Sorting, Approximate Sorting, and Searching in Rounds

Abstract

The worst case number of comparisons needed for sorting or selecting in rounds is considered. The following results are obtained. (a) For every fixed $k\geqq 2$, $\Omega ( n^{1 + 1/k} ( \log n )^{1/k} )$ comparisons are required to sort n elements in k rounds. ($O ( n^{1 + 1 / k} \log n )$ are known to be sufficient.) This improves the previous known bounds by a factor of $( \log n )^{1/k} $, which separates deterministic algorithms from randomized ones, as there are randomized algorithms whose expected number of comparisons is $O ( n^{1 + 1/k} )$. (b) For every fixed $k\geqq 2$, $\Omega ( n^{1 + 1/( 2^k - 1 )} ( \log n )^{2/( 2^k - 1 )} )$ comparisons are required to select the median from n elements in k rounds. ($O ( n^{1 + 1/ ( 2^k - 1 ) } ( \log n )^{2 - 2/ ( 2^k - 1 ) } )$ are known to be sufficient.) This improves the previous known bounds by a factor of $( \log n )^{2/( 2^k - 1 )} $ and separates the problem of finding the median from that of finding the minimum, as $O( n^{1 + 1/( 2^k - 1 ) } )$ c...