Computing the Union Join and Subset Graph of Acyclic Hypergraphs in Subquadratic Time
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We investigate the two problems of computing the union join graph as well as computing the subset graph for acyclic hypergraphs and their subclasses. In the union join graph G of an acyclic hypergraph H, each vertex of G represents a hyperedge of H and two vertices of G are adjacent if there exits a join tree T for H such that the corresponding hyperedges are adjacent in T. The subset graph of a hypergraph H is a directed graph where each vertex represents a hyperedge of H and there is a directed edge from a vertex u to a vertex v if the hyperedge corresponding to u is a subset of the hyperedge corresponding to v.
For a given hypergraph H=(V,E), let n=|V|, m=|E|, and N=∑E∈ε|E|. We show that, if the Strong Exponential Time Hypothesis is true, both problems cannot be solved in O(N2−ε) time for α-acyclic hypergraphs and any constant ε>0, even if the created graph is sparse. Additionally, we present algorithms that solve both problems in O(N2/logN+|G|) time forα-acyclic hypergraphs, in O(N log(n+m)+|G|) time for β-acyclic hypergraphs, and in O(N+|G|) time for γ-acyclic hypergraphs as well as for interval hypergraphs, where |G| is the size of the computed graph.
Leitert A. (2021) Computing the Union Join and Subset Graph of Acyclic Hypergraphs in Subquadratic Time. In: Lubiw A., Salavatipour M. (eds) Algorithms and Data Structures. WADS 2021. Lecture Notes in Computer Science, vol 12808. Springer, Cham. https://doi.org/10.1007/978-3-030-83508-8_41
Workshop on Algorithms and Data Structures
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