Back - up 2 - Center on a 1 Path / Tree / Cycle / Unicycle Network

This paper addresses the problem of locating two facilities 11 in vertex-weighted tree/cycle/unicycle networks, where each facility can 12 fail with a given probability [16]. It is assumed that the two facilities 13 do not fail simultaneously, and when a facility fails, the other facility is 14 required to service all the vertices. We show that the back-up 2-center on 15 a weighted path (resp. tree, cycle, unicyclic) network can be computed in 16 O(n) (resp. O(n logn), O(n), O(n logn)) time, where n is the number 17 of vertices. This is the first sub-quadratic time result for vertex-weighted 18 tree networks. For vertex-unweighted trees, it runs in O(n) time, match19 ing the best known result [19]. The algorithm remains linear when there 20 are only a constant number of distinct weights. 21