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 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].
Reachability Shortcuts
Width
DAG
Theory of computation~Graph algorithms analysis
79:1-79:15
Regular Paper
This project is funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 949083), and by the Israeli Science Foundation (ISF), grant No. 2084/18.
Shimon
Kogan
Shimon Kogan
Weizmann Institute of Science, Rehovot, Israel
Merav
Parter
Merav Parter
Weizmann Institute of Science, Rehovot, Israel
10.4230/LIPIcs.ESA.2024.79
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Shimon Kogan and Merav Parter
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