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C.d. Yang, D.t. Lee, C.k. Wong, "The Smallest Pair of Noncrossing Paths in a Rectilinear Polygon," IEEE Transactions on Computers, vol. 46, no. 8, pp. 930941, August, 1997.  
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@article{ 10.1109/12.609280, author = {C.d. Yang and D.t. Lee and C.k. Wong}, title = {The Smallest Pair of Noncrossing Paths in a Rectilinear Polygon}, journal ={IEEE Transactions on Computers}, volume = {46}, number = {8}, issn = {00189340}, year = {1997}, pages = {930941}, doi = {http://doi.ieeecomputersociety.org/10.1109/12.609280}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Computers TI  The Smallest Pair of Noncrossing Paths in a Rectilinear Polygon IS  8 SN  00189340 SP930 EP941 EPD  930941 A1  C.d. Yang, A1  D.t. Lee, A1  C.k. Wong, PY  1997 KW  Rectilinear polygon KW  shortest rectilinear path KW  minimum bend path KW  noncrossing paths KW  computational geometry KW  VLSI routing. VL  46 JA  IEEE Transactions on Computers ER   
Abstract—Smallest rectilinear paths are rectilinear paths with a minimum number of bends and with a minimum length simultaneously. In this paper, given two pairs of terminals within a rectilinear polygon, we derive an algorithm to find a pair of noncrossing rectilinear paths within the polygon such that the total number of bends and the total length are both minimized. Although a smallest rectilinear path between two terminals in a rectilinear polygon always exists, we show that such a smallest pair may not exist for some problem instances. In that case, the algorithm presented will find, among all noncrossing paths with a minimum total number of bends, a pair whose total length is the shortest, or find, among all noncrossing paths with a minimum total length, a pair whose total number of bends is minimized. We provide a simple linear time and space algorithm based on the fact that there are only a limited number of configurations of such a solution pair.
[1] B. Chazelle, "Triangulating a Simple Polygon in Linear Time," Discrete&Computational Geometry, vol. 6, pp. 485524, 1991.
[2] K.L. Clarkson, S. Kapoor, and P.M. Vaidya, "Rectilinear Shortest Paths Through Polygonal Obstacles in O(nlog3/2n) Time," unpublished manuscript, 1988.
[3] M. de Berg, M. van Kreveld, B.J. Nilsson, and M.H. Overmars, "Shortest Path Queries in Rectilinear Worlds," Int'l J. Computational Geometry&Applications, vol. 2, no. 3, pp. 287309, Sept. 1992.
[4] M. de Berg, "On Rectilinear Link Distance," Computational Geometry Theory&Application, vol. 1, no. 1, pp. 1334, July 1991.
[5] P.J. deRezende, D.T. Lee, and Y.F. Wu, "Rectilinear Shortest Paths with Rectangular Barriers," Discrete&Computational Geometry, vol. 4, pp. 4153, 1989.
[6] H.N. Djidjev, A. Lingas, and J. Sack, "An O(nlog n) Algorithm for Computing the Link Center in a Simple Polygon," Discrete&Computational Geometry, vol. 8, pp. 131152, 1992.
[7] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NPCompleteness.New York: W.H. Freeman, 1979.
[8] D.B. Johnson, "Efficient Algorithms for Shortest Paths in Sparse Networks," J. ACM, vol. 24, pp. 113, 1977.
[9] Y. Ke, "An Efficient Algorithm For LinkDistance Problems," Proc. Fifth ACM Symp. Computational Geometry, pp. 6978, 1989.
[10] R.C. Larson and V.O. Li, "Finding Minimum Rectilinear Distance Paths in the Presence of Barriers," Networks, vol. 11, pp. 285304, 1981.
[11] D.T. Lee, C.D. Yang, and T.H. Chen, "Shortest Rectilinear Paths Among Weighted Obstacles," Int'l J. Computational Geometry&Applications, vol. 1, no. 2, pp. 109124, June 1991.
[12] D.T. Lee, C.D. Yang, and C.K. Wong, "On Bends and Distances of Paths Among Obstacles in TwoLayer Interconnection Model," IEEE Trans. Computers, vol. 43, no. 6, pp. 711724, June 1994.
[13] W. Lenhart, R. Pollack, J. Sack, R. Seidel, M. Sharir, S. Ruri, G. Toussaint, S. Whitesides, and C. Yap, "Computing the Link Center of a Simple Polygon," Discrete&Computational Geometry, vol. 3, pp. 281293, 1988.
[14] K.M. McDonald and J.G. Peters, "Smallest Paths in Simple Rectilinear Polygons," IEEE Trans. ComputerAided Design of Integrated Circuits and Systems, vol. 11, no. 7, pp. 864875, July 1992.
[15] J.S.B. Mitchell, "Shortest Rectilinear Paths among Obstacles" Algorithmica, vol. 8, pp. 5588, 1992.
[16] J.S.B. Mitchell, "A New Algorithm for Shortest Paths among Obstacles in the Plane," Annals of Math. and Artificial Intelligence, vol. 3, pp. 83106, 1991.
[17] J.S.B. Mitchell, G. Rote, and G. Woginger, "Computing the Minimum Link Path among a Set of Obstacles in the Planes," Proc. Sixth ACM Symp. Computational Geometry, pp. 6372, 1990.
[18] T. Ohtsuki, "The Two Disjoint Path Problem and Wire Routing Design," Lecture Notes in Computer Science, vol. 108, pp. 207216. SpringerVerlag, 1981.
[19] T. Ohtsuki, "Gridless Routers—New Wire Routing Algorithm Based on Computational Geometry," Proc. Int'l Conf. Circuits and Systems,China, 1985.
[20] Y. Perl and Y. Shiloach, "Finding Two Disjoint Paths Between Two Pairs of Vertices in a Graph," J. ACM, vol. 25, no. 1, pp. 19, 1978.
[21] Y. Shiloach, "A Polynominal Solution to the Undirected Two Pahts Problem," J. ACM, vol. 27, pp. 445456, 1980.
[22] S. Suri, "A Linear Time Algorithm for Minimum Link Paths Inside a Simple Polygon," Computer Vision, Graphics, and Image Processing, vol. 35, pp. 99110, 1986.
[23] J.W. Suurballe, "The SingleSource, AllTerminals Problem for Disjoint Paths," unpublished technical memorandum, Bell Laboratories, 1982.
[24] J. Takahashi, H. Suzuki, and T. Nishizeki, "Algorithms for Finding Noncrossing Paths with Minimum Total Length in Plane Graphs," Proc. ISAAC '92, Lecture Notes in Computer Science, vol. 650, pp. 400409, SpringerVerlag, 1992.
[25] J. Takahashi, H. Suzuki, and T. Nishizeki, "Finding Shortest Noncrossing Rectilinear Paths in Plane Regions," Proc. ISAAC '93, Lecture Notes in Computer Science, vol. 762, pp. 98107, SpringerVerlag, 1993.
[26] R. Tarjan, "Data Structures and Network Algorithms," SIAM,Philadelphia, Penn., 1983.
[27] P. Widmayer, "Network Design Issues in VLSI," manuscript, 1989.
[28] 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. Computers, vol. 36, pp. 321331, 1987.
[29] C.D. Yang, D.T. Lee, and C.K. Wong, "On Bends and Lengths of Rectilinear Paths: A GraphTheoretic Approach," Int'l J. Computational Geometry&Applications, vol. 2, no. 1, pp. 6174, Mar. 1992.