A Simple Algorithm for Computing the Zone of a Line in an Arrangement of Lines

Document Type

Conference Paper

Journal/Book Title/Conference

Symposium on Simplicity in Algorithms (SOSA)

Publisher

Society for Industrial and Applied Mathematics

Publication Date

2022

Award Number

NSF, Division of Computing and Communication Foundations (CCF) 2005323

Funder

NSF, Division of Computing and Communication Foundations (CCF)

First Page

79

Last Page

86

Abstract

Let L be a set of n lines in the plane. The zone Z(ℓ) of a line ℓ in the arrangement A(L) of L is the set of faces of A(L) whose closure intersects ℓ. It is known that the combinatorial size of Z(ℓ) is O(n). Given L and ℓ, computing Z(ℓ) is a fundamental problem. Linear-time algorithms exist for computing Z(ℓ) if A(L) has already been built, but building A(L) takes O(n2) time. On the other hand, O(n log n)-time algorithms are also known for computing Z(ℓ) without relying on A(L), but these algorithms are relatively complicated. In this paper, we present a simple algorithm that can compute Z(ℓ) in O(n log n) time. More specifically, once the sorted list of the intersections between ℓ and the lines of L is known, the algorithm runs in O(n) time. A big advantage of our algorithm, which mainly involves a Graham's scan style procedure, is its simplicity.

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