On Minimum Link Monotone Path Problems

The problem of finding monotone paths between two given points has useful applications in path planning, and in particular, it is useful to look for minimum link paths. We are given a simple polygon P or a polygonal domain D with n vertices and a triplet of input parameters: (s, t, d), where s and t are two points in the plane and d is any direction. We show how to answer a query for the existence of a d-monotone path between s and t inside P (or D) in logarithmic time after preprocessing P in O(En) time, or D in O(En + ERlogR) time, where E is the size of the visibility graph of P (or D), and R is the number of reflex vertices in D. Our approach is based on the novel idea utilizing the dual graph of the trapezoidal map of P (or D). For polygonal domains, our approach uses a trapezoidal map associated with each visibility edge of D, and we show how to compute this large set of trapezoidal maps efficiently. Furthermore, we show how to output a minimum linkd-monotone path between points s and t, for an arbitrary input triplet (s, t, d).

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

[2]  Joseph S. B. Mitchell,et al.  Minimum-link paths among obstacles in the plane , 2005, Algorithmica.

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

[4]  Nancy M. Amato,et al.  Computing faces in segment and simplex arrangements , 1995, STOC '95.

[5]  Marc J. van Kreveld,et al.  Linear-Time Reconstruction of Delaunay Triangulations with Applications , 1997, ESA.

[6]  David M. Mount,et al.  An output sensitive algorithm for computing visibility graphs , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[7]  Jorge Urrutia,et al.  Computing shortest heterochromatic monotone routes , 2008, Oper. Res. Lett..

[8]  Atlas F. Cook,et al.  Link distance and shortest path problems in the plane , 2011, Comput. Geom..

[9]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[10]  Leonidas J. Guibas,et al.  The complexity and construction of many faces in arrangements of lines and of segments , 1990, Discret. Comput. Geom..

[11]  Simeon C. Ntafos,et al.  On Decomposing Polygons into Uniformly Monotone Parts , 1988, Inf. Process. Lett..

[12]  Matti Nykänen,et al.  Finding Lowest Common Ancestors in Arbitrarily Directed Trees , 1994, Inf. Process. Lett..

[13]  Oscar Castillo,et al.  Intelligent Control and Planning of Autonomous Mobile Robots Using Fuzzy Logic and Multiple Objective Genetic Algorithms , 2007, Analysis and Design of Intelligent Systems using Soft Computing Techniques.

[14]  Ovidiu Daescu,et al.  On Geometric Path Query Problems , 1997, WADS.

[15]  Mikkel Thorup.  Compact oracles for reachability and approximate distances in planar digraphs , 2004, JACM.

[16]  Esther M. Arkin,et al.  Optimal link path queries in a simple polygon , 1992, SODA '92.

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

[18]  Leonidas J. Guibas,et al.  Optimal Point Location in a Monotone Subdivision , 1986, SIAM J. Comput..

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

[20]  David G. Kirkpatrick,et al.  Optimal Search in Planar Subdivisions , 1983, SIAM J. Comput..

[21]  Radovan Kovacevic,et al.  Automated torch path planning using polygon subdivision for solid freeform fabrication based on welding , 2004 .