Improved deterministic parallel integer sorting

Abstract

We consider the problem of deterministic sorting of integers on a parallel RAM (PRAM). The best previous result ( T. Hagerup, 1987 , Inform. and Comput. 75 , 39–51) states that n integers of size polynomial in n can be sorted in time O (log n ) on a Priority CRCW PRAM with O( n log log n log n ) processors. We prove that n integers drawn from a set {0, …, m −1} can be sorted on an Arbitrary CRCW PRAM in time O( log n log log n + log log m) with a time-processor product of O ( n log log m ). In particular, if m = n (log n ) O (1) , the time and number of processors used are O( log n log log n ) and O( n( log log n) 2 log n ) , respectively. This improves the previous result in several respects: The new algorithm is faster, it works on a weaker PRAM model, and it is closer to optimality for input numbers of superpolynomial size. If log log m = O( log n log log n ) , the new algorithm is optimally fast, for any polynomial number of processors, and if log log m = (1 + Ω(1)) log log n and log log m = 0 ( log n ), it has optimal speedup relative to the fastest known sequential algorithm. The space needed is O ( nm ε ), for arbitrary but fixed ε > 0. The sorting algorithm derives its speed from a fast solution to a special list ranking problem of possible independent interest, the monotonic list ranking problem . In monotonic list ranking, each list element has an associated key, and the keys are known to increase monotonically along the list. We show that monotonic list ranking problems of size n can be solved optimally in time O( log n log log n ) . We also discuss and attempt to solve some of the problems arising in the precise description and implementation of parallel recursive algorithms. As part of this effort, we introduce a new PRAM variant, the allocated PRAM .