Rectilinear paths among rectilinear obstacles

  • D. T. Lee
Session 1: Invited Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 650)


Given a set of obstacles and two distinguished points in the plane the problem of finding a collision-free path subject to a certain optimization function is a fundamental problem that arises in many fields, such as motion planning in robotics, wire routing in VLSI and logistics in operations research. In this survey we emphasize its applications to VLSI design and limit ourselves to the rectilinear domain in which the goal path to be computed and the underlying obstacles are all rectilinearly oriented, i.e., the segments are either horizontal or vertical. We consider different routing environments, and various optimization criteria pertaining to VLSI design, and provide a survey of results that have been developed in the past, present current results and give open problems for future research.


Short Path Computational Geometry Query Point Simple Polygon Space Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    T. Asano, T. Asano, L. Guibas, J. Hershberger and H. Imai, “Visibility of disjoint polygons,” Algorithmica, 1, 1986, 49–63.Google Scholar
  2. 2.
    T. Asano, “Generalized Manhattan path algorithm with applications,” IEEE Trans. CAD, 7, 1988, 797–804.Google Scholar
  3. 3.
    T. Asano, M. Sato and T. Ohtsuki, “Computational geometry algorithms,” in Layout Design and Verification, (T. Ohtsuki Ed.), North-Holland, 1986, 295–347.Google Scholar
  4. 4.
    M. J. Atallah and D. Z. Chen, “Parallel rectilinear shortest paths with rectangular obstacles,” Computational Geometry: Theory and Applications, 1, 2, 1991, 79–113.CrossRefGoogle Scholar
  5. 5.
    B. Chazelle, “Triangulating a simple polygon in linear time,” Proc. 31st FOCS, Oct. 1990, 220–230.Google Scholar
  6. 6.
    K. L. Clarkson, S. Kapoor and P. M. Vaidya, “Rectilinear shortest paths through polygonal obstacle in O(n log2 n) time” Proc. 3rd ACM Symp. on Computational Geometry, 1987, 251–257.Google Scholar
  7. 7.
    K. L. Clarkson, S. Kapoor and P. M. Vaidya, “Rectilinear shortest paths through polygonal obstacles in O(n log3/2 n) time,” submitted for publication.Google Scholar
  8. 8.
    M. de Berg, M. van Kreveld, B. J. Nilsson, M. H. Overmars, “Finding shortest paths in the presence of orthogonal obstacles using a combined L1 and link metric,” Proc. SWAT '90, Lect. Notes in Computer Science, 447, Springer-Verlag, 1990, 213–224.Google Scholar
  9. 9.
    M. de Berg, “On rectilinear link distance,” Computational Geometry: Theory and Applications”, 1, 1, 1991, 13–34.Google Scholar
  10. 10.
    P. J. deRezende, D. T. Lee and Y. F. Wu, “Rectilinear shortest paths with rectangular barriers,” Discrete and Computational Geometry, 4, 1989, 41–53.Google Scholar
  11. 11.
    E. W. Dijkstra, “A note on two problems in connection with graphs,” Numerische Mathematik, 1, p.269.Google Scholar
  12. 12.
    M. L. Fredman and R. E. Tarjan, “Fibonacci heaps and their uses in improved network optimization algorithms,” J. ACM, 34, 3, July 1987, 596–615.CrossRefGoogle Scholar
  13. 13.
    M. L. Fredman and D. E. Willard, “Trans-dichotomous Algorithms for Minimum Spanning Trees and Shortest Paths,” Proc. 31st IEEE FOCS, 1990, 719–725.Google Scholar
  14. 14.
    L. J. Guibas and J. Hershberger, “Optimal shortest path queries in a simple polygon,” Proc. 3rd ACM Symp. on Computational Geometry, 1987, 50–63.Google Scholar
  15. 15.
