← 研踪 PaperTrailarXiv:2610.40521 · v1
Directed Cut-Query Complexity of Finding a King
首次提交 2026-10-30 01:29
版本更新 2026-10-30 01:29
雷达观测 2026-10-30 13:28
摘要
A king in a tournament is a vertex from which every other vertex is reachable by a directed path of length at most two. We study the deterministic query complexity of finding a king when a query specifies a set of vertices and returns the number of edges directed from that set to its complement. We prove that this complexity is $Θ(n)$ for $n$-vertex tournaments. To show this, we first show that every correct deterministic algorithm in this model must output a maximum-outdegree vertex, even though a king need not have maximum outdegree. We then give an adversary whose answers, after too few queries, remain consistent with a tournament in which the algorithm's output does not have maximum outdegree.
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v1 · 2026-10-30 01:29 · 最新观测版本
Directed Cut-Query Complexity of Finding a King
A king in a tournament is a vertex from which every other vertex is reachable by a directed path of length at most two. We study the deterministic query complexity of finding a king when a query specifies a set of vertices and returns the number of edges directed from that set to its complement. We prove that this complexity is $Θ(n)$ for $n$-vertex tournaments. To show this, we first show that every correct deterministic algorithm in this model must output a maximum-outdegree vertex, even though a king need not have maximum outdegree. We then give an adversary whose answers, after too few queries, remain consistent with a tournament in which the algorithm's output does not have maximum outdegree.
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