Optimal parallel algorithms for multiselection on mesh-connected computers

Abstract

Multiselection is the problem of selecting multiple elements at specified ranks from a set of arbitrary elements. In this paper, we first present an efficient algorithm for single-element selection that runs in <$>O(sqrt{p} +(n/p) log p log (kp/n))<$> time for selecting the kth smallest element from n elements on a <$>sqrt{p} times sqrt{p}<$> mesh-connected computer of <$>p leq n<$> processors, where the first component is for communication and second is for computation (data comparisons). Our algorithm is more computationally efficient than the existing result when <$>p geq nˆ{1/2 + varepsilon}<$> for any <$>0 lt varepsilon lt 1/2<$>. Combining our result for <$>p = Omega (sqrt{n})<$> with the existing result for <$>p = O(sqrt{n})<$> yields an improved computation time complexity for the selection problem on mesh <$>t_{rm comp}ˆ{rm sel} = O(min {(n/p) log plog (kp/n), (n/p + p) log(n/p)})<$>. Using this algorithm as a building block, we then present two efficient parallel algorithms for multiselection on the mesh-connected computers. For selecting r elements from a set of n elements on a <$>sqrt{p} times sqrt{p}<$> mesh, <$>p, r leq n<$>, our first algorithm runs in time <$>O(pˆ{1/2} + t_{rm comp}ˆ{rm sel} min {r log r, log p})<$> with processors operating in the SIMD mode, which is time-optimal when <$>p le r<$>. Allowing processors to operate in the MIMD mode, our second algorithm runs in <$>O(pˆ{1/2} + t_{rm comp}ˆ{rm sel} log r)<$> time and is time-optimal for any r and p.