Waste makes haste: tight bounds for loose parallel sorting

Conventional parallel sorting requires the n input keys to be output in an array of size n, and is known to take Omega (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. The authors give very fast randomised CRCW PRAM algorithms for several padded-sorting problems. Applying only pairwise comparisons to the input and using kn processors, where 2 <or= k <or= n, they can padded-sort n keys in O(log n/log k) time with high probability (WHP), which is the best possible (expected) run time for any comparison-based algorithm. They also show how to padded-sort n independent random numbers in O(log/sub */n) time WHP with O(n) work, which matches a recent lower bound, and how to padded-sort n integers in the range 1..n in constant time whp using n processors. If the integer sorting is required to be stable, they can still solve the problem in O(log log n/log k) time WHP using kn processors, for any k with 2 <or= k <or= log n. The integer sorting results require the nonstandard OR PRAM. As an application of the padded-sorting algorithms, they can solve approximate prefix summation problems of size n with O(n) work in constant time WHP on the OR PRAM, and in O(log log n) time WHP on standard PRAM variants.<<ETX>>