DPTL plus : Efficient Parallel Triangle Listing on Batch -Dynamic Graphs

Triangle listing is an important topic in many practical applications. We have observed that this problem has not yet been studied systematically in the context of batchdynamic graphs. In this paper, we aim to fill this gap by developing novel and efficient parallel solutions. Specifically, given a graph G and a batch-update of edges B, we report the updated triangles (deleted triangles and new triangles) resulting from the batch of updates. We notice that it is cost expensive to directly apply state-of-the-art triangle listing algorithms because they are designed to enumerate the complete set of triangles from a given graph, whereas only the updated ones are the relevant output for our problem setting. In this paper, we developed an efficient algorithm, namely DPTL, based on a newly designed orientation technique, which only outputs the updated triangles while ensuring that each triangle solution is identified without any duplicate solutions. We follow up by taking advantage of a graph's degree distributions and designed a more sophisticated algorithm, namely DPTL+. We show that DPTL+ can achieve the best performance in terms of both practical performance and theoretical time complexity. Our comprehensive experiments over 28 real-life large graphs show the superior performance of the DPTL+ algorithm when compared against DPTL and two baseline solutions. Theoretically, we also show that DPTL+ has a time complexity of 8(E min{deg (u), deg (v)} m) where deg (x) is the degree of a vertex x, and m is the number of edges adjacent to the vertices in the batch-update. This time complexity is more promising than that of other solutions.