Shortest Paths and Networks

We survey various forms of the problem, primarily in two and three dimensions, for motion of a single point, since most results have focused on these cases. We discuss shortest paths in a simple polygon (Section 31.1), shortest paths among obstacles (Section 31.2), and other metrics for length (Section 31.3). We also survey other related geometric network optimization problems (Section 31.4). Higher dimensions are discussed in Section 31.5.

[1]  Subhash Suri,et al.  Matrix searching with the shortest path metric , 1993, SIAM J. Comput..

[2]  Joseph S. B. Mitchell,et al.  The weighted region problem: finding shortest paths through a weighted planar subdivision , 1991, JACM.

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

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

[5]  Michiel H. M. Smid,et al.  Lower bounds for computing geometric spanners and approximate shortest paths , 1996, Discret. Appl. Math..

[6]  Amit Kumar,et al.  Maximum Coverage Problem with Group Budget Constraints and Applications , 2004, APPROX-RANDOM.

[7]  John Hershberger,et al.  Optimal parallel algorithms for triangulated simple polygons , 1992, SCG '92.

[8]  Esther M. Arkin,et al.  Minimum-link watchman tours , 2003, Inf. Process. Lett..

[9]  Simeon C. Ntafos,et al.  Optimum Watchman Routes , 1988, Inf. Process. Lett..

[10]  Sergei Bespamyatnikh,et al.  An O( n log n ) algorithm for the zoo-keeper's problem , 2003 .

[11]  Sariel Har-Peled,et al.  Constructing approximate shortest path maps in three dimensions , 1998, SCG '98.

[12]  Subhash Suri,et al.  On some link distance problems in a simple polygon , 1990, IEEE Trans. Robotics Autom..

[13]  Jürgen Sellen,et al.  Direction weighted shortest path planning , 1995, Proceedings of 1995 IEEE International Conference on Robotics and Automation.

[14]  Jon M. Kleinberg,et al.  An improved approximation ratio for the minimum latency problem , 1996, SODA '96.

[15]  Joseph S. B. Mitchell,et al.  Approximation algorithms for TSP with neighborhoods in the plane , 2001, SODA '01.

[16]  Danny Ziyi Chen,et al.  On the all-pairs Euclidean short path problem , 1995, SODA '95.

[17]  Satish Rao,et al.  Approximating geometrical graphs via “spanners” and “banyans” , 1998, STOC '98.

[18]  John F. Canny,et al.  New lower bound techniques for robot motion planning problems , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[19]  Subhash Suri,et al.  Curvature-Constrained Shortest Paths in a Convex Polygon , 2002, SIAM J. Comput..

[20]  Naveen Garg,et al.  Saving an epsilon: a 2-approximation for the k-MST problem in graphs , 2005, STOC '05.

[21]  Joseph S. B. Mitchell,et al.  New results on shortest paths in three dimensions , 2004, SCG '04.

[22]  Boris Aronov,et al.  The furthest-site geodesic voronoi diagram , 1993, SCG '88.

[23]  Moshe Dror,et al.  Touring a sequence of polygons , 2003, STOC '03.

[24]  Joseph S. B. Mitchell,et al.  A new algorithm for computing shortest paths in weighted planar subdivisions (extended abstract) , 1997, SCG '97.

[25]  Bernard Chazelle,et al.  Sublinear geometric algorithms , 2003, STOC '03.

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

[27]  Chee-Keng Yap,et al.  Approximate Euclidean Shortest Paths in 3-Space , 1997, Int. J. Comput. Geom. Appl..

[28]  Anna Lubiw,et al.  Shortest Descending Paths through Given Faces , 2009, CCCG.

[29]  Hazel Everett,et al.  The Aquarium Keeper's Problem , 1991, SODA '91.

[30]  Kenneth L. Clarkson,et al.  Rectilinear shortest paths through polygonal obstacles in O(n(logn)2) time , 1987, SCG '87.

