C Levcopoulos
ABSTRACT
We consider the problem of partitioning isothetic polygons into rectangles by drawing edges of minimum total length. The problem has various applications [LPRS], eg. in VLSI design when dividing routing regions into channels ([Riv1], [Riv2]). If the polygons contain holes, the problem in NP-hard [LPRS]. In this paper it is shown how solutions within a constant factor of the optimum can be computed in time Ο(n log n), thus improving the previous Ο(n2) time bound. An unusual divide-and-conquer technique is employed, involving alternating search from two opposite directions, and further efficiency is gained by using a fast method to sort subsets of points. Generalized Voronoi diagrams are used in combination with plane-sweeping in order to detect all “well bounded” rectangles, which are essential for the heuristic.
- ChewD.Chew, L.P., and Drysdale, R.L., "Voronoi Diagrams Based on Convex Distance Functions", Proc. First ACM Symposium on Computational Geometry, Baltimore, June 1985. Google Scholar
Digital Library
- Fort.Fortune, S.J., "A Fast Algorithm for Polygon Containment by Translation", Proc. 12th ICALP, Nafplion, Greece, 1985. Google ScholarDigital Library
- GonZ.Gonzalez, T., and S.Q. Zheng, "Bounds for Partitioning Rectilinear Polygons' Proc. First ACM Symposium on Computational Geometry, Baltimore, June 1985. Google ScholarDigital Library
- ImAs.Imai, H., and Asano, T., "Dynamic intersection Search with Applications", Proc. 25th iEEE FOCS Symp., Florida, 1984.Google Scholar
- KeilS.Keil, J.M., and J.R. Sack, "Minimum Decompositions of Polygonal Objects", SCS-TR-42, Carleton University, Ottawa.Google Scholar
- Kirk.Kirkpatrick, D.G., "An upper bound for sorting integers in restricted ranges", Proc. 18th Allerton Conf. on Comm. Control and Computing, Illinois, (Oct. 1980).Google Scholar
- Lev1.Levcopoulos, C., "Minimum Length and 'Thickest- First' Rectangular Partitions of Polygons", Proc. 23rd Allerton Conf. on Comm. Control and Computing, Illinois, October 1985.Google Scholar
- Lev2.Levcopoulos, C., "Heuristics for Decomposing Polygons into Rectangles" (preliminary title), in prepara~ tion, LinkSping.Google Scholar
- LevLin.Levcopoulos, C., and A. Lingas, "Bounds on the Length of Convex Partitions of Polygons", Proc. 4th Conf. on Found. of Software Technology and Theoretical Computer Science, Bangalore, India, 1984 (Springer Verlag). Google ScholarDigital Library
- Lin.Lingas, A., "Heuristics for Minimum Edge Length Rectangular Partitions of Rectilinear Figures", Proc. 6th GI-Conference, Dortmund, January 1983 (Springer Verlag). Google ScholarDigital Library
- Lloyd.Lloyd, E.L., "On Triangulations of a Set of Points in the Plane", Proc. of 18th IEEE Conf. on Foundations of Comp. Sc., Providence, 1977.Google ScholarDigital Library
- LPRS.Lingas, A., R.Y. Pinter, R.L. Rivest, and A. Shamir, "Minimum Edge Length Partitioning of Rectilinear Polygons", Proc. 20th Allerton Conf. on Comm. Control and Compt., Illinois, 1982.Google Scholar
- PrepS.Preparata, F.P., and M.I. Shamos, "Computational Geometry", New York Springer-Verlag, 1985. Google ScholarDigital Library
- Riv1.Rivest, R., "The PI (Placement and Interconnect) System", Proc. 19th Design Automation Conference, June 1982. Google ScholarDigital Library
- Riv2.Rivest, R., Private Communication, October 1985.Google Scholar
Index Terms
- Fast heuristics for minimum length rectangular partitions of polygons
Recommendations
Minimum rectangular partition problem for simple rectilinear polygons
An O( n log log n ) algorithm is proposed for minimally rectangular partitioning a simple rectilinear polygon. For any simple rectilinear polygon P , a vertex-edge visible pair is a vertex and an edge that can be connected by a horizontal or vertical ...
Minimum-perimeter intersecting polygons
LATIN'10: Proceedings of the 9th Latin American conference on Theoretical InformaticsGiven a set ${\mathcal S}$ of segments in the plane, a polygon P is an intersecting polygon of ${\mathcal S}$ if every segment in ${\mathcal S}$ intersects the interior or the boundary of P. The problem MPIP of computing a minimum-perimeter intersecting ...
Comments