Generalization of the runner-up opening

Investigate whether the runner-up opening used in the 28-comparison algorithm for selecting the 7th smallest of 16 elements can be extended to a family of effective selection algorithms, particularly for central ranks of $2^k$ elements.

Background

The paper describes a 15-comparison runner-up opening that produces a structured partial order and is shared by several of the new algorithms. The authors ask whether this construction reflects a broader algorithmic pattern that can be systematically extended to larger instances, with central ranks of powers of two given as an example.

References

Finally, can the runner-up opening be extended into a family of good selection algorithms, for example for central ranks of $2k$ elements?

— Selecting the 7th Smallest of 16 Elements Takes Exactly 28 Comparisons  (2610.18740 - Cremer, 13 Oct 2026) in Section “Conclusion and open problems,” paragraph “Open problems”