[31]  Joseph S. B. Mitchell,et al.  Path planning in 0/1/ weighted regions with applications , 1988, SCG '88.

[32]  Jörg-Rüdiger Sack,et al.  Approximation algorithms for geometric shortest path problems , 2000, STOC '00.

[33]  Kenneth L. Clarkson,et al.  Approximation algorithms for shortest path motion planning , 1987, STOC.

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

[35]  Joseph S. B. Mitchell,et al.  Geometric Shortest Paths and Network Optimization , 2000, Handbook of Computational Geometry.

[36]  Bruce Randall Donald,et al.  Kinodynamic motion planning , 1993, JACM.

[37]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1984, JACM.

[38]  John Hershberger,et al.  An optimal visibility graph algorithm for triangulated simple polygons , 1989, Algorithmica.

[39]  Esther M. Arkin,et al.  Angewandte Mathematik Und Informatik Universit at Zu K Oln Approximation Algorithms for Lawn Mowing and Milling Ss Andor P.fekete Center for Parallel Computing Universitt at Zu Kk Oln D{50923 Kk Oln Germany Approximation Algorithms for Lawn Mowing and Milling , 2022 .

[40]  Alok Aggarwal,et al.  The angular-metric traveling salesman problem , 1997, SODA '97.

[41]  Mikhail J. Atallah,et al.  On Parallel Rectilinear Obstacle- Avoiding Paths , 1993, Comput. Geom..

[42]  Esther M. Arkin,et al.  Computing a shortest k-link path in a polygon , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[43]  Jean-Daniel Boissonnat,et al.  A polynomial-time algorithm for computing a shortest path of bounded curvature amidst moderate obstacles (extended abstract) , 1996, SCG '96.

[44]  Subhash Suri,et al.  Practical methods for approximating shortest paths on a convex polytope in R3 , 1998, SODA '95.

[45]  Marek Chrobak,et al.  Computing simple paths among obstacles , 2000, Comput. Geom..

[46]  Michel Pocchiola,et al.  The visibility complex , 1993, SCG '93.

[47]  Pinaki Mitra,et al.  Orthogonal shortest route queries among axis parallel rectangular obstacles , 1994, Int. J. Comput. Geom. Appl..

[48]  Sariel Har-Peled,et al.  Approximate shortest paths and geodesic diameters on convex polytopes in three dimensions , 1997, SCG '97.

[49]  Joseph S. B. Mitchell,et al.  Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems , 1999, SIAM J. Comput..

[50]  Joseph S. B. Mitchell,et al.  Two-point Euclidean shortest path queries in the plane , 1999, SODA '99.

[51]  Christos H. Papadimitriou,et al.  An Algorithm for Shortest-Path Motion in Three Dimensions , 1985, Inf. Process. Lett..

[52]  Joachim Gudmundsson,et al.  TSP with neighborhoods of varying size , 2005, J. Algorithms.

[53]  Leonidas J. Guibas,et al.  Optimal Shortest Path Queries in a Simple Polygon , 1989, J. Comput. Syst. Sci..

[54]  Bruce Randall Donald,et al.  Provably good approximation algorithms for optimal kinodynamic planning for Cartesian robots and open chain manipulators , 1990, SCG '90.

[55]  Jörg-Rüdiger Sack,et al.  Approximating Shortest Paths on Weighted Polyhedral Surfaces , 2001, Algorithmica.

[56]  Joachim Gudmundsson,et al.  A Fast Approximation Algorithm for TSP with Neighborhoods , 1999, Nord. J. Comput..

[57]  Joseph S. B. Mitchell,et al.  The Discrete Geodesic Problem , 1987, SIAM J. Comput..

[58]  Oded Schwartz,et al.  On the complexity of approximating tsp with neighborhoods and related problems , 2003, computational complexity.

[59]  Joseph S. B. Mitchell,et al.  A new algorithm for shortest paths among obstacles in the plane , 1991, Annals of Mathematics and Artificial Intelligence.

[60]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[61]  Chee Yap,et al.  Rectilinear geodesics in 3-space , 1995 .

