Computing the Maximum Detour and Spanning Ratio of Planar Paths, Trees, and Cycles

Langerman, Stefan; Morin, Pat; Soss, Michael

doi:10.1007/3-540-45841-7_20

Cited by 18 publications

(17 citation statements)

References 7 publications

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“…Therefore we briefly consider the problem of computing the stretch factor of a given graph with positive edge weights. This problem has recently received considerable attention; see, for example, [13,22,27].…”

Section: Three Simple Algorithmsmentioning

confidence: 99%

“…This can be done by running Dijkstra's algorithm -implemented using Fibonacci heaps-n times, resulting in an O(mn+n 2 log n)-time algorithm using linear space. This approach is quite slow, and we would like to be able to compute the stretch factor more efficiently, but no faster algorithm is known for any graphs except planar graphs, paths, cycles, stars, and trees [13,22,27]. Applying the stated bound to the problem of computing the exact stretch factor of G gives that G P can be computed in time O(n 3 (m + n log n)) using linear space.…”

Section: Exact Algorithmsmentioning

confidence: 99%

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Improving the Stretch Factor of a Geometric Network by Edge Augmentation

Farshi¹,

Giannopoulos²,

Gudmundsson³

2008

SIAM J. Comput.

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Abstract. Given a Euclidean graph G in R d with n vertices and m edges, we consider the problem of adding an edge to G such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a graph with positive edge weights runs in O(nm + n 2 log n) time, resulting in a trivial O(n 3 m + n 4 log n)-time algorithm for computing the optimal edge. First, we show that a simple modification yields the optimal solution in O(n 4 ) time using O(n 2 ) space. To reduce the running time we consider several approximation algorithms.Key words. computational geometry, approximation algorithms, geometric networks AMS subject classifications. 65D18, 68U05, 68Q25 DOI. 10.1137/050635675 1. Introduction. Consider a set V of n points in R d . A network on V can be modeled as an undirected graph G with vertex set V of size n and an edge set E of size m where every edge (u, v) has a positive weight w (u, v). A Euclidean network is a geometric network where the weight of the edge (u, v) is equal to the Euclidean distance |uv| between its two endpoints u and v.For two vertices u, v in a weighted graph G we use δ G (u, v) to denote a shortest path between u and v in G, and the length of the path is denoted by d G (u, v). Consider a weighted graph G = (V, E) and a graph G = (V, E ) on the same vertex set but with edge set E ⊆ E. We say that G is a t-spanner of G if for each pair of vertices (u, v). The minimum t such that G is a t-spanner for V is called the stretch factor, or dilation, of G.We say that a Euclidean network G = (V, E) is a t-spanner if G = (V, E) is a t-spanner of the complete network on V . In other words, for any two points p, q ∈ V the graph distance in G is at most t times the Euclidean distance between the two points.Complete graphs represent ideal communication networks, but they are expensive to build; sparse spanners represent low-cost alternatives. The weight of the spanner network is a measure of its sparseness; other sparseness measures include the number of edges, the maximum degree, and the number of Steiner points. Spanners for complete Euclidean graphs as well as for arbitrary weighted graphs find applications in robotics, network topology design, distributed systems, design of parallel machines, and many other areas. Recently spanners found interesting practical applications

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Section: Three Simple Algorithmsmentioning

confidence: 99%

Section: Exact Algorithmsmentioning

confidence: 99%

Improving the Stretch Factor of a Geometric Network by Edge Augmentation

Farshi¹,

Giannopoulos²,

Gudmundsson³

2008

SIAM J. Comput.

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“…Preliminary versions of this work appeared in [2,20]; the 2-dimensional algorithm described in [20] is significantly different from the one presented here.…”

Section: New Resultsmentioning

confidence: 99%

Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

Agarwal

Klein

Knauer

et al. 2007

Discrete Comput Geom

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The detour and spanning ratio of a graph G embedded in E d measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(n log n) time algorithms for computing the detour and spanning ratio of a planar polygonal path. By generalizing these algorithms, we 18 Discrete Comput Geom (2008) 39: 17-37 obtain O(n log 2 n)-time algorithms for computing the detour or spanning ratio of planar trees and cycles. Finally, we develop subquadratic algorithms for computing the detour and spanning ratio for paths, cycles, and trees embedded in E 3 , and show that computing the detour in E 3 is at least as hard as Hopcroft's problem.

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“…Dilation is a concept that has been studied extensively in computational geometry [2,3,6,7,10,12,14]. The basic problem is to compute -for a given set of points -a graph in which a shortest path between any two points is close to their Euclidean distance.…”

Section: Detoursmentioning

confidence: 99%

Feed-links for network extensions

Aronov

Buchin

et al. 2008

Proceedings of the 16th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems

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Road network data is often incomplete, making it hard to perform network analysis. This paper discusses the problem of extending partial road networks with reasonable links, using the concept of dilation (also known as crow flight conversion coefficient). To this end, we study how to connect a point (relevant location) inside a polygon (face of the known part of the road network) to the boundary so that the dilation from that point to any point on the boundary is not too large. We provide algorithms and heuristics, and give a computational and experimental analysis.

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Computing the Maximum Detour and Spanning Ratio of Planar Paths, Trees, and Cycles

Cited by 18 publications

References 7 publications

Improving the Stretch Factor of a Geometric Network by Edge Augmentation

Improving the Stretch Factor of a Geometric Network by Edge Augmentation

Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D

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