HERO: A Hierarchical Set Partitioning and Join Framework for Speeding up the Set Intersection Over Graphs

As one of the most primitive operators in graph algorithms, such as the triangle counting, maximal clique enumeration, and subgraph listing, a set intersection operator returns common vertices between any two given sets of vertices in data graphs. It is therefore very important to accelerate the set intersection, which will benefit a bunch of tasks that take it as a built-in block. Existing works on the set intersection usually followed the merge intersection or galloping-search framework, and most optimization research focused on how to leverage the SIMD hardware instructions. In this paper, we propose a novel multi-level set intersection framework, namely hierarchical set partitioning and join (HERO), by using our well-designed set intersection bitmap tree (SIB-tree) index, which is independent of SIMD instructions and completely orthogonal to the merge intersection framework. We recursively decompose the set intersection task into small-sized subtasks and solve each subtask using bitmap and boolean AND operations. To sufficiently achieve the acceleration brought by our proposed intersection approach, we formulate a graph reordering problem, prove its NP-hardness, and then develop a heuristic algorithm to tackle this problem. Extensive experiments on real-world graphs have been conducted to confirm the efficiency and effectiveness of our HERO approach. The speedup over classic merge intersection achieves up to 188x and 176x for triangle counting and maximal clique enumeration, respectively.