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

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.