[62]  Micha Sharir,et al.  An Optimal-Time Algorithm for Shortest Paths on a Convex Polytope in Three Dimensions , 2006, SCG '06.

[63]  Micha Sharir,et al.  Approximating shortest paths on a convex polytope in three dimensions , 1997, JACM.

[64]  Sven Schuierer,et al.  An optimal data structure for shortest rectilinear path queries in a simple rectilinear polygon , 1996, Int. J. Comput. Geom. Appl..

[65]  Ioannis G. Tollis,et al.  Path planning in the presence of vertical obstacles , 1990, IEEE Trans. Robotics Autom..

[66]  S. Rao Kosaraju,et al.  A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields , 1995, JACM.

[67]  J. Reif,et al.  Approximate Kinodynamic Planning Using L2-norm Dynamic Bounds , 1990 .

[68]  David M. Mount,et al.  An Output Sensitive Algorithm for Computing Visibility Graphs , 1987, FOCS.

[69]  Gerhard J. Woeginger,et al.  The geometric maximum traveling salesman problem , 2002, JACM.

[70]  Joseph S. B. Mitchell,et al.  Approximation algorithms for geometric tour and network design problems (extended abstract) , 1995, SCG '95.

[71]  Sanjiv Kapoor,et al.  Efficient computation of geodesic shortest paths , 1999, STOC '99.

[72]  D. T. Lee.  Rectilinear Paths among Rectilinear Obstacles , 1992, ISAAC.

[73]  Roberto Tamassia,et al.  Optimal Shortest Path and Minimum-Link Path Queries Between Two Convex Polygons Inside a Simple Polygonal Obstacle , 1997, Int. J. Comput. Geom. Appl..

[74]  Satish Rao,et al.  Paths, trees, and minimum latency tours , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[75]  Jörg-Rüdiger Sack,et al.  Determining approximate shortest paths on weighted polyhedral surfaces , 2005, JACM.

[76]  D. T. Lee,et al.  A New Approach for the Geodesic Voronoi Diagram of Points in a Simple Polygon and Other Restricted Polygonal Domains , 1998, Algorithmica.

[77]  D. T. Lee,et al.  Finding an Approximate Minimum-Link Visibility Path Inside a Simple Polygon , 1995, Inf. Process. Lett..

[78]  D. T. Lee,et al.  Euclidean shortest paths in the presence of rectilinear barriers , 1984, Networks.

[79]  Leonidas J. Guibas,et al.  Ray Shooting in Polygons Using Geodesic Triangulations , 1991, ICALP.

[80]  Hongyan Wang,et al.  The complexity of the two dimensional curvature-constrained shortest-path problem , 1998 .

[81]  Michael T. Goodrich,et al.  Dynamic Ray Shooting and Shortest Paths in Planar Subdivisions via Balanced Geodesic Triangulations , 1997, J. Algorithms.

[82]  Jack Snoeyink,et al.  Testing Homotopy for Paths in the Plane , 2002, SCG '02.

[83]  Joseph S. B. Mitchell,et al.  An Efficient Algorithm for Euclidean Shortest Paths Among Polygonal Obstacles in the Plane , 1997, Discret. Comput. Geom..

[84]  John Hershberger,et al.  A New Data Structure for Shortest Path Queries in a Simple Polygon , 1991, Inf. Process. Lett..

[85]  Joonsoo Choi,et al.  Monotonicity of rectilinear geodesics in d-space , 1996, SoCG 1996.

[86]  Esther M. Arkin,et al.  On monotone paths among obstacles with applications to planning assemblies , 1989, SCG '89.

[87]  John Hershberger,et al.  Computing Minimum Length Paths of a Given Homotopy Class , 1994, Comput. Geom..

[88]  Micha Sharir,et al.  On shortest paths amidst convex polyhedra , 1987, SCG '86.

[89]  Subhash Suri,et al.  An Optimal Algorithm for Euclidean Shortest Paths in the Plane , 1999, SIAM J. Comput..

