Efficient Trace Frequency Queries in Sparse Graphs

Understanding how a vertex relates to a set of vertices is a fundamental task in graph analysis. Given a graph G and a vertex set X ⊆ V(G) , consider the collection of subsets of the form N(u) ∩ X where u ranges over all vertices outside X. These intersections, which we call the traces of X, capture all ways vertices in G connect to X, and in this paper we consider the problem of listing these traces efficiently, and the related problem of recording the multiplicity (frequency) of each trace.For a given query set X, both problems have obvious algorithms with running time O(|N(X)| · |X|) and conditional lower bounds suggest that, on general graphs, one cannot expect better. However, in certain sparse graph classes, more efficient algorithms are possible: Drange et al.(IPEC 2023) used a data structure that answers trace queries in d-degenerate graphs with linear initialisation time and query time that only depends on the query set X and d. However, the query time is exponential in |X|, which makes this approach impractical. By using a stronger parameter than degeneracy, namely the strong 2-colouring number s_2 , we construct a data structure in O(d ·‖ G‖ ) time, which answers subsequent trace frequency queries in time O ((d^2 + s_2^d+2)|X| ) , where ‖ G‖ is the number of edges of G, s_2 is the strong 2-colouring number and d the degeneracy of a suitable ordering of G. We demonstrate that this data structure is indeed practical and that it beats the simple, obvious alternative in almost all tested settings, using a collection of 217 real-world networks with up to 1.1M edges. As part of this effort, we demonstrate that computing an ordering with a small strong 2-colouring number is feasible with a simple heuristic.