The Smallest Pair of Noncrossing Paths in a Rectilinear Polygon

Smallest rectilinear paths are rectilinear paths with simultaneous minimum numbers of bends and minimum lengths. Given two pairs of terminals within a rectilinear polygon, the authors 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, they 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. They 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]  Chak-Kuen Wong,et al.  On Bends and Distances of Paths Among Obstacles in Two-Layer Interconnection Model , 1994, IEEE Trans. Computers.

[2]  Mark de Berg,et al.  On Rectilinear Link Distance , 1991, Comput. Geom..

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  D. T. Lee,et al.  Rectilinear shortest paths with rectangular barriers , 1985, SCG '85.

[5]  Joseph S. B. Mitchell,et al.  Minimum-link paths among obstacles in the plane , 1990, SCG '90.

[6]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[7]  D. T. Lee,et al.  Shortest rectilinear paths among weighted obstacles , 1990, SCG '90.

[8]  D. T. Lee,et al.  Shortest rectilinear paths among weighted obstacles , 1991, Int. J. Comput. Geom. Appl..

[9]  Chak-Kuen Wong,et al.  On Bends and Lengths of Rectilinear Paths: A Graph-Theoretic Approach , 1991, WADS.

[10]  Takao Nishizeki,et al.  Algorithms for Finding Non-Crossing Paths with Minimum Total Length in Plane Graphs , 1992, ISAAC.

[11]  Yossi Shiloach,et al.  A Polynomial Solution to the Undirected Two Paths Problem , 1980, JACM.

[12]  Joseph G. Peters,et al.  Smallest paths in simple rectilinear polygons , 1992, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[14]  R Kerridge,et al.  A Single Source , 1998, Science.

[15]  Yan Ke,et al.  An efficient algorithm for link-distance problems , 1989, SCG '89.

[16]  Chak-Kuen Wong,et al.  Rectilinear Shortest Paths and Minimum Spanning Trees in the Presence of Rectilinear Obstacles , 1987, IEEE Transactions on Computers.

[17]  Mark de Berg,et al.  Shortest path queries in rectilinear worlds , 1992, Int. J. Comput. Geom. Appl..

[18]  S. Suri.  A linear time algorithm with minimum link paths inside a simple polygon , 1986 .

[19]  Joseph S. B. Mitchell,et al.  Shortest Rectilinear Paths Among Obstacles , 1987 .

[20]  Chak-Kuen Wong,et al.  On bends and lengths of rectilinear paths: a graph theoretic approach , 1992, Int. J. Comput. Geom. Appl..

[21]  Jörg-Rüdiger Sack,et al.  An O(n log n) Algorithm for Computing a Link Center in a Simple Polygon , 1989, STACS.

[22]  D. T. Lee,et al.  Rectilinear shortest paths in the presence of rectangular barriers , 1989, Discret. Comput. Geom..

[23]  Micha Sharir,et al.  Computing the link center of a simple polygon , 1987, SCG '87.

[24]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[25]  Yehoshua Perl,et al.  Finding Two Disjoint Paths Between Two Pairs of Vertices in a Graph , 1978, JACM.

[26]  Takao Nishizeki,et al.  Finding Shortest Non-Crossing Rectilinear Paths in Plane Regions , 1993, ISAAC.

[27]  Tatsuo Ohtsuki,et al.  The two disjoint path problem and wire routing design , 1980, Graph Theory and Algorithms.

[28]  Donald B. Johnson,et al.  Efficient Algorithms for Shortest Paths in Sparse Networks , 1977, J. ACM.

[29]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[30]  Bernard Chazelle.  Triangulating a simple polygon in linear time , 1991, Discret. Comput. Geom..

[31]  Andrzej Lingas,et al.  AnO(n logn) algorithm for computing the link center of a simple polygon , 1992, Discret. Comput. Geom..

[32]  Victor O. K. Li,et al.  Finding minimum rectilinear distance paths in the presence of barriers , 1981, Networks.