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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.WADS.2025.9

Online Routing in Directed Yao₄^∞ Graphs

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

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract

The x⃑{Yao₄^∞} and x⃑{Yao₄} graphs are two families of directed geometric graphs whose vertices are points in the plane, and each vertex has up to four outgoing edges. Consider a horizontal and a vertical line through each vertex v, defining four quadrants around v. Then v has an outgoing edge to the closest vertex in each of its four quadrants. When the distance is measured using the Euclidean norm, the resulting graph is the x⃑{Yao₄} graph, whereas with the L_∞-norm, we obtain the x⃑{Yao^{∞}₄} graph, which is a sub-graph of the well-known L_∞-Delaunay graph. In this paper, we provide a local routing algorithm with routing ratio at most 85.22 for x⃑{Yao^{∞}₄} graphs. Prior to this work, no constant spanning or routing ratios for x⃑{Yao₄^∞} graphs were previously known. Now, x⃑{Yao₄^∞} graphs are the sparsest family of directed planar graphs supporting a competitive local routing strategy. Furthermore, we show that no local routing algorithm for x⃑{Yao₄^∞} graphs can have a routing ratio lower than 7+√2≈ 8.41. Moreover, we prove that the spanning ratio is at least 5+√2≈ 6.41 in the worst case. The techniques we develop in this paper also allow us to prove lower bounds of 7-√3+√{5-2√3}≈ 6.51 and 7+√2 for the spanning and routing ratios of x⃑{Yao₄}, respectively.

Cite as

Prosenjit Bose, Jean-Lou De Carufel, and John Stuart. Online Routing in Directed Yao₄^∞ Graphs. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 9:1-9:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bose_et_al:LIPIcs.WADS.2025.9,
  author =	{Bose, Prosenjit and De Carufel, Jean-Lou and Stuart, John},
  title =	{{Online Routing in Directed Yao₄^∞ Graphs}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{9:1--9:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.9},
  URN =		{urn:nbn:de:0030-drops-242404},
  doi =		{10.4230/LIPIcs.WADS.2025.9},
  annote =	{Keywords: Geometric Spanners, Yao Graphs, Local Routing Algorithms}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2024.14

Routing from Pentagon to Octagon Delaunay Graphs

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

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

Abstract

The standard Delaunay triangulation is a geometric graph whose vertices are points in the plane, and two vertices share an edge if they lie on the boundary of an empty disk. If the disk is replaced with a homothet of a fixed convex shape C, then the resulting graph is called a C-Delaunay graph. We study the problem of local routing in C-Delaunay graphs where C is a regular polygon having five to eight sides. In particular, we generalize the routing algorithm of Chew for square-Delaunay graphs (Chew. SCG 1986, 169-177) in order to obtain the following approximate upper bounds of 4.640, 6.429, 8.531 and 4.054 on the spanning and routing ratios for pentagon-, hexagon-, septagon-, and octagon-Delaunay graphs, respectively. The exact expression for the upper bounds of the routing ratio is Ψ(n):= √{1+((cos(2π/n)+n-1)/sin(2π/n))^2} (if n ∈ {5,6,7}), √{1+((cos(π/8)cos(3π/8)+3)/(cos(π/8)sin(3π/8)))^2} (if n = 8). We show that these bounds are tight for the output of our routing algorithm by providing a point set where these bounds are achieved. We also include lower bounds of 1.708 and 1.995 on the spanning and routing ratios of the pentagon-Delaunay graph. Our upper bounds yield a significant improvement over the previous routing ratio upper bounds for this problem, which previously sat at around 400 for the pentagon, septagon, and octagon as well as 18 for the hexagon. Our routing ratios also provide significant improvements over the previously best known spanning ratios for pentagon-, septagon- and octagon-Delaunay graphs, which were around 45.

Cite as

Prosenjit Bose, Jean-Lou De Carufel, and John Stuart. Routing from Pentagon to Octagon Delaunay Graphs. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bose_et_al:LIPIcs.ISAAC.2024.14,
  author =	{Bose, Prosenjit and De Carufel, Jean-Lou and Stuart, John},
  title =	{{Routing from Pentagon to Octagon Delaunay Graphs}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{14:1--14:16},
  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.2024.14},
  URN =		{urn:nbn:de:0030-drops-221411},
  doi =		{10.4230/LIPIcs.ISAAC.2024.14},
  annote =	{Keywords: Geometric Spanners, Generalized Delaunay Graphs, Local Routing Algorithms}
}

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Documents authored by Stuart, John

The Spanning Ratio of the Directed Θ₆-Graph Is 5

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Online Routing in Directed Yao₄^∞ Graphs

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Routing from Pentagon to Octagon Delaunay Graphs

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