Optimally Locating a Structured Facility of a Specified Length in a Weighted Tree Network

including the one proposed in [5]. Minieka showed that all the eight problems can be solved in polynomial time except the one of finding the maximum distancesum tree of a specified length, which was proved to be NP-hard later in [8]. Recently, in [6,7], Peng and Lo have proposed efficient parallel algorithms on the EREW PRAM for optimally locating in a tree network a path and a tree of a specified length. They considered all the eight problems studied in Minieka's paper. In [6,7], it was assumed that the tree network is unweighted. Peng and Lo's results are summarized in Table 1. Note that under the assumption that the tree network is unweighted, the problem of finding a maximum distancesum tree of a specified length is not NP-hard. In this paper, we propose efficient parallel algorithms on the EREW PRAM for optimally locating in a weighted tree network a tree-shaped facility of a specified length. Two optimization criteria are considered: minimum eccentricity and minimum distancesum. Let n be the number of vertices in the tree network. Both the two algorithms we shall propose take O(log nloglog n) time using O(n) work.