The 1-Center and 1-Highway problem revisited

In this paper we extend the Rectilinear 1-center problem as follows: given a set $$S$$ S of $$n$$ n demand points in the plane, simultaneously locate a facility point $$f$$ f and a rapid transit line (i.e. highway) $$h$$ h that together minimize the expression $$\max _{p\in S}T_{h}(p,f)$$ max p ∈ S T h ( p , f ) , where $$T_{h}(p,f)$$ T h ( p , f ) denotes the travel time between $$p$$ p and $$f$$ f . A point of $$S$$ S uses $$h$$ h to reach $$f$$ f if $$h$$ h saves time: every point $$p\in S$$ p ∈ S moves outside $$h$$ h at unit speed under the $$L_1$$ L 1 metric, and moves along $$h$$ h at a given speed $$v>1$$ v > 1 . We consider two types of highways: (1) a turnpike in which the demand points can enter/exit the highway only at the endpoints; and (2) a freeway problem in which the demand points can enter/exit the highway at any point. We solve the location problem for the turnpike case in $$O(n^2)$$ O ( n 2 ) or $$O(n\log n)$$ O ( n log n ) time, depending on whether or not the highway’s length is fixed. In the freeway case, independently of whether the highway’s length is fixed or not, the location problem can be solved in $$O(n\log n)$$ O ( n log n ) time.