Fast approximation algorithms for p-centres in large delta-hyperbolic graphs

We provide a quasi-linear time algorithm for the p-center clustering problem with an additive error less than or equal to 3 times the input graph's hyperbolic constant. Specifically, for the graph G = (V;E) with n vertices, m edges and hyperbolic constant delta, we construct an algorithm for p-center clustering that runs in time O(p(delta + 1)(n + m) log(n)) with radius not exceeding r_p +delta_ when p is less than or equal to 2 and r_p + 3delta when p is greater than 3, where r_p is the optimal radius of the p-center clusters. Prior work had identified p-centers with accuracy r_p+delta but with cubic time complexity, O((n^3 log n + n^2 m) log(diam(G))), which is impractical for large graphs. We also provide computational results for networks with 10Ks to 2M edges showing the proposed algorithm not only runs considerably faster than the prior work as shown by theory, but also provides similar clustering results in practice.