    W. Heyns, W. Sansen, and H. Beke, “A line-expansion algorithm for the general routing problem with a guaranteed solution,” Proc. 17th Design Automation Conf., 1980, 243–249.Google Scholar
  16. 16.
    D. W. Hightower, “A solution to line-routing problem on the continuous plane,” Proc. 6th Design Automation Workshop,, 1969, 1–24.Google Scholar
  17. 17.
    H. Imai and T. Asano, “Dynamic segment intersection search with applications,” Proc. 25th IEEE FOCS, 1984, 393–402.Google Scholar
  18. 18.
    Y. Ke, “An efficient algorithm for link-distance problems,” Proc. 5th ACM Sympo. on Computational Geometry, 1989, 69–78.Google Scholar
  19. 19.
    R. C. Larson and V. O. Li, “Finding minimum rectilinear distance paths in the presence of barriers,” Networks, 11, 1981, 285–304.Google Scholar
  20. 20.
    C. Y. Lee, “An algorithm for path connections and its application,” IRE Trans. on Electronic Computers, V.EC-10, 1961, 346–365.Google Scholar
  21. 21.
    D. T. Lee, “Proximity and reachability in the plane,” PhD. thesis, Univ. of Illinois, 1978.Google Scholar
  22. 22.
    D. T. Lee and F. P. Preparata, “Euclidean shortest paths in the presence of rectilinear barriers,” Networks, 14, 1984, 393–410.Google Scholar
  23. 23.
    D. T. Lee, C. D. Yang and T. H. Chen, “Shortest rectilinear paths among weighted obstacles,” Int'l J. Computational Geometry & Applications, 1, 2, 1991, 109–124.Google Scholar
  24. 24.
    D. T. Lee, C. D. Yang and C. K. Wong, “Problem transformation for finding rectilinear paths among obstacles in two-layer interconnection model”, submitted for publication.Google Scholar
  25. 25.
    D. T. Lee, C. D. Yang and C. K. Wong, “Rectilinear paths among rectilinear obstacles,” Tech. Rep., Dept. EE/CS, Northwestern University, Aug. 1992.Google Scholar
  26. 26.
    W. Lipski, “Finding a Manhattan path and related problems,” Networks, 13, 1983, 399–409.Google Scholar
  27. 27.
    W. Lipski, “An O(n log n) Manhattan path algorithm,” Inform. Process. Lett., 19, 1984, 99–102.CrossRefGoogle Scholar
  28. 28.
    T. Lozano-Perez and M. A. Wesley, “An algorithm for planning collision-free paths among polyhedral obstacles,” Comm. ACM, 22, 1979, 560–570.CrossRefGoogle Scholar
  29. 29.
    A. Margarino, A. Romano, A. De Gloria, F. Curatelli and P. Antognetti, “A tileexpansion router,” IEEE Trans. CAD, 6, 4, 1987, 507–517.Google Scholar
  30. 30.
    K. M. McDonald and J. G. Peters, “Smallest paths in simple rectilinear polygons,” IEEE Trans. CAD, 11,7 July 1992, 864–875.Google Scholar
  31. 31.
    K. Mikami and K. Tabuchi, “A computer program for optimal routing of printed circuit connectors,” IFIPS Proc., Vol. H47, 1968, 1475–1478.Google Scholar
  32. 32.
    J. S. B. Mitchell, “Shortest rectilinear paths among obstacles,” Technical report NO. 739, School of OR/IE, Cornell University, April 1987. A revised version titled “L1 Shortest paths Among Polygon Obstacles in the Plane,” Algorithmica, to appear.Google Scholar
  33. 33.
    J. S. B. Mitchell, “A new algorithm for shortest paths among obstacles in the plane,” Technical report NO. 832, School of OR/IE, Cornell University, Oct. 1988.Google Scholar
  34. 34.
    J. S. B. Mitchell, “An optimal algorithm for shortest rectilinear paths among obstacles,” Abstracts of 1st Canadian Conference on Computational Geometry, 1989, p.22.Google Scholar
  35. 35.
