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Documents authored by Hill, Darryl

Document
DOI: 10.4230/LIPIcs.SoCG.2026.20
The Spanning Ratio of the Directed Θ₆-Graph Is 5

Authors: Prosenjit Bose, Jean-Lou De Carufel, John Stuart, and Darryl Hill

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Given a finite set P ⊂ ℝ², the directed Theta-6 graph, denoted Θ₆(P), is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The Θ₆(P)-graph is defined as follows: the plane around each point u ∈ P is partitioned into 6 equiangular cones with apex u, and in each cone, u is joined to the point whose projection on the bisector of the cone is closest. Equivalently, the Θ₆(P)-graph contains an edge from u to v exactly when the interior of ∇_u^v is disjoint from P, where ∇_u^v is the unique equilateral triangle containing u on a corner, v on the opposite side, and whose sides are parallel to the cone boundaries. It was previously shown that the spanning ratio of the Θ₆(P)-graph is between 4 and 7 in the worst case (Akitaya, Biniaz, and Bose Comput. Geom., 105-106:101881, 2022). We close this gap by showing a tight spanning ratio of 5. This is the first tight bound proven for the spanning ratio of any Θ_k(P)-graph. Our lower bound models a long path by mapping it to a converging series. Our upper bound proof uses techniques novel to the area of spanners. We use linear programming to prove that among several candidate paths, there exists a path satisfying our bound.
Cite as

Prosenjit Bose, Jean-Lou De Carufel, John Stuart, and Darryl Hill. The Spanning Ratio of the Directed Θ₆-Graph Is 5. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bose_et_al:LIPIcs.SoCG.2026.20,
  author =	{Bose, Prosenjit and De Carufel, Jean-Lou and Stuart, John and Hill, Darryl},
  title =	{{The Spanning Ratio of the Directed \Theta₆-Graph Is 5}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{20:1--20:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.20},
  URN =		{urn:nbn:de:0030-drops-258268},
  doi =		{10.4230/LIPIcs.SoCG.2026.20},
  annote =	{Keywords: Geometric Spanners, Theta Graphs, Directed Theta Graphs, Spanning Ratio, Computational Geometry}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2024.15
On the Spanning and Routing Ratios of the Yao-Four Graph

Authors: Prosenjit Bose, Darryl Hill, Michiel Smid, and Tyler Tuttle

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
The Yao graph is a geometric spanner that was independently introduced by Yao [SIAM J. Comput., 1982] and Flinchbaugh and Jones [SIAM J. Algebr. Discret. Appl., 1981]. We prove that for any two vertices of the undirected version of the Yao graph with four cones, there is a path between them with length at most 13 + 5/√2 ≈ 16.54 times the Euclidean distance between the vertices, improving the previous best bound of approximately 54.62. We also present an online routing algorithm for the directed Yao graph with four cones that constructs a path between any two vertices with length at most 17 + 9/√2 ≈ 23.36 times the Euclidean distance between the vertices. This is the first routing algorithm for a directed Yao graph with fewer than six cones. The algorithm uses knowledge of the coordinates of the current vertex, the (up to) four neighbours of the current vertex, and the destination vertex to make a routing decision. It also uses one additional bit of memory. We show how to dispense with this single bit at the cost of increasing the length of the path to √{331 + 154√2} ≈ 23.43 times the Euclidean distance between the vertices.
Cite as

Prosenjit Bose, Darryl Hill, Michiel Smid, and Tyler Tuttle. On the Spanning and Routing Ratios of the Yao-Four Graph. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bose_et_al:LIPIcs.ISAAC.2024.15,
  author =	{Bose, Prosenjit and Hill, Darryl and Smid, Michiel and Tuttle, Tyler},
  title =	{{On the Spanning and Routing Ratios of the Yao-Four Graph}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{15:1--15:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.15},
  URN =		{urn:nbn:de:0030-drops-221422},
  doi =		{10.4230/LIPIcs.ISAAC.2024.15},
  annote =	{Keywords: Yao graph, online routing, geometric spanners}
}
Document
DOI: 10.4230/LIPIcs.ESA.2018.22
Improved Routing on the Delaunay Triangulation

Authors: Nicolas Bonichon, Prosenjit Bose, Jean-Lou De Carufel, Vincent Despré, Darryl Hill, and Michiel Smid

Published in: LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Abstract
A geometric graph G=(P,E) is a set of points in the plane and edges between pairs of points, where the weight of an edge is equal to the Euclidean distance between its two endpoints. In local routing we find a path through G from a source vertex s to a destination vertex t, using only knowledge of the current vertex, its incident edges, and the locations of s and t. We present an algorithm for local routing on the Delaunay triangulation, and show that it finds a path between a source vertex s and a target vertex t that is not longer than 3.56|st|, improving the previous bound of 5.9|st|.
Cite as

Nicolas Bonichon, Prosenjit Bose, Jean-Lou De Carufel, Vincent Despré, Darryl Hill, and Michiel Smid. Improved Routing on the Delaunay Triangulation. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 22:1-22:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bonichon_et_al:LIPIcs.ESA.2018.22,
  author =	{Bonichon, Nicolas and Bose, Prosenjit and De Carufel, Jean-Lou and Despr\'{e}, Vincent and Hill, Darryl and Smid, Michiel},
  title =	{{Improved Routing on the Delaunay Triangulation}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{22:1--22:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Azar, Yossi and Bast, Hannah and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.22},
  URN =		{urn:nbn:de:0030-drops-94857},
  doi =		{10.4230/LIPIcs.ESA.2018.22},
  annote =	{Keywords: Delaunay, local routing, geometric, graph}
}
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