An Ω(n3) Lower Bound on the Number of Cell Crossings for Weighted Shortest Paths in 3-Dimensional Polyhedral Structures

Abstract

A new lower bound of Ω(n3) on the maximum number of cell crossings for weighted shortest paths in 3-dimensional polyhedral structures consisting of a linear number of O(n) polyhedral cells and cell faces is derived. This is a generalization and sharpening of the formerly known Ω(n2) lower bound on the maximum number of cell crossings for weighted shortest path in 2-dimensional polyhedral structures and has been a long-standing open problem for the 3-dimensional case.