Engineering Faster Sorters for Small Sets of Items

Abstract

Sorting a set of items is a task that can be useful by itself or as a building block for more complex operations. That is why a lot of effort has been put into finding sorting algorithms that sort large sets as efficiently as possible. But the more sophisticated and fast the algorithms become asymptotically, the less efficient they are for small sets of items due to large constant factors.A relatively simple sorting algorithm that is often used as a base case sorter is insertion sort, because it has small code size and small constant factors influencing its execution time.We aim to determine if there is a faster way to sort small sets of items to provide an efficient base case sorter. We looked at sorting networks, at how they can improve the speed of sorting few elements, and how to implement them in an efficient manner by using conditional moves. Since sorting networks need to be implemented explicitly for each set size, providing networks for larger sizes becomes less efficient due to increased code sizes. To also enable the sorting of slightly larger base cases, we adapted sample sort to Register Sample Sort, to break down those larger sets into sizes that can in turn be sorted by sorting networks.From our experiments we found that when sorting only small sets, the sorting networks outperform insertion sort by a factor of at least 1.76 for any array size between six and sixteen, and by a factor of 2.72 on average across all machines and array sizes.When integrating sorting networks as a base case sorter into Quicksort, we achieved far less performance improvements over using insertion sort, which is probably due to the networks having a larger code size and cluttering the L1 instruction cache.The same effect occurs when including Register Sample Sort as a base case sorter for IPS 4 o. But for x86 machines that have a larger L1 instruction cache of 64 KiB or more, we obtained speedups of 12.7% when using sorting networks as a base case sorter in std::sort, and of 5-6% when integrating Register Sample Sort as a base case sorter into IPS 4 o, each in comparison to using insertion sort as the base case sorter.In conclusion, the desired improvement in speed could only be achieved under special circumstances, but the results clearly show the potential of using conditional moves in the field of sorting algorithms.

Engineering In-place (Shared-memory) Sorting Algorithms

“…For small inputs, the base case sorter becomes also more relevant. Here we could profit from several results on fast sorting for very small inputs [10,13,16]. Also, we would like to speed up the branchless decision tree with vector instructions.…”