Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams

Abstract

An O(n log n) algorithm for planning a purely translational motion for a simple convex object amidst polygonal barriers in two-dimensional space is given. The algorithm is based on a new generalization of Voronoi diagrams (similar to that proposed by Chew and Drysdale [1] and by Fortune [2]), and adapts and uses a recent technique of Yap for the efficient construction of these diagrams.

Submatrix Maximum Queries in Monge Matrices and Partial Monge Matrices, and Their Applications

“…An easier problem that has been studied more extensively is that of finding the largest-area P -empty axis-parallel rectangle contained in B. Notice that the largest P -empty square is easier to compute, because its center is a Voronoi vertex in the L ∞ -Voronoi diagram of P (and of the edges of B), which can be found in O(n log n) time [17,32]. There have been several studies on finding the largest-area maximal empty rectangle [5,14,37] in B; the fastest algorithm to date, by Aggarwal and Suri [4], takes O(n log 2 n) time and O(n) space.…”

On the construction of abstract voronoi diagrams

“…Our algorithm runs in time O(n log n). Other applications are the Voronoi diagrams for circles under the Laguerre distance [11], [1], [2] and for disjoint convex polygons under a convex distance function [20].…”

Extremal polygon containment problems

“…The merging step can be done in time proportional to the length of Γ, which, by [LS1] and the comments in Section 2, is O(λ 4 (kn)). Hence the calculation of the lower envelope Ψ L;O takes O(λ 4 (kn) log kn) time, so all these envelopes can be computed in overall time O(knλ 4 (kn) log kn).…”

“…We employ the parametric search technique of Megiddo [Me], and the fixed size polygon placement algorithms developed by Leven and Sharir [LS,LS1], to obtain an algorithm that runs in time O(k 2 nλ 4 (kn) log 3 (kn) log log(kn)). We also present several other efficient algorithms for restricted variants of the extremal polygon containment problem, using the same ideas.…”