FERRARI: Flexible and efficient reachability range assignment for graph indexing

Abstract

Abstract-In this paper, we propose a scalable and highly efficient index structure for the reachability problem over graphs. We build on the well-known node interval labeling scheme where the set of vertices reachable from a particular node is compactly encoded as a collection of node identifier ranges. We impose an explicit bound on the size of the index and flexibly assign approximate reachability ranges to nodes of the graph such that the number of index probes to answer a query is minimized. The resulting tunable index structure generates a better range labeling if the space budget is increased, thus providing a direct control over the trade off between index size and the query processing performance. By using a fast recursive querying method in conjunction with our index structure, we show that, in practice, reachability queries can be answered in the order of microseconds on an off-the-shelf computer -even for the case of massive-scale real world graphs. Our claims are supported by an extensive set of experimental results using a multitude of benchmark and real-world web-scale graph datasets.

Reachability querying: an independent permutation labeling approach

“…The query time is either O(k) using the label only or O(n + m) when it needs to do DFS. Seufert et al propose Ferrari [24]. Like GRAIL, Ferrari computes up to k intervals for every vertex over an optimal spanning tree computed by [1].…”

“…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.…”

“…Here, GRAIL and Ferrari are two state-of-art Label+G approaches, and their index size and construction time are in linear. We test Ferrari using the Ferrari-G index given in [24], because Ferrari-G is scalable to handle massive-scale graphs. GRAIL * is the improved version of GRAIL [30] and ScaGRAIL is the implementation of GRAIL in the SCARAB framework [17].…”

“…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.…”

“…By the name, the approaches in this category use labels computed where possible, and conduct depth-first-search (DFS) at run-time, if the reachability queries cannot be answered using the labels only. Such approaches include Tree+SSPI [7], GRIPP [26], GRAIL [29], and Ferrari [24].…”