Crossing Patterns in Nonplanar Road Networks

Eppstein, David; Gupta, Siddharth

doi:10.1145/3139958.3139999

Cited by 22 publications

(24 citation statements)

References 26 publications

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“…We note that k-planar graphs are known to be (k + 1)-quasiplanar [4,34]. Furthermore, we investigate the relationship between k-gap-planar graphs and d-degenerate crossing graphs, a class of graphs recently introduced by Eppstein and Gupta [25]. • The complete graph K n is 1-gap-planar if and only if n ≤ 8 (Section 5).…”

Section: Introductionmentioning

confidence: 99%

Gap-planar graphs

Bae

Baffier

Jin

et al. 2018

Theoretical Computer Science

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We introduce the family of k-gap-planar graphs for k ≥ 0, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most k of its crossings. This definition is motivated by applications in edge casing, as a k-gap-planar graph can be drawn crossing-free after introducing at most k local gaps per edge. We present results on the maximum density of k-gap-planar graphs, their relationship to other classes of beyond-planar graphs, characterization of k-gap-planar complete graphs, and the computational complexity of recognizing kgap-planar graphs.

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Section: Introductionmentioning

confidence: 99%

Gap-planar graphs

Bae

Baffier

Jin

et al. 2018

Theoretical Computer Science

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“…By construction of the network, crossings are practically impossible in street networks [5]. Crossings are scarce in road networks and imply bridges and tunnels [6]. In syntactic dependency networks, C, the number of crossings, has been shown to be low with respect to random linear arrangements of the words of the sentences [7] and predictable to a large extent by the Euclidean distance between syntactically related words: crossings are more likely for dependencies involving distant words, for the range of distances that is typically found in real sentences [8].…”

mentioning

confidence: 99%

Edge crossings in random linear arrangements

Alemany-Puig¹,

Ferrer-i-Cancho²

2020

J. Stat. Mech.

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The interest in spatial networks where vertices are embedded in a one-dimensional space is growing. Remarkable examples of these networks are syntactic dependency trees and RNA structures. In this setup, the vertices of the network are arranged linearly and then edges may cross when drawn above the sequence of vertices. Recently, two aspects of the distribution of the number of crossings in uniformly random linear arrangements have been investigated: the expectation and the variance. While the computation of the expectation is straightforward, that of the variance is not. Here we present fast algorithms to calculate that variance in arbitrary graphs and forests. As for the latter, the algorithm calculates variance in linear time with respect to the number of vertices. This paves the way for many applications that rely on an exact but fast calculation of that variance. These algorithms are based on novel arithmetic expressions for the calculation of the variance that we develop from previous theoretical work.

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“…First, we build a separator hierarchy of the graph. This hierarchy can be constructed in O(n) time and space in planar graphs [32] and graphs with sparse crossing graphs [26]. However, we do not need the construction to take linear time, as this is not the bottleneck of the preprocessing.…”

Section: Preprocessingmentioning

confidence: 99%

“…Bounding the degeneracy of the crossing graph instead of the maximum degree accounts for, e.g., long tunnels that go under many street-level roads. Like planar graphs, the class of graphs with sparse crossing graphs is also hereditary and has O(n 0.5 )-size separators [26]. This is fortunate, because it means that we can use our data structures in applications dealing with road networks.…”

Section: Introductionmentioning

confidence: 99%

Reactive Proximity Data Structures for Graphs

Eppstein

Goodrich

Mamano

2018

LATIN 2018: Theoretical Informatics

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We consider data structures for graphs where we maintain a subset of the nodes called sites, and allow proximity queries, such as asking for the closest site to a query node, and update operations that enable or disable nodes as sites. We refer to a data structure that can efficiently react to such updates as reactive. We present novel reactive proximity data structures for graphs of polynomial expansion, i.e., the class of graphs with small separators, such as planar graphs and road networks. Our data structures can be used directly in several logistical problems and geographic information systems dealing with real-time data, such as emergency dispatching. We experimentally compare our data structure to Dijkstra's algorithm in a system emulating random queries in a real road network.

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Crossing Patterns in Nonplanar Road Networks

Cited by 22 publications

References 26 publications

Gap-planar graphs

Gap-planar graphs

Edge crossings in random linear arrangements

Reactive Proximity Data Structures for Graphs

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