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Sifting Edges to Accelerate the Computation of Absolute 1-Center in Graphs

  • Wei DingEmail author
  • Ke Qiu
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

Given an undirected connected graph \(G = (V, E, w)\), where V is the set of n vertices, E is the set of m edges and each edge \(e \in E\) has a positive weight \(w(e) > 0\), a subset \(\mathcal {T} \subseteq V\) of p terminals and a subset \(\mathcal {E} \subseteq E\) of candidate edges, the absolute 1-center problem (A1CP) asks for a point on some edge in \(\mathcal {E}\) to minimize the distance from it to \(\mathcal {T}\). We prove that a vertex 1-center (V1C) is just an absolute 1-center (A1C) if the all-pairs shortest paths distance matrix from the vertices covered by the edges in \(\mathcal {E}\) to \(\mathcal {T}\) has a (global) saddle point. Furthermore, we define the local saddle point of an edge and conclude that the candidate edge having a local saddle point can be sifted. By combining the tool of sifting edges with the framework of Kariv and Hakimi’s algorithm, we design an \(O(m + p m^*+ n p \log p)\)-time algorithm for A1CP, where \(m^*\) is the number of the remaining candidate edges. Applying our algorithm to the classic A1CP takes \(O(m + m^*n + n^2 \log n)\) time when the distance matrix is known and \(O(m n + n^2 \log n)\) time when the distance matrix is unknown, which are smaller than \(O(mn + n^2 \log n)\) time and \(O(mn + n^3)\) time of Kariv and Hakimi’s algorithm, respectively.

Keywords

Absolute 1-center Sifting edges Saddle point 

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Zhejiang University of Water Resources and Electric PowerHangzhouChina
  2. 2.Department of Computer ScienceBrock UniversitySt. CatharinesCanada

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