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

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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