Exact value of the median-selection complexity for 16 elements

Determine whether the minimum worst-case number of comparisons required to select the 8th smallest of 16 distinct elements, V_8(16), is 28 or 29.

Background

The paper proves the upper bound V_8(16)≤29 and uses known lower bounds to establish 28≤V_8(16)≤29. Thus, the exact complexity remains unresolved. A 28-comparison certificate would settle the problem constructively, whereas proving V_8(16)=29 would require excluding every 28-comparison selection algorithm. The paper further notes that a lower bound based only on the total number of decision-tree leaves cannot establish 29, so a successful proof of the value 29 would need to exploit comparison-tree depth or path information.

References

Is $V_8(16)$ equal to 28 or 29? A 28-comparison certificate would settle it constructively; a proof of 29 must exclude all such algorithms and, by Remark~\ref{rem:leafcount}, needs more than a total leaf count.

— 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”