  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.
