Eccentricity Heuristics through Sublinear Analysis Lenses

The eccentricity of a node in a graph is its maximal shortestpath distance to any other node. Computing all eccentricities is a basic task in large-scale graph mining. Shun (KDD 2015) empirically studied two simple heuristics for this task: k-BFS1, based on parallel BFS from a small sample of nodes, was shown to work well on a variety of graphs; kBFS2, a two-phase version, was shown to outperform stateof-the-art algorithms by up to orders of magnitude. This empirical success stands in apparent contrast to recent theoretical hardness results on approximating all eccentricities (Backurs et al., STOC 2018). This paper aims to formally explain the performance of these heuristics, by studying them through computational models designed for sublinear time or sublinear space algorithms. We use the proposed framework to derive improved variants, which retain their practicality while having better performance and formal guarantees. 1. For k-BFS1, we draw a connection to diameter property testing (Parnas and Ron, Random Struct Alg. 2002). It is not hard to observe that k-BFS1 essentially tests the values of all eccentricities simultaneously, in the classical property testing sense. We show that the same guarantee is achieved by a more efficient algorithm, whose work is nearly linear in the number of nodes and independent of the number of edges. By utilizing the connection in the opposite direction, we also obtain some results on classical testing of the graph radius and diameter. 2. For k-BFS2, we draw a connection to the streaming Set Cover algorithm of Demaine et al. (DISC 2014). We use it to suggest a variant of k-BFS2 with similar work and depth bounds, which is guaranteed to compute almost all eccentricities exactly, if the graph satisfies a condition we call small eccentric cover. The condition can be ascertained for all real-world graph used in Shun (KDD 2015) and in our experiments. Our experimental results on real-world graphs demonstrate the validity of our analysis and the empirical advantage of the proposed algorithms.