Crossing patterns in Nonplanar Road networks

Crossing Patterns in Nonplanar Road
Networks
David Eppstein and Siddharth Gupta
Dept. of Computer Science
Univ. of California, Irvine
Conference: ACM SIGSPATIAL - 2017
November 7-10, Redondo Beach, California, USA
Presented By: Ajinkya Ghadge

Overview
● Problem Statement
● Definitions and Basic concepts
● Introduction and Main idea
● Past Work
● Crossing Graph
● Properties of Sparse Graphs and Definitions
● Experiments
● Results
● Analysis
● Theoretical analysis of graphs with sparse crossing
● Conclusion
● References

Problem Statement
● Study the sparsity property of "crossing graphs" of real world road networks
● To show that in large road networks crossing graphs have connected components
which are primarily trees and the remaining non tree components are typically
sparse.
● To theoretically prove that when an embedded graph has a sparse "crossing graph", it
has other desirable properties that lead to fast algorithms of shortest path and other
algos important in GIS. All the graphs considered have polynomial expansion, meaning
all their subgraphs have small separators.

Definitions and Basic Concepts
● Planar and Non Planar Graph
● Degeneracy of a Graph (d) : A graph G is said to be k-degenerate if every subgraph of G has
a vertex of degree at most k that is, some vertex in the subgraph touches k or fewer of the
subgraph's edges.

Introduction and Main Idea
● Most of the road networks are modelled graph theoretically
● Most of the past work have assumed the road networks to be planar
● Unfortunately, the author discovered that road networks are not planar but they contain
many crossings! [8]
● We desire a model that is near planar ! Why?
● Authors developed a mathematical model for nonplanar road networks in terms of
sparseness of the crossing graph. They analyzed the Urban Road Network Dataset and
proved that the networks in this model have polynomial expansion and due to this
different algos can be adapted to work on these networks.

Past Work
The past work by the same author tried to model the non-planarities in planar road
networks in two different ways.
1) Studying (non planar) road network through an algorithmic lens ACM SIGSPATIAL - 2008
● Posited that road networks are subgraphs of the intersection graphs of systems of
disks (the disks centered at each intersection of roads with radius equal to half the
length of the longest segment of roads meeting at that intersection)
2) Going off road: Transversal complexity in road networks ACM SIGSPATIAL - 2008
● Tried to justify low number of crossings in the road networks by showing that
randomly chosen lines (modeling, for instance, a highway cutting across an older city
grid) typically have a sublinear number of crossings with other roads.

● This paper studies the road networks using a new auxiliary graph called as a crossing graph
Crossing Graph!

Properties of Sparse Graphs and Definitions
● A degree of a vertex in a graph is the maximum number of edges touching that vertex.
● A family F of graphs has bounded degree if all the graphs in F have maximum degree of
O(1).
● A hop count from vertex u to vertex v is the minimum number of edges between them.
● A radius r, is the smallest number in the subgraph such that there exists a vertex within
the hop count of r of all other vertices.
● A Separator is small sets of vertices that partition the graph and are the basis of many
divide and conquer algorithms.
● According to planar separator theorem, a separator hierarchy is formed by taking
separators recursively until all remaining components have size O(1).
● For planar graphs computing shortest path can be done in linear time [19] hence, for a
family F of graphs that is closed under taking subgraphs, F has polynomial expansion if
and only if F has sublinear separators. [7]

Classification of non-planarities
The authors distinguish in two types of non-planarities in road networks:
Definition: An embedding is mapping of vertices to points and its edges to curves, such that
vertices at the ends of each edge are mapped to the corresponding points at the end of the
curve.
A crossing is where two edge curves intersect, that is not a common endpoint for both the
curves.
1) A removable crossing is a crossing between two edges in an embedding that can be
removed by only making local re-routings in the embedding.
2) An essential crossing is a crossing between two edges in an embedding of a graph that
represent disjoint (non-intersecting) road segments and that cannot be removed by local
changes.