[90]  Michiel H. M. Smid,et al.  Planar Spanners and Approximate Shortest Path Queries among Obstacles in the Plane , 1996, ESA.

[91]  Andrzej Lingas,et al.  Fast Approximation Schemes for Euclidean Multi-connectivity Problems , 2000, ICALP.

[92]  Stephen G. Kobourov,et al.  Computing homotopic shortest paths efficiently , 2002, Comput. Geom..

[93]  Joseph O'Rourke.  Computational Geometry Column 35 , 1999, Int. J. Comput. Geom. Appl..

[94]  Petr Slavik,et al.  The Errand Scheduling Problem , 1997 .

[95]  Michel Pocchiola,et al.  Topologically sweeping visibility complexes via pseudotriangulations , 1996, Discret. Comput. Geom..

[96]  Mikhail J. Atallah,et al.  Parallel Rectilinear Shortest Paths with Rectangular Obstacles , 1991, Comput. Geom..

[97]  Esther M. Arkin,et al.  Approximation Algorithms for the Geometric Covering Salesman Problem , 1994, Discret. Appl. Math..

[98]  Danny Ziyi Chen,et al.  Shortest path queries among weighted obstacles in the rectilinear plane , 1995, SCG '95.

[99]  Boris Aronov,et al.  Star Unfolding of a Polytope with Applications , 1997, SIAM J. Comput..

[100]  Esther M. Arkin,et al.  Logarithmic-time link path queries in a simple polygon , 1995, Int. J. Comput. Geom. Appl..

[101]  Sándor P. Fekete,et al.  On Simple Polygonalizations with Optimal Area , 2000, Discret. Comput. Geom..

[102]  Sergei N. Bespamyatnikh.  Computing homotopic shortest paths in the plane , 2003, SODA '03.

[103]  Wei-Pand Chin,et al.  Shortest watchman routes in simple polygons , 1990, Discret. Comput. Geom..

[104]  Danny Z. Chen,et al.  Efficient Algorithms for Geometric Shortest Path Query Problems , 1998 .

[105]  D. T. Lee,et al.  Minimal Link Visibility Paths Inside a Simple Polygon , 1993, Comput. Geom..

[106]  Khaled M. Elbassioni,et al.  On Approximating the TSP with Intersecting Neighborhoods , 2006, ISAAC.

[107]  Leonidas J. Guibas,et al.  Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons , 1987, Algorithmica.

[108]  Yijie Han,et al.  Shortest paths on a polyhedron , 1990, SCG '90.

[109]  Danny Ziyi Chen,et al.  Rectilinear Short Path Queries Among Rectangular Obstacles , 1996, Inf. Process. Lett..

[110]  Jörg-Rüdiger Sack,et al.  Link Distance Problems , 2000, Handbook of Computational Geometry.

[111]  Ovidiu Daescu,et al.  On Geometric Path Query Problems , 1997, Int. J. Comput. Geom. Appl..

[112]  Pankaj K. Agarwal,et al.  Approximation algorithms for curvature-constrained shortest paths , 1996, SODA '96.

[113]  Joseph S. B. Mitchell,et al.  L1 shortest paths among polygonal obstacles in the plane , 1992, Algorithmica.

[114]  Steven J. Gortler,et al.  Fast exact and approximate geodesics on meshes , 2005, ACM Trans. Graph..

[115]  Joseph S. B. Mitchell,et al.  Shortest paths among obstacles in the plane , 1993, SCG '93.

[116]  Joseph S. B. Mitchell,et al.  Bicriteria Shortest Path Problems in the Plane ( extended abstract ) , 2022 .

[117]  Chee-Keng Yap,et al.  Precision-sensitive Euclidean shortest path in 3-space (extended abstract) , 1995, SCG '95.

[118]  Sanjeev Arora,et al.  Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems , 1998, JACM.

[119]  Pankaj K. Agarwal,et al.  Approximating shortest paths on a nonconvex polyhedron , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.