Selection Algorithms with Small Groups
Ke Chen, Adrian Dumitrescu · International Journal of Foundations of Computer Science · 2020
We revisit the selection problem, namely that of computing the [Formula: see text]th order statistic of [Formula: see text] 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 [Formula: see text] 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 [Formula: see text]. 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 [Formula: see text] 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.