaches the construction of the transform for large data sets is still challenging. In this paper we present a new semi external memory algorithm capable of constructing the transform in time O(n log2 log n) on average if su cie...

In the contiguous variant of the Scheduling with Interval Conflicts problem, there is a universe U consisting of elements being consecutive positive integers. An input is a sequence of conflicts in the form of intervals of length at most σ. For each conflict, an algorithm has to choose at most one surviving element, with the ultimate goal of maximizing the number of elements that survived all c...

In this paper, we look at the complexity of designing algorithms without any bank conflicts in the shared memory of Graphical Processing Units (GPUs). Given input of size n, w processors and w memory banks, we study three fundamental problems: sorting, permuting and w-way partitioning (defined as sorting an input containing exactly n/w copies of every integer in [w]). We solve sorting in optima...

Conventional parallel sorting requires the n input keys to be output in an array of size n, and is known to take fl(log n/log log n) time using any polynomial number of processors. The lower bound does not apply to the more "wasteful" convention of padded sorting, which requires the keys to be output in sorted order in an array of size (1 + o(1))n. We give very fast randomized CRCW PRAM algorit...

We consider the complexity for computing the approximate sum a1 + a2 + · · · + an of a sorted list of numbers a1 ≤ a2 ≤ · · · ≤ an. We show an algorithm that computes an (1 + ǫ)-approximation for the sum of a sorted list of nonnegative numbers in an O( 1 ǫ min(log n, log(max xmin )) · (log 1 ǫ + log log n)) time, where xmax and xmin are the largest and the least positive elements of the input l...

We study the complexity of one-dimensional extrema testing: given one input number, determine if it is properly contained in the interval spanned by the remaining n input numbers. We assume that each number is given as a finite stream of bits, in decreasing order of significance. Our cost measure, referred to as the leading-input-bits-cost (or LIB-cost for short), for an algorithm solving such ...