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International Workshop on Graph-Theoretic Concepts in Computer Science

WG 2015: Graph-Theoretic Concepts in Computer Science pp 122–137Cite as

Efficient Farthest-Point Queries in Two-terminal Series-parallel Networks

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9224))

Abstract

Consider the continuum of points along the edges of a network, i.e., a connected, undirected graph with positive edge weights. We measure the distance between these points in terms of the weighted shortest path distance, called the network distance. Within this metric space, we study farthest points and farthest distances. We introduce a data structure supporting queries for the farthest distance and the farthest points on two-terminal series-parallel networks. This data structure supports farthest-point queries in \(O(k + \log n)\) time after \(O(n \log p)\) construction time, where \(k\) is the number of farthest points, \(n\) is the size of the network, and \(p\) parallel operations are required to generate the network.

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Notes

  1. 1.

    Observe that \(p \notin V\) when \(\lambda \in (0,1)\) in which case none of the sub-edges \(up\) and \(pv\) are edges in \(E\). When \(\lambda = 0\) or \(\lambda = 1\), the point \(p\) coincides with \(u\) and \(v\), respectively.

  2. 2.

    The final graph is simple even if intermediate graphs have loops and multiple edges.

  3. 3.

    More precisely, we consider extended shortest path trees [15] which result from splitting each non-tree edge \(st\) of a shortest path tree into two sub-edges \(sx\) and \(xt\), where all points on \(sx\) reach the root through \(s\) and all points on \(xt\) reach the root through \(t\).

  4. 4.

    We consider paths of length \(w_{p-1}\) instead of \(w_p\), because we treat \(P_p\) separately.

  5. 5.

    The priority queue in Dijkstra’s algorithm manages never more than \(p\) entries.

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Author information

Authors and Affiliations

  1. Fakultät für Informatik, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany

    Carsten Grimm

  2. School of Computer Science, Carleton University, Ottawa, ON, Canada

    Carsten Grimm

Authors
  1. Carsten Grimm
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Correspondence to Carsten Grimm .

Editor information

Editors and Affiliations

  1. TU München Institut für Informatik, Garching, Germany

    Ernst W. Mayr

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Grimm, C. (2016). Efficient Farthest-Point Queries in Two-terminal Series-parallel Networks. In: Mayr, E. (eds) Graph-Theoretic Concepts in Computer Science. WG 2015. Lecture Notes in Computer Science(), vol 9224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53174-7_10

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  • DOI: https://doi.org/10.1007/978-3-662-53174-7_10

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53173-0

  • Online ISBN: 978-3-662-53174-7

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