Partial Order Multiway Search

Abstract

Partial order multiway search(POMS) is a fundamental problem that finds applications in crowdsourcing, distributed file systems, software testing, and more. This problem involves an interaction between an algorithm 𝒜 and an oracle, conducted on a directed acyclic graph 𝒢 known to both parties. Initially, the oracle selects a vertextin 𝒢 called thetarget. Subsequently, 𝒜 must identify the target vertex by probing reachability. In eachprobe, 𝒜 selects a setQof vertices in 𝒢, the number of which is limited by a pre-agreed valuek. The oracle then reveals, for each vertexq∈Q, whetherqcan reach the target in 𝒢. The objective of 𝒜 is to minimize the number of probes. We propose an algorithm to solve POMS in\(O(\log _{1+k} n + \frac{d}{k} \log _{1+d} n)\)probes, wherenrepresents the number of vertices in 𝒢, andddenotes the largest out-degree of the vertices in 𝒢. The probing complexity is asymptotically optimal. Our study also explores two new POMS variants: The first one, namedtaciturn POMS, is similar to classical POMS but assumes a weaker oracle, and the second one, namedEM POMS, is a direct extension of classical POMS to theexternal memory(EM) model. For both variants, we introduce algorithms whose performance matches or nearly matches the corresponding theoretical lower bounds.