Select with Groups of 3 or 4

We revisit the selection problem, namely that of computing the ith order statistic of n given elements, in particular the classical deterministic algorithm by grouping and partition due to Blum, Floyd, Pratt, Rivest, and Tarjan (1973). While 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 groups of 3 or 4 will force the worst-case running time to become superlinear, namely Ω(n log n). We first point out that the arguments existent 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 3 or 4 while maintaining the worst case linear running time. To this end we introduce two simple variants of the classical algorithm, the repeated step algorithm and the shifting target algorithm, both running in linear time.