Recursive Graph Deduction and Reachability Queries

In this paper, we discuss an adjustable strategy for the transitive closure compression to support graph reachability queries, asking whether a given node u in a directed graph G is reachable from another node v through a path. The main idea behind it is to define a series of graph deductions G0(V0, E0) (= G), G1(V1, E1), ..., Gk(Vk, Ek) with ni > ni+1 (i = 0, ..., k 1), where ni = |Vi|. Each node v will be associated with an interval sequence [ v 0 α , v 0 β ), ..., [ v j α , v j β ) (j ≤ k 1) with each [ v i α , v i β ) used to check reachability in Gi. Together with a subgraph structure (called the core graph of G) and a small matrix (called the core matrix), the reachability checking can be done in O(k) time. The size of the compressed transitive closure is bounded by O(kn0 + bknk), where bk is the width of Gk, defined to be the size of a largest node subset U of Gk such that for every pair of nodes u, v ∈ U, there does not exist a path from u to v or from v to u in Gk. The time for generating such a compressed transitive closure is bounded by O(e0 + kn0 + 2 k n + bknk k b ), where e0 = |E0|. For different applications, we can adjust k to different constants to get effective space reduction, but still have constant query time.