An Ω(nd) lower bound on the number of cell crossings for weighted shortest paths in d-dimensional polyhedral structures

Abstract

The d-dimensional weighted region shortest path problem asks for finding a shortest path between two given points s and t in a d-dimensional polyhedral structure consisting of polyhedral cells having individual positive weights. It is a generalization of the d-dimensional unweighted (Euclidean) shortest path problem for polyhedral structures.In the unweighted (Euclidean) setting, a shortest path visits, due to cell convexity, each polyhedral cell at most once. Surprisingly, this is no longer true for the weighted setting, which is a strong motivation for studying the complexity of weighted shortest paths in polyhedral structures.Previously, Ω(n2), respectively Ω(n3) lower bounds on the maximum number of cell crossings for weighted shortest paths in 2-dimensional, respectively 3-dimensional polyhedral structures have been obtained.In this paper, a new lower bound of Ω(nd) is derived on the maximum number of cell crossings a weighted shortest path could take in d-dimensional polyhedral structures consisting of a linear number of O(n) polyhedral cells and cell faces. This new result is a generalization and sharpening of the formerly known lower bounds and has been a long-standing open problem for the general d-dimensional case.