Simple, fast, and scalable reachability oracle

Abstract

A reachability oracle (or hop labeling) assigns each vertex v two sets of vertices: Lout(v) and Lin (v), such that u reaches v iff Lout(u) ∩ Lin(v) = ∅. Despite their simplicity and elegance, reachability oracles have failed to achieve efficiency in more than ten years since their introduction: the main problem is high construction cost, which stems from a set-cover framework and the need to materialize transitive closure. In this paper, we present two simple and efficient labeling algorithms, Hierarchical-Labeling and Distribution-Labeling, which can work on massive real-world graphs: their construction time is an order of magnitude faster than the setcover based labeling approach, and transitive closure materialization is not needed. On large graphs, their index sizes and their query performance can now beat the state-of-the-art transitive closure compression and online search approaches. large graphs, their index sizes and their query performance beat the state-of-the-art transitive closure compression and online search approaches [21,35,23,35,12,37]. Using these two algorithms, the power of hop labeling is finally unleashed and a fast, compact and scalable reachability oracle becomes a reality.

Reachability querying: an independent permutation labeling approach

“…), where h is the number of backbones, and g(·) is the cost of computing 2-Hop label for v. Also, in [19], Jin et al propose DL (Distribution Labeling). It is based on a list of vertices in an order Label+G: Tree+SSPI [7] and GRIPP [26] are two works that use an interval label for every vertex over a spanning tree, and attempt to reduce depth-first-search (DFS) time if needed at run-time.…”

“…Given the linear construction time and index size, the query time for Tree+SSPI and GRIPP is O(m − n) and the query time for GRAIL and Ferrari can be up to O(n + m), which means it cannot deal with large dense graphs. In [10,19], the authors indicate that TF-Label, HL, and DL outperform GRAIL, in terms of query time in practice.…”

“…We denote our approach as IP+ (Algorithm 2), which uses IP labels with two additional Level labels and HVLabel. And we compare IP+ with the state-of-the-art reachability approaches including GRAIL [29], GRAIL * [30], ScaGRAIL [17,29], PWAH8 [27], TF-Label [10], HL [19], DL [19], and Ferrari [24]. Here, GRAIL and Ferrari are two state-of-art Label+G approaches, and their index size and construction time are in linear.…”

“…The real applications that need this operation are many and can be found among online social networks, biological networks, ontology, transportation networks, etc. The reachability query has been extensively studied over a decade [1,16,13,23,11,28,7,26,12,8,21,20,29,27,9,10,19,24,31], and the early work can be traced back to 1989 to compute transitive closure (TC) over a graph. However, it is still an unsolved question whether we can do faster to answer reachability queries online over even larger and/or even denser graphs with the possible minimum cost (time/space) for offline precomputing and preparation.…”

“…One is called Label-Only. By the name, the approaches in this category only use the labels computed to answer reachability queries, and include Chain-Cover [16,8], Tree-Cover [1], Dual-Label [28], 2-Hop [13], Path-Tree [21], 3-Hop [20], PWAH8 [27], TF-Label [10], HL [19], and DL [19]. The other is called Label+G.…”