Finding the conditional location of a median path on a tree

In this paper, we study the problem of locating a median path of limited length on a tree under the condition that some existing facilities are already located. The existing facilities may be located at any subset of vertices. Upper and lower bounds are proposed for both the discrete and continuous models. In the discrete model, a median path is not allowed to contain partial edges. In the continuous model, a median path may contain partial edges. The proposed upper bounds for these two models are O(n log n) and O(n log nα(n)), respectively. They improve the previous known bounds from O(n log2n) and O(n2), respectively. The proposed lower bounds are both Ω(n log n).