Computing Covers of Plane Forests
Luis Barba, Alexis Beingessner, Prosenjit Bose, Michiel H. M. Smid

TL;DR
This paper introduces a method to compute covers of plane forests based on a function $$, providing efficient algorithms for specific functions like convex hulls and bounding boxes, with applications in geometric covering problems.
Contribution
The paper defines conditions for a function $$ to produce well-defined covers and presents algorithms to compute these covers efficiently for certain $$, such as convex hulls and bounding boxes.
Findings
The $$-cover can be computed in $O(n ext{log}^2 n)$ time for specific functions $$.
The paper establishes properties that ensure the $$-cover is well-defined.
Efficient algorithms are provided for convex hull and bounding box functions.
Abstract
Let be a function that maps any non-empty subset of to a non-empty subset of . A -cover of a set of pairwise non-crossing trees in the plane is a set of pairwise disjoint connected regions such that each tree is contained in some region of the cover, and each region of the cover is either (1) for some , or (2) , where and are constructed by either (1) or (2), and . We present two properties for the function that make the -cover well-defined. Examples for such functions are the convex hull and the axis-aligned bounding box. For both of these functions , we show that the -cover can be computed in time, where is the total number of vertices of the trees in .
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Algorithms and Data Compression · Advanced Graph Theory Research
