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On optimal cuts of hyperrectangles

Über optimale Schnitte von Hyperreckecken

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Abstract

We are given a set ofn d-dimensional (possibly intersecting) isothetic hyperrectangles. The topic of this paper is the separation of these rectangles by means of a cutting isothetic hyperplane. Thereby we assume that a rectangle which is intersected by the cutting plane iscut into two non-overlapping hyperrectangles. We investigate the behavior of several kinds of balancing functions, as well as their linear combination and present optimal and practical algorithms for computing the corresponding balanced cuts. In addition, we give tight worst-case bounds for the quality of the balanced cuts.

Zusammenfassung

Gegeben sei eine Menge vonn (ggf. überlappenden) isothetischen Hyperrechtecken imd-dimensionalen Raum. Diese Arbeit beschäftigt sich mit Zerlegungen dieser Hyperrechteckmenge durch Schnitthyperebenen, wobei wir annehmen, daß jedes von einer Hyperebene geschnittene Hyperrechteck in zwei nicht-überlappende Hyperrechtecke zerschnitten wird. Wir untersuchen das Verhalten einiger Balancierungskriterien für Schnitte und präsentieren optimale and praktikable Algorithmen zur Berechnung der entsprechenden balancierten Schnitte. Schließlich geben wir auch scharfe Worst-case-Schranken für die bestmöglich erreichbare Qualität der balancierten Schnitte an.

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References

  1. d'Amore, F., Franciosa, P. G.: Separating sets of hyperrectangles. Int. J. Comp. Geometry Appl.3, 155–165, 345 (1993).

    Google Scholar 

  2. d'Amore, F., Franciosa, P. G.: On the optimal binary plane partition for sets of isothetic rectangles. IPL44, 255–259 (1992).

    Google Scholar 

  3. d'Amore, F., Roos, T., Widmayer, P.: An optimal algorithm for computing a best cut of a set of hyperrectangles. In: Graphics, design and visualization. IFIP Transactions B-9 (Pattanaik, S. N., Mudur, S. P., eds.), pp. 215–224. Amsterdam: Elsevier 1993.

    Google Scholar 

  4. Ben-Or, M.: Lower bounds for algebraic computation trees. Proc. 15th ACM STOC, pp. 80–86 (1983).

  5. Bentley, J. L.: Multidimensional binary search trees used for associative searching. Comm. ACM,18, 509–517 (1975).

    Google Scholar 

  6. de Berg, M., de Groot, M., Overmars, M.: Perfect binary space partitions. Proc. 5th Canad. Conf. Comput. Geom., Waterloo, Ontario, pp. 109–114 (1993).

  7. de Berg, M., de Groot, M., Overmars, M.: New results on binary space partitions in the plane. Proc. SWAT'94. Lecture Notes in Computer Science Vol.824, 61–72 (1994).

    Google Scholar 

  8. de Berg, M., de Groot, M.: Binary space partitions for sets of cubes, Abstracts 10th European Workshop Comput. Geom. (CG'94), pp. 84–88 (1994).

  9. Fredman, M. L., Weide, B.: On the complexity of computing the measure of U[ai, bj]. Comm. ACM21, 540–544 (1978).

    Google Scholar 

  10. Fuchs, H., Kedem, Z., Naylor, B.: On visible surface generation by a priori tree structures, Comput. Graphics.14, 124–133 (1980).

    Google Scholar 

  11. Guttman, A.: R-trees.: a dynamic index structure for spatial searching Proc. ACM SIGMOD, pp. 47–57 (1984).

  12. Amatodes, J. A., Naylor, B., Thibault, W.: Merging BSP trees yields polyhedral set operations. Comp. Graphics24, 115–124 (1990).

    Google Scholar 

  13. Nievergelt, J., Hinterberger, H., Sevcik, K. C.: The grid file: An adaptable, symmetric multikey file structure. ACM Trans. Database Syst.9, 38–71 (1984).

    Google Scholar 

  14. Nguyen, V. H., Ross, T., Widmayer, P.: Balanced cuts of a set of hyperrectangles. Proc. 5th Canad. Conf. Comput. Geom., Waterloo, Ontario, pp. 121–126 (1993).

  15. Paterson, M. S., Yao, F. F.: Efficient binary space partitions for hidden-surface removal and solid modeling. Discrete Comput. Geom.5, 485–503 (1990).

    Google Scholar 

  16. Paterson, M. S., Yao, F. F.: Optimal binary space partitions for orthogonal objects. J. Algorithms13, 99–113 (1992).

    Google Scholar 

  17. Samet, H.: The design and analysis of spatial data structures. Reading: Addison-Wesley 1990.

    Google Scholar 

  18. Thibault, W. C., Naylor, B. F.: Set operations on polyhedra using binary space partitioning trees. Proc. SIGGRAPH'87, pp. 153–162 (1987).

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Authors and Affiliations

  1. Department Informatik, ETH Zürich ETH Zentrum, CH-8092, Zürich, Switzerland

    V. H. Nguyen,  T. Roos &  P. Widmayer

  2. Dipartimento di Informatica e Sistemistica, Università di Roma ‘La Sapienza’, via Salaria 113, I-00198, Roma, Italia

    F. d'Amore

Authors
  1. F. d'Amore
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  2. V. H. Nguyen
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  3. T. Roos
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  4. P. Widmayer
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d'Amore, F., Nguyen, V.H., Roos, T. et al. On optimal cuts of hyperrectangles. Computing 55, 191–206 (1995). https://doi.org/10.1007/BF02238431

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  • DOI: https://doi.org/10.1007/BF02238431

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