GRAIL: a scalable index for reachability queries in very large graphs

Abstract

Given a large directed graph, rapidly answering reachability queries between source and target nodes is an important problem. Existing methods for reachability tradeoff indexing time and space versus query time performance. However, the biggest limitation of existing methods is that they do not scale to very large real-world graphs. We present a simple yet scalable reachability index, called GRAIL, that is based on the idea of randomized interval labeling and that can effectively handle very large graphs. Based on an extensive set of experiments, we show that while more sophisticated methods work better on small graphs, GRAIL is the only index that can scale to millions of nodes and edges. GRAIL has linear indexing time and space, and the query time ranges from constant time to being linear in the graph order and size. Our reference C++ implementations are open source and available for download at

An early look at the LDBC social network benchmark's business intelligence workload

“…This optimization is an example of an exploitation of CP-7.1 Path pattern reuse, which for this query would result in an improvement of ≈ 1.8 − 2× over simple top-k pushdown, as shown in Table 1. Reachability indexes are a form of reusing patterns and are an active research area [34]. This chokepoint was partially designed to stimulate such research efforts.…”

FERRARI: Flexible and efficient reachability range assignment for graph indexing

“…The recently proposed GRAIL reachability index [22], [23] further extends this idea by labeling the vertices with approximate intervals: Suppose that for every node v we replace the set I(v) by a single interval…”

“…However, especially in the case of positive queries, a large portion of the graph still has to be expanded. While extensions have been proposed to GRAIL to improve performance on positive queries [23], the processing time in these cases remains high. Furthermore, while an increase of the number of intervals assigned to the nodes potentially reduces false positive elements, no guarantee can be made due to the heuristic nature of the underlying algorithm.…”

“…Similar to previously proposed index structures [1], [19], [22], [23], we use intervals to encode reachability relationships of the vertices. However, in contrast to existing approaches, FERRARI can be regarded as an adaptive transitive closure compression algorithm.…”

“…While many proposed index structures can easily handle small to medium-size graphs comprising hundreds of thousands of nodes (e.g., [1], [4], [6], [11], [12], [13], [16], [18], [19], [20]), massive problem instances still remain a challenge to most of them. The only technique that can cope with webscale graphs while satisfying the requirements of restricted index size and fast query processing time, employs guided online search [22], [23], leading to an index structure that is competitive in terms of its construction time and storage space consumption, yet speeds up reachability query answering significantly when compared to a simple DFS/BFS traversal of the graph. However, it suffers from two major drawbacks.…”