Locating a single facility (the 1-median problem) on a deterministic tree network is very efficiently solved by Goldman's algorithm. This paper explores the possibility of efficiently solving the 2-median problem for both deterministic and probabilistic tree networks. Interesting properties (Theorems 1-4) are derived that relate the location of l-median to the location of pairs of 2-medians. Two simple algorithms, the "Improved Link-Deletion'' algorithm for the deterministic case and the "Selective Enumeration'' algorithm for the probabilistic case, are presented which somewhat improve existing methods to determine 2-medians on tree networks.
When deterministic assumptions in ‘classical’ location problems are relaxed some new theoretical and algorithmic problems arise. In this paper, the definition of medians is generalized to the cases: (1) when generated demands and travel costs are random, and (2) when the demand induced at the facilities is random because of competition. Under a particular set of assumptions, the well-known theorems on ‘optimality of node locations' hold for the underlying ‘probabilistic networks’. For case (1) the model is shown to be a generalization of the probabilistic and deterministic median location problems on networks. The relationship of the model to a multicommodity location problem is also pointed out. Additionally, within the framework of the problem formulation, certain parallels with multiobjective location problems are drawn. For case (2), the competitive median location problem is formulated, and some results presented. For the special structure of probabilistic tree networks some interesting localization properties for the 1-median and 2-median are discussed.
The p-median problem on general networks has been studied since the 1960s. Kariv and Hakimi [10] showed that this problem is NP-hard even if the network is a planar graph of maximum degree 3. In the case of tree networks the p-median problem is solvable in polynomial time. Kariv and Hakimi [10] developed an algorithm that computes a solution in O(p 2 n 2) time. The running time was improved to O(pn 2) by Tamir [16] and later to O(n lg p+2 n) by Benkoczi and Bhattacharya [2]. Better bounds are known for the special cases (on trees) where p = 1 or 2. Goldman [6] gave an O(n) algorithm for the 1-median problem on trees. The 2-median problem was studied by Mirchandani and Oudjit [11], whose localization properties were later used to improve the O(n 2) bound (derived from the general tree case) to O(n lg n)-see papers by Hämäläinen [9] and Gavish and Sridhar [5]. We present a framework for solving the 2-median problem on trees, building on earlier work. Our framework leads to several algorithms with o(n lg n) runtime, i.e., better than the current best-known O(n lg n) runtime, in common special cases. The time bounds are: (i) O(n lg wmax/ lg n), where wmax is the largest largest node weight, which is linear when node weights are bounded by a polynomial in n; (ii) O(n lg nL), where nL is the number of leaves in in the tree; (iii) O(ndmax), where dmax is the maximum edge length, which is linear when edge lengths are bounded by a constant; and (iv) O(n lg), where is the number of nodes on the trunk, an easily identified path that is guaranteed to contain at least one of the two medians.
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