    J. S. B. Mitchell and C. H. Papadimitriou, “The weighted region problem: finding shortest paths through a weighted planar subdivision”, Journal of the ACM, 38, 1, January 1991, 18–73.CrossRefGoogle Scholar
  36. 36.
    J. S. B. Mitchell, G. Rote and G. WŌginger, “Minimum link path among a set of obstacles in the planes,” Proc. 6th ACM Symp. on Computational Geometry, 1990, 63–72.Google Scholar
  37. 37.
    E. F. Moore, “The shortest path through a maze,” Annals of the Harvard Computation Laboratory, Vol. 30, Pt.II, 1959, 185–292.Google Scholar
  38. 38.
    T. Ohtsuki and M. Sato, “Gridless routers for two layer interconnection,” IEEE Int'l Conference on CAD, 1984, 76–78.Google Scholar
  39. 39.
    T. Ohtsuki, “Gridless routers — new wire routing algorithm based on computational geometry,” Int'l Conference on Circuits and Systems, China, 1985.Google Scholar
  40. 40.
    T. Ohtsuki, “Maze running and line-search algorithms,” in Layout Design and Verification, (T. Ohtsuki Ed.), North-Holland, 1986, pp.99–131.Google Scholar
  41. 41.
    F. P. Preparata and M. I. Shamos, Computational Geometry, Springer-Verlag, NY, 1985.Google Scholar
  42. 42.
    M. Sato, J. Sakanaka and T. Ohtsuki, “A fast line-search method based on a tile plane,” in Proc. IEEE ISCAS, 1987, 588–591.Google Scholar
  43. 43.
    S. Suri, “A Linear Time algorithm for minimum link paths inside a simple polygon,” Computer Vision, Graphics and Image Processing, 35, 1986, 99–110.Google Scholar
  44. 44.
    S. Suri, “On some link distance problems in a simple polygon,” IEEE Trans. on Robotics and Automation, 6, 1990, 108–113.Google Scholar
  45. 45.
    E. Welzl, “Constructing the visibility graph for n line segments in O(n 2) time,” Info. Proc. Lett., 1985, 167–171.Google Scholar
  46. 46.
    P. Widmayer, Y. F. Wu, C. K. Wong, “On some distance problems in fixed orientations,” SIAM J. Computing, 16, 1987, 728–746.CrossRefGoogle Scholar
  47. 47.
    Y. F. Wu, P. Widmayer, M. D. F. Schlag, and C. K. Wong, “Rectilinear shortest paths and minimum spanning trees in the presence of rectilinear obstacles,” IEEE Trans. Comput., 1987, 321–331.Google Scholar
  48. 48.
    C. D. Yang, T. H. Chen and D. T. Lee, “Shortest rectilinear paths among weighted rectangles,” J. of Information Processing, 13, 4, 1990, 456–462.Google Scholar
  49. 49.
    C. D. Yang, D. T. Lee and C. K. Wong, “On Bends and Lengths of Rectilinear Paths: A Graph-Theoretic Approach,” Int'l Journal of Computational Geometry & Application, 2, 1, March 1992, 61–74.Google Scholar
  50. 50.
    C. D. Yang, D. T. Lee and C. K. Wong, “On Bends and Distances of Paths Among Obstacles in Two-Layer Interconnection Model,” manuscript, 1991.Google Scholar
  51. 51.
    C. D. Yang, D. T. Lee and C. K. Wong, “Rectilinear path problems among rectilinear obstacles revisited,” manuscript, 1992.Google Scholar
  52. 52.
    C. D. Yang, D. T. Lee and C. K. Wong, “On minimum-bend shortest path among weighted rectangles,” manuscript, 1992.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • D. T. Lee
    • 1
  1. 1.Department of Electrical Engineering and Computer ScienceNorthwestern UniversityEvanston

Personalised recommendations