Triply-Logarithmic Parallel Upper and Lower Bounds for Minimum and Range Minima over Small Domains

We consider the problem of computing the minimumof n values and several well known generalizations pre x minima range minima and all nearest smaller values or ansv for input elements drawn from the integer domain s where s n In this paper we give simple and e cient algorithms for all of the above problems These algorithms all take O log log log s time using an optimal number of processors and O ns space for constant on the common crcw pram The best known upper bounds for the range minima and ansv problems were previously O log logn using algorithms for unbounded domains For the pre x minima and for the mini mum problems the improvement is with regard to the model of computation We also prove a lower bound of log logn for domain size s logn log logn Since for s at the lower end of this range log logn log log log s this demon strates that any algorithm running in o log log log s time must restrict the range of s on which it works A preliminary version of this paper was presented at the Third Workshop on Algorithms and Data Structures WADS Montr eal Canada August Tel Aviv Academic College Antokolsky St Tel Aviv Israel omer math tau ac ilPart of this work was carried out at the Dept of Computer Science King s College London The Strand London WC R LS England AT T Bell Laboratories Mountain Ave Murray Hill NJ USA matias research att com Dep of Computer Science University of Waterloo Waterloo Ontario Canada N L G plragde plg uwaterloo ca