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Minimum Cell Connection in Line Segment Arrangements

Alt, H., Cabello, S., Giannopoulos, P. ORCID: 0000-0002-6261-1961 and Knauer, C. (2018). Minimum Cell Connection in Line Segment Arrangements. International Journal of Computational Geometry & Applications, 27(03), pp. 159-176. doi: 10.1142/s0218195917500017

Abstract

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points a and b in different cells of the induced arrangement:

[(i)] compute the minimum number of segments one needs to remove so that there is a path connecting a to b that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell.

We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting a
to b must stay inside a given polygon P with a constant number of holes, the segments are contained in P, and the endpoints of the segments are on the boundary of P. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution.

Publication Type: Article
Additional Information: Electronic version of an article published as Alt, H., Cabello, S., Giannopoulos, P. and Knauer, C. (2018). Minimum Cell Connection in Line Segment Arrangements. International Journal of Computational Geometry & Applications, 27(03) DOI: 10.1142/S0218195917500017] © copyright World Scientific Publishing Company https://www.worldscientific.com/worldscinet/ijcga
Publisher Keywords: Cell connection, segment arrangement, path homotopy
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Departments: School of Mathematics, Computer Science & Engineering > Computer Science
URI: http://openaccess.city.ac.uk/id/eprint/21510
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