Giving Some Slack: Shortcuts and Transitive Closure Compressions

We consider the fundamental problems of reachability shortcuts and compression schemes of the transitive closure (TC) of n-vertex directed acyclic graphs (DAGs) G when we are allowed to neglect the distance (or reachability) constraints for an ϵ fraction of the pairs in the transitive closure of G, denoted by TC(G). Shortcuts with Slack. For a directed graph G = (V, E), a d-reachability shortcut is a set of edges H ⊆ TC(G), whose addition decreases the directed diameter of G to be at most d. We introduce the notion of shortcuts with slack which provide the desired distance bound d for all but a small fraction ϵ of the vertex pairs in TC(G). For ϵ ∈ (0, 1), a (d, ϵ)-shortcut H ⊆ TC(G) is a subset of edges with the property that distG∪H(u, v) ≤ d for at least (1 − ϵ) fraction of the (u, v) pairs in TC(G). Our constructions hold for any DAG G and their size bounds are parameterized by the width of the graph G defined by the smallest number of directed paths in G that cover all vertices in G. For every ϵ ∈ (0, 1] and integer d ≥ 5, every n-vertex DAG G of width ω admits a (d, ϵ)-shortcut of size O^e(ω²/(ϵd)+n). A more delicate construction yields a (3, ϵ)-shortcut of size O^e(ω²/(ϵd)+n/ϵ), hence of linear size for ω ≤ √n. We show that without a slack (i.e., for ϵ = 0), graphs with ω ≤ √n cannot be shortcut to diameter below n^1/6 using a linear number of shortcut edges. There exists an n-vertex DAG G for which any (3, ϵ = 1/2 ^√log ^ω)-shortcut set has Ω(ω²/2 ^√log ^ω + n) edges. Hence, for d = O^e(1), our constructions are almost optimal. Approximate TC Representations. A key application of our shortcut's constructions is a (1−ϵ)-approximate all-successors data structure which given a vertex v, reports a list containing (1 − ϵ) fraction of the successors of v in the graph. We present a O^e(ω²/ϵ + n)-space data structure with a near linear (in the output size) query time. Using connections to Error Correcting Codes, we also present a near-matching space lower bound of Ω(ω² + n) bits (regardless of the query time) for constant ϵ. This improves upon the state-of-the-art space bounds of O(ω · n) for ϵ = 0 by the prior work of Jagadish [ACM Trans. Database Syst., 1990].