Sesquickselect: One and a half pivots for cache-efficient selection

TL;DR

Sesquickselect is a novel Quickselect algorithm that adaptively uses one or two pivots, optimizing memory transfers and outperforming other variants, especially with small samples, by reducing memory bottlenecks in modern CPU architectures.

Contribution

The paper introduces Sesquickselect, an adaptive multiway Quickselect variant that effectively uses one or two pivots to minimize memory transfers, advancing selection algorithms for modern hardware.

Findings

Two pivots are beneficial for Quickselect performance.

Sesquickselect outperforms other variants in memory transfer efficiency.

Adaptive pivot selection improves selection performance with small samples.

Abstract

Because of unmatched improvements in CPU performance, memory transfers have become a bottleneck of program execution. As discovered in recent years, this also affects sorting in internal memory. Since partitioning around several pivots reduces overall memory transfers, we have seen renewed interest in multiway Quicksort. Here, we analyze in how far multiway partitioning helps in Quickselect. We compute the expected number of comparisons and scanned elements (approximating memory transfers) for a generic class of (non-adaptive) multiway Quickselect and show that three or more pivots are not helpful, but two pivots are. Moreover, we consider "adaptive" variants which choose partitioning and pivot-selection methods in each recursive step from a finite set of alternatives depending on the current (relative) sought rank. We show that "Sesquickselect", a new Quickselect variant that uses either one or two pivots, makes better use of small samples w.r.t. memory transfers than other Quickselect variants.