Computing and Listing st-Paths in Subway Networks
Résumé
Given a set of paths (called lines) L, a subway network is a graph GL = (V, A) where V contains exactly the vertices and arcs of every line l ∈ L. An st-route is a pair (π, γ) where γ = (l1,. .. , l h) is a line sequence and π is an st-path in GL which is the concatena-tion of subpaths of the lines l1,. .. , l h , in this order. We study three related problems concerning traveling from s to t in GL. We present an efficient (i.e., polynomial-time) algorithm for computing an st-route (π, γ) where |γ| (i.e., the number of line changes plus one) is minimum among all st-routes. We show for the problem of finding an st-route (π, γ) that minimizes the number of different lines in γ, even computing an o(log |V |)-approximation is NP-hard. Finally, given an integer β, we present an algorithm for enumerating all st-paths π for which a route (π, γ) with |γ| ≤ β exists, and show that the running time of this algorithm is polynomial with respect to both input and output size.
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