The continuous p-median of a network.

*(English)*Zbl 0675.90026Summary: The distance between an edge and a point of a network N is defined as the maximum distance from that point to any point on that edge. A continuous median of N is a point of N such that the sum of the distances between all edges and that point is minimum. A continuous p-median is a set of p points of N such that the sum for all edges of the distance to the closest point of that set is minimum. It is shown that the set of vertices and middle points of edges always contains a continuous p- median. Therefore, powerful algorithms for the usual p-median problem can be brought to bear. Moreover, algorithms requiring \(O(m^ 2)\) operations in worst case for determining the set of all continuous and conditional continuous medians of N are obtained. A linear algorithm for the set of all continuous medians of a tree is also provided.

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\textit{P. Hansen} and \textit{M. Labbé}, Networks 19, No. 5, 595--606 (1989; Zbl 0675.90026)

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