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/lp/association-for-computing-machinery/partitioning-a-polygonal-region-into-trapezoids-0mWT8fnFQl
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- Publisher
- Association for Computing Machinery
- Copyright
- Copyright © 1986 by ACM Inc.
- ISSN
- 0004-5411
- DOI
- 10.1145/5383.5387
- Publisher site
- See Article on Publisher Site
Abstract
The problem of partitioning a polygonal region into a minimum number of trapezoids with two horizontal sides is discussed. A triangle with a horizontal side is considered to be a trapezoid with two horizontal sides one of which is degenerate. First, a method of achieving a minimum partition is presented. The number M * of the trapezoids in the minimum partition of a polygonal region P is shown to be M * = n + w - h - d - 1, where n , w , and h are the number of vertices, windows (holes), and horizontal edges of P , respectively, and d is the cardinality of a maximum independent set of the straight-lines-in-the-plane graph associated with P . Next, this problem is shown to be polynomially equivalent to the problem of finding a maximum independent set of a straight-lines-in-the-plane graph, and consequently, it is shown to be NP-complete. However, for a polygonal region without windows, an O ( n 2 )-time algorithm for partitioning it into a minimum number of trapezoids is presented. Finally, an O ( n log n )-time approximation algorithm with the performance bound 3 is presented.
Journal
Journal of the ACM (JACM) – Association for Computing Machinery
Published: Apr 1, 1986