Exact (n + 2) Comparison Complexity for the N-Repeated Element Problem

Abstract

This paper establishes the exact comparison complexity of finding an element repeated $n$ times in a $2n$-element array containing $n+1$ distinct values, under the equality-comparison model with $O(1)$ extra space. We present a simple deterministic algorithm performing exactly $n+2$ comparisons and prove this bound tight: any correct algorithm requires at least $n+2$ comparisons in the worst case. The lower bound follows from an adversary argument using graph-theoretic structure. Equality queries build an inequality graph $I$; its complement $P$ (potential-equalities) must contain either two disjoint $n$-cliques or one $(n+1)$-clique to maintain ambiguity. We show these structures persist up through $n+1$ comparisons via a "pillar matching" construction and edge-flip reconfiguration, but fail at $n+2$. This result provides a concrete, self-contained demonstration of exact lower-bound techniques, bridging toy problems with nontrivial combinatorial reasoning.