Experiments
● The authors measure the sparsity of crossing graphs experimentally on real world
networks. URN dataset is used. Self loops and parallel edges were removed before
processing the data. Data contains essential and removable crossings.
● Crossings identified using Plane Sweep Algorithm
● Identified whether the crossing is essential or removable heuristically.
● Heuristic is that when there is a crossing between two road segments, neither of which is a
bridge or tunnel then it is removable
● Used NetworkX Python package to study the structure of crossing graphs they
constructed.
Hypothesis
● Based on the intuitions of bridges and tunnels, authors expected the crossing graphs to
include some vertices of moderate degree, but otherwise to be very sparse.
● For instance, they considered it to be possible that all of the connected components of the
crossing graph would be trees.

Analysis
● Table shows that the graphs are sparse and the hypothesis that the degeneracy is
significantly smaller than the maximum degree, helped in the experiments.
● Although it is not true that all the components of the crossing graphs are trees, most
of them are and remaining non tree components have low degeneracy.

Theoretical analysis of graphs with sparse
crossings
● Based on the experimental results, we can see that road networks have sparse crossing
graphs.
● What does this indicate about graph theoretic properties of road networks?
● Do they have bounded degeneracy? Polynomial expansion?
● Let's understand the notion of a graph with a sparse crossing graph.
The authors define a nice embedding to be a mapping of vertices of graph to points in the plane
and edges to curve, with the following conditions:
● Each edge is mapped to a Jordan arc (a non-self-intersecting curve) whose endpoints are
the images of the endpoints of the edge.
● If an edge and a vertex are disjoint in the graph, their images in the plane are disjoint
● If two edges are mapped to curves that intersect, then that intersection consists of a single
point, and is either a shared endpoint of both edges or a point where their two curves cross.
● No three edges have curves that all cross at the same point

Lemmas
● Let Cd denote the family of embedded graphs with nice embeddings, such that the
crossing graphs of these embeddings have degeneracy at most d.
● We say that a graph G is d-crossing-degenerate if it belongs to Cd
● Lemma: Every subgraph of a graph in Cd also belongs to Cd .
● After this, they examine the number of crossings that a graph in this family can have,
so by Crossing number inequality, they show that the road networks with d-
degenerate crossing graphs are sparse.
● Lemma: Every n-vertex embedded graph G in Cd has O(n √ d) edges
● Lemma: For the graphs in Cd , every n-vertex graph G has a planarization with
O(nd3/2 ) vertices and edges.

● Using these lemmas, they prove that the given graphs have sublinear separators.
● Based on the methods mentioned in paper "Faster-shortest path algorithm for planar
graphs " [19] We obtain the Corollary: If we are given the planarization of a graph G in
Cd , we can compute shortest paths in G itself in linear time.
● According to "Linear-time algorithms for geometric graphs with sublinearly many crossings"
[9] a linear time algorithm can be used to find shortest paths when the number of
crossings is significantly smaller than the overall number of road segments (as it was in
our experiments)

Conclusions
● Computed study of the removable crossings in large-scale planarized road network
data.
● These crossings form a crossing graph that have high degree of vertices, most
connected components of crossing graphs are trees and a few remaining components
have max degeneracy 6.
● Based on this model, they developed a model for nearly planar graphs.
● For constant values of d, these graphs have a linear number of crossings, a linear
number of edges, and separators of size proportional to the square root of the number
of vertices.
● In addition, a separator hierarchy for these graphs can be constructed in linear time,
and applied in separator-based divide and conquer algorithms for shortest paths and
other computational problems on road networks.

References
1. Degeneracy - https://en.wikipedia.org/wiki/Degeneracy_(graph_theory)
2. Planar Separator Theorem - https://en.wikipedia.org/wiki/Planar_separator_theorem
3. Crossing number inequality - https://en.wikipedia.org/wiki/Crossing_number_inequality
4. [8] - Studying (non-planar) road networks through an algorithmic lens -
https://dl.acm.org/citation.cfm?doid=1463434.1463455
5. Going off road: Transversal complexity in road networks -
https://doi.org/10.1145/1653771.1653778
6. [19]- Faster shortest-path algorithms for planar graphs -
https://dl.acm.org/citation.cfm?doid=195058.195092
7. [7] - Strongly sublinear separators and polynomial expansion.
https://arxiv.org/abs/1504.04821
8. [9] - Linear-time algorithms for geometric graphs with sublinearly many crossings.

Crossing patterns in Nonplanar Road networks

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