Giving Some Slack: Shortcuts and Transitive Closure Compressions

Abstract

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 dist_{G ∪ 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 Õ({ω}²/(ε d)+n). A more delicate construction yields a (3,ε)-shortcut of size Õ({ω}²/(ε d)+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({ω}²/2^{√{log ω}}+n) edges. Hence, for d = Õ(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 Õ({ω}²/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({ω}²+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].