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Documents authored by Hoppenworth, Gary

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.27
The Discrepancy of Shortest Paths

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

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

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].
Cite as

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)

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@InProceedings{bodwin_et_al:LIPIcs.ICALP.2024.27,
  author =	{Bodwin, Greg and Deng, Chengyuan and Gao, Jie and Hoppenworth, Gary and Upadhyay, Jalaj and Wang, Chen},
  title =	{{The Discrepancy of Shortest Paths}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{27:1--27:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.27},
  URN =		{urn:nbn:de:0030-drops-201705},
  doi =		{10.4230/LIPIcs.ICALP.2024.27},
  annote =	{Keywords: Discrepancy, hereditary discrepancy, shortest paths, differential privacy}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.28
Additive Spanner Lower Bounds with Optimal Inner Graph Structure

Authors: Greg Bodwin, Gary Hoppenworth, Virginia Vassilevska Williams, Nicole Wein, and Zixuan Xu

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
We construct n-node graphs on which any O(n)-size spanner has additive error at least +Ω(n^{3/17}), improving on the previous best lower bound of Ω(n^{1/7}) [Bodwin-Hoppenworth FOCS '22]. Our construction completes the first two steps of a particular three-step research program, introduced in prior work and overviewed here, aimed at producing tight bounds for the problem by aligning aspects of the upper and lower bound constructions. More specifically, we develop techniques that enable the use of inner graphs in the lower bound framework whose technical properties are provably tight with the corresponding assumptions made in the upper bounds. As an additional application of our techniques, we improve the corresponding lower bound for O(n)-size additive emulators to +Ω(n^{1/14}).
Cite as

Greg Bodwin, Gary Hoppenworth, Virginia Vassilevska Williams, Nicole Wein, and Zixuan Xu. Additive Spanner Lower Bounds with Optimal Inner Graph Structure. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bodwin_et_al:LIPIcs.ICALP.2024.28,
  author =	{Bodwin, Greg and Hoppenworth, Gary and Vassilevska Williams, Virginia and Wein, Nicole and Xu, Zixuan},
  title =	{{Additive Spanner Lower Bounds with Optimal Inner Graph Structure}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{28:1--28:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.28},
  URN =		{urn:nbn:de:0030-drops-201715},
  doi =		{10.4230/LIPIcs.ICALP.2024.28},
  annote =	{Keywords: Additive Spanners, Graph Theory}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.61
The Fine-Grained Complexity of Median and Center String Problems Under Edit Distance

Authors: Gary Hoppenworth, Jason W. Bentley, Daniel Gibney, and Sharma V. Thankachan

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
We present the first fine-grained complexity results on two classic problems on strings. The first one is the k-Median-Edit-Distance problem, where the input is a collection of k strings, each of length at most n, and the task is to find a new string that minimizes the sum of the edit distances from itself to all other strings in the input. Arising frequently in computational biology, this problem provides an important generalization of edit distance to multiple strings and is similar to the multiple sequence alignment problem in bioinformatics. We demonstrate that for any ε > 0 and k ≥ 2, an O(n^{k-ε}) time solution for the k-Median-Edit-Distance problem over an alphabet of size O(k) refutes the Strong Exponential Time Hypothesis (SETH). This provides the first matching conditional lower bound for the O(n^k) time algorithm established in 1975 by Sankoff. The second problem we study is the k-Center-Edit-Distance problem. Here also, the input is a collection of k strings, each of length at most n. The task is to find a new string that minimizes the maximum edit distance from itself to any other string in the input. We prove that the same conditional lower bound as before holds. Our results also imply new conditional lower bounds for the k-Tree-Alignment and the k-Bottleneck-Tree-Alignment problems studied in phylogenetics.
Cite as

Gary Hoppenworth, Jason W. Bentley, Daniel Gibney, and Sharma V. Thankachan. The Fine-Grained Complexity of Median and Center String Problems Under Edit Distance. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 61:1-61:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hoppenworth_et_al:LIPIcs.ESA.2020.61,
  author =	{Hoppenworth, Gary and Bentley, Jason W. and Gibney, Daniel and Thankachan, Sharma V.},
  title =	{{The Fine-Grained Complexity of Median and Center String Problems Under Edit Distance}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{61:1--61:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.61},
  URN =		{urn:nbn:de:0030-drops-129278},
  doi =		{10.4230/LIPIcs.ESA.2020.61},
  annote =	{Keywords: Edit Distance, Median String, Center String, SETH}
}
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