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 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 algorithms for several padded-sorting problems. Applying only pairwise comparisons to the input and using kn processors, where 2 :s; k :s; n, we can padded-sort n keys in O(logn/logk) time with high probability (whp), which is the best possible (expected) run time for any comparison-based algorithm. We also show how to padded-sort n independent random numbers in O(log*n) time whp with O(n) work, which matches arecent 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, we can still solve the problem in o (log log n/log k) time whp using kn processors, for any k with 2 :s; k :s; log n. The integer sorting results require the nonstandard OR PRAM; alternative implementations on standard PRAM variants run in O(log log n) time whp. As an application of our padded-sorting algorithms, we can solve approximate prefix summation problems of size n with O(n) work in constant time whp on the OR PRAM, and in O(loglog n) time whp on standard PRAM variants.