We investigate the average-case complexity of decision problems for finitely generated groups, in particular the word and membership problems. Using our recent results on “generic-case complexity” we show that if a finitely generated group G has word problem solvable in subexponential time and has a subgroup of finite index which possesses a non-elementary word-hyperbolic quotient group, then t...

The subset sum problem (SSP) can be briefly stated as: given a target integer E and a set A containing n positive integer aj , find a subset of A summing to E. The density d of an SSP instance is defined by the ratio of n to m, where m is the logarithm of the largest integer within A. Based on the structural and statistical properties of subset sums, we present an improved enumeration scheme fo...

The average complexity of any algorithm whatsoever (provided it always terminates) under the universal distribution is of the same order of magnitude as the worst-case complexity. This holds both for time complexity and for space complexity. To focus our discussion, we use as illustrations the particular case of sorting algorithms, and the general case of the average case complexity of NP-compl...

For every polynomial q, we present worst-case to average-case (almost-linear-time) reductions for a class of problems in P that are widely conjectured not to be solvable in time q. These classes contain, for example, the problems of counting the number of k-cliques in a graph, for any fixed k ≥ 3. In general, we consider the class of problems that consist of counting the number of local neighbo...

We prove a connection of the worst-case complexity and the average-case complexity for the Closest Vector Problem (CVP) for lattices. Assume that there is an eecient algorithm which can solve approximately a random instance of CVP for lattices under a certain natural distribution, at least with a non-trivial success probability over this distribution, we show that one can approximately solve se...

The all-du-paths structural testing criterion is one of the most discriminating of the data ow testing criteria. Unfortunately, in the worst case, the criterion requires an intractable number of test cases. In a case study of an industrial software system, we nd that the worst case scenario is rare. Eighty percent of the subroutines require ten or fewer test cases. Only one subroutine out of 14...

Herman’s self-stabilization algorithm, introduced 25 years ago, is a well-studied synchronous randomized protocol for enabling a ring of N processes collectively holding any odd number of tokens to reach a stable state in which a single token remains. Determining the worst-case expected time to stabilization is the central outstanding open problem about this protocol. It is known that there is ...

The subset sum problem (SSP) can be briefly stated as: given a target integer E and a set A containing n positive integer aj , find a subset of A summing to E. The density d of an SSP instance is defined by the ratio of n to m, where m is the logarithm of the largest integer within A. Based on the structural and statistical properties of subset sums, we present an improved enumeration scheme fo...

The Novelli–Pak–Stoyanovskii algorithm is a sorting algorithm for Young tableaux of a fixed shape that was originally devised to give a bijective proof of the hooklength formula. We obtain new asymptotic results on the average case and worst case complexity of this algorithm as the underlying shape tends to a fixed limit curve. Furthermore, using the summation package Sigma we prove an exact fo...

Angiosarcomas are endothelial cell neoplasms in the lining of the blood vessel wall and account for about 0.04% of all breast malignancies with a high rate of error in primary diagnosis. The breast angiosarcoma is a rare and uncommon pathology and has been described mostly as case reports. Indeed, only a limited number of cases have been published. Accordingly, the natural history of this tumor...