Partial Order Multiway Search

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 vertex t in G called the target. Subsequently, A must identify the target vertex by probing reachability. In each probe, A selects a set Q of vertices in G, the number of which is limited by a pre-agreed value k. The oracle then reveals, for each vertex q is an element of Q, whether q can 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 + d/k log(1+d) n) probes, where n represents the number of vertices in G, andd denotes 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, named taciturn POMS, is similar to classical POMS but assumes a weaker oracle, and the second one, named EM POMS, is a direct extension of classical POMS to the external memory (EM) model. For both variants, we introduce algorithms whose performance matches or nearly matches the corresponding theoretical lower bounds.