TY - GEN

T1 - Triangles and girth in disk graphs and transmission graphs

AU - Kaplan, Haim

AU - Klost, Katharina

AU - Mulzer, Wolfgang

AU - Roditty, Liam

AU - Seiferth, Paul

AU - Sharir, Micha

N1 - Publisher Copyright:
© Haim Kaplan, Katharina Klost, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir.

PY - 2019/9

Y1 - 2019/9

N2 - Let S ⊂ ℝ2 be a set of n sites, where each s ∈ S has an associated radius rs > 0. The disk graph D(S) is the undirected graph with vertex set S and an undirected edge between two sites s, t ∈ S if and only if |st| ≤ rs + rt, i.e., if the disks with centers s and t and respective radii rs and rt intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph T(S) is the directed graph with vertex set S and a directed edge from a site s to a site t if and only if |st| ≤ rs, i.e., if t lies in the disk with center s and radius rs. We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in D(S) and in T(S). These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in D(S) and in T(S) can be found in O(n log n) expected time, and that the (weighted) girth of D(S) can be found in O(n log n) expected time. For this, we develop new tools for batched range searching that may be of independent interest.

AB - Let S ⊂ ℝ2 be a set of n sites, where each s ∈ S has an associated radius rs > 0. The disk graph D(S) is the undirected graph with vertex set S and an undirected edge between two sites s, t ∈ S if and only if |st| ≤ rs + rt, i.e., if the disks with centers s and t and respective radii rs and rt intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph T(S) is the directed graph with vertex set S and a directed edge from a site s to a site t if and only if |st| ≤ rs, i.e., if t lies in the disk with center s and radius rs. We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in D(S) and in T(S). These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in D(S) and in T(S) can be found in O(n log n) expected time, and that the (weighted) girth of D(S) can be found in O(n log n) expected time. For this, we develop new tools for batched range searching that may be of independent interest.

KW - Disk graph

KW - Girth

KW - Transmission graph

KW - Triangle

UR - http://www.scopus.com/inward/record.url?scp=85074820557&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ESA.2019.64

DO - 10.4230/LIPIcs.ESA.2019.64

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AN - SCOPUS:85074820557

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 27th Annual European Symposium on Algorithms, ESA 2019

A2 - Bender, Michael A.

A2 - Svensson, Ola

A2 - Herman, Grzegorz

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 27th Annual European Symposium on Algorithms, ESA 2019

Y2 - 9 September 2019 through 11 September 2019

ER -