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 complex the algorithms become, 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 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 of integers, the sorting networks outperform insertion sort by a factor of at least 1.76 for any array size between six and 16, 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 IPS4o. 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 IPS4o, 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.

Vectorized and performance‐portable quicksort

“…We used the Highway cross‐platform abstraction layer for implementing vqsort (vectorized quicksort) and utilizing the most efficient instructions available on the current CPU. The algorithm features a new recursive sorting network for up to 256 elements that mitigates the previously reported 23 problem of excessive code size, and also a vector‐friendly pivot sampling, with the surprising result that robustness versus adversarial input increases CPU cost by less than 2%.…”

“…We now handle small arrays separately as a "base case" of the recursion, a common optimization for quicksort. 6,23 Sorting networks built upon vector instructions can have much lower constant factors than other algorithms because they execute fewer instructions and avoid conditional branches. For moderate input sizes, this outweighs their higher O(n ⋅ log 2 (n)) complexity.…”

Vectorized and performance-portable Quicksort

“…We now handle small arrays separately as a 'base case' of the recursion, a common optimization for Quicksort [6,23]. Sorting networks built upon vector instructions can have much lower constant factors than other algorithms because they execute fewer instructions and avoid conditional branches.…”