The complexity of manipulating hierarchically defined sets of rectangles

  • Jon Louis Bentley
  • Thomas Ottmann
Invited Lectures
Part of the Lecture Notes in Computer Science book series (LNCS, volume 118)


Algorithms that manipulate sets of rectangles are of rectangles are of great practical importance in VLST design systems and other applications. Although much theoretical work has appeared recently on the complexity of rectangle problems, it has assumed that the inputs are given as a list of rectangles. In this paper we study the complexity of rectangle problems when the inputs are given in a hierarchical language that allows the designer to build large designs by replicating small designs. We will see that while most of the problems are NP-hard in the general case, there are O(N log N) algorithms that process inputs obeying certain restrictions.


VLSI Design Query Object Hierarchical Design Symbol Number Consistent Design 


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  1. Bentley, J.L. [1977]: Solution to Klee's rectangle problems, unpublished manuscript, Dept. of Computer Science, Carnegie-Mellon University, 1977.Google Scholar
  2. Bentley, J.L. and Wood, D. [1980]: An optimal worst-case algorithm for reporting intersections of rectangles, IEEE Transactions on Computers, Vol. C-29, 1980, 572–577.Google Scholar
  3. Bentley, J.L., Haken, D., and Hon, R. [1980]: Statistics on VLSI Designs, Dept. of Computer Science, Carnegie-Mellon University, Technical Report CMU-CS-80.Google Scholar
  4. Edelsbrunner, H., van Leeuwen, J., Ottmann Th., and Wood, D. [1980]: Connected Components of Orthogonal Geometric Objects, Computer Science Technical Report, 1981, McMaster University, Hamilton, Ontario, Canada.Google Scholar
  5. Garey, M.R. and Johnson, D.S. [1979]: Computers and Intractability, A Guide to the Theory of NP-Completeness, Freeman, San Francisco, 1979.Google Scholar
  6. Haken, D. [1980]: A geometric design rule checker, VLSI Document V053, Carnegie-Mellon University, 9 June 80.Google Scholar
  7. Hon, R. [1980]: The Hierarchical Analysis of VLSI Designs, Thesis proposal, Carnegie-Mellon University, Dec. 1980.Google Scholar
  8. Klee, V. [1977]: Can the Measure of U[ai,bi] be computed in less than O(n log n) steps, Research Probl. Sect., Amer. Math. Monthly 84, 1977, 284–285.Google Scholar
  9. Lauther [1980]: A Data Structure for Gridless Routing, 17th Design Automation Conference, Minneapolis 1980, 1–7.Google Scholar
  10. van Leeuwen, J. and Wood, D. [1979]: The Measure Problem for Rectangular Ranges in d-Space, Technical Report, RUU-CS-79-6, July 1979.Google Scholar
  11. McCreight, E. M. [1980]: Efficient Algorithms for Enumerating Intersecting Intervals and Rectangles. XEROX Palo Alto Research Center, 1980, Report CSL-80-9.Google Scholar
  12. Mead, C. and Conway, L. [1980]: Introduction to VLSI Systems, Addison-Wesley.Google Scholar
  13. Nievergelt, J. and Preparata, F.P. [1980]: Planesweep algorithms for intersecting geometric figures, Technical Report (in preparation), Institut für Informatik, ETH Zürich.Google Scholar
  14. Ottmann, Th. and Widmayer, P. [1981]: Reasonable encodings make Rectangle Problems Hard, Forschungsbericht des Instituts für Angewandte Informatik und Formale Beschreibungsverfahren, Universität Karlsruhe, 1981.Google Scholar
  15. Vaishnavi, V. and Wood, D. [1980]:Rectilinear line segment intersection, layered segment trees and dynamization, Computer Science Technical Report, 80-CS-8, McMaster University, Hamilton, Ontario, Canada.Google Scholar
  16. Vitanyi, P.M.B. and Wood, D. [1979]: Computing the Perimeter of a Set of Rectangles, Computer Science Technical Report, 79-CS-23, McMaster University, Hamilton, Ontario, Canada.Google Scholar
  17. Whitney, T. [1980]: Description of the Hierarchical Design Rule Filter, Caltech SSP File 4027, Oct. 1980.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • Jon Louis Bentley
    • 1
  • Thomas Ottmann
    • 2
  1. 1.Department of Computer Science and MathematicsCarnegie-Mellon-UniversityPittsburghU.S.A.
  2. 2.Institut für Angewandte Informatik und Formale BeschreibungsverfahrenUniversität KarlsruheKarlsruhe 1West-Germany

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