The Discrepancy of Shortest Paths

Authors Greg Bodwin, Chengyuan Deng, Jie Gao , Gary Hoppenworth, Jalaj Upadhyay, Chen Wang



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Author Details

Greg Bodwin
  • Computer Science and Engineering, University of Michigan, Ann Arbor, MI, USA
Chengyuan Deng
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Jie Gao
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Gary Hoppenworth
  • Computer Science and Engineering, University of Michigan, Ann Arbor, MI, USA
Jalaj Upadhyay
  • Management Science and Information Systems, Rutgers University, Piscataway, NJ, USA
Chen Wang
  • Department of Computer Science, Rice University, Houston, TX, USA
  • Computer Science and Engineering, Texas A&M University, College Station, TX, USA

Cite AsGet BibTex

Greg Bodwin, Chengyuan Deng, Jie Gao, Gary Hoppenworth, Jalaj Upadhyay, and Chen Wang. The Discrepancy of Shortest Paths. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 27:1-27:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.27

Abstract

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].

Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
  • Theory of computation → Computational geometry
Keywords
  • Discrepancy
  • hereditary discrepancy
  • shortest paths
  • differential privacy

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