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Title: The Discrepancy of Shortest Paths
The hereditary discrepancy of a set system is a quantitative measure of the pseudorandom properties of the system. Roughly speaking, hereditary discrepancy measures how well one can 2-color the elements of the system so that each set contains approximately the same number of elements of each color. Hereditary discrepancy has numerous applications in computational geometry, communication complexity and derandomization. More recently, the hereditary discrepancy of the set system of shortest paths has found applications in differential privacy [Chen et al. SODA 23]. The contribution of this paper is to improve the upper and lower bounds on the hereditary discrepancy of set systems of unique shortest paths in graphs. In particular, we show that any system of unique shortest paths in an undirected weighted graph has hereditary discrepancy O(n^{1/4}), and we construct lower bound examples demonstrating that this bound is tight up to polylog n factors. Our lower bounds hold even for planar graphs and bipartite graphs, and improve a previous lower bound of Ω(n^{1/6}) obtained by applying the trace bound of Chazelle and Lvov [SoCG'00] to a classical point-line system of Erdős. As applications, we improve the lower bound on the additive error for differentially-private all pairs shortest distances from Ω(n^{1/6}) [Chen et al. SODA 23] to Ω̃(n^{1/4}), and we improve the lower bound on additive error for the differentially-private all sets range queries problem to Ω̃(n^{1/4}), which is tight up to polylog n factors [Deng et al. WADS 23].  more » « less
Award ID(s):
2118953 2208663 2220271
PAR ID:
10524535
Author(s) / Creator(s):
; ; ; ; ;
Editor(s):
Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Volume:
297
ISSN:
1868-8969
ISBN:
978-3-95977-322-5
Page Range / eLocation ID:
297-297
Subject(s) / Keyword(s):
Discrepancy hereditary discrepancy shortest paths differential privacy Theory of computation → Shortest paths Theory of computation → Computational geometry
Format(s):
Medium: X Size: 20 pages; 916165 bytes Other: application/pdf
Size(s):
20 pages 916165 bytes
Right(s):
Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
Sponsoring Org:
National Science Foundation
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