Selection Algorithms with Small Groups

TL;DR

This paper challenges the belief that small group sizes in selection algorithms necessarily lead to superlinear worst-case time, demonstrating that linear time can be maintained with groups smaller than five.

Contribution

It introduces three new variants of the classic selection algorithm that operate in linear time using smaller group sizes than previously thought possible.

Findings

Small group sizes can be used without losing linear worst-case time

Three new algorithms maintain linear time with groups smaller than five

Theoretical analysis refutes previous assumptions about superlinear complexity

Abstract

We revisit the selection problem, namely that of computing the $i$th order statistic of $n$ given elements, in particular the classic deterministic algorithm by grouping and partition due to Blum, Floyd, Pratt, Rivest, and Tarjan (1973). Whereas the original algorithm uses groups of odd size at least $5$ and runs in linear time, it has been perpetuated in the literature that using smaller group sizes will force the worst-case running time to become superlinear, namely $\Omega(n \log{n})$. We first point out that the usual arguments found in the literature justifying the superlinear worst-case running time fall short of proving this claim. We further prove that it is possible to use group size smaller than $5$ while maintaining the worst case linear running time. To this end we introduce three simple variants of the classic algorithm, the repeated step algorithm, the shifting target algorithm, and the hyperpair algorithm, all running in linear time.