Memory-constrained algorithms for simple polygons

A constant-work-space algorithm has read-only access to an input array and may use only O(1) additional words of O(log n) bits, where n is the input size. We show how to triangulate a plane straight-line graph with n vertices in O(n2) time and constant workspace. We also consider the problem of preprocessing a simple n-gon P for shortest path queries, where P is given by the ordered sequence of its vertices. For this, we relax the space constraint to allow s words of work-space. After quadratic preprocessing, the shortest path between any two points inside P can be found in O(n2/s) time.

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

[2]  S. Muthukrishnan,et al.  Data streams: algorithms and applications , 2005, SODA '03.

[3]  Raimund Seidel.  Convex Hull Computations , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[4]  Robert E. Tarjan,et al.  Triangulating a Simple Polygon , 1978, Inf. Process. Lett..

[5]  Luis Barba,et al.  Computing the Visibility Polygon Using Few Variables , 2011, ISAAC.

[6]  Timothy M. Chan,et al.  Towards in-place geometric algorithms and data structures , 2004, SCG '04.

[7]  Mark de Berg,et al.  Computational Geometry: Algorithms and Applications, Second Edition , 2000 .

[8]  Bernard Chazelle,et al.  A theorem on polygon cutting with applications , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[9]  Leonidas J. Guibas,et al.  Optimal shortest path queries in a simple polygon , 1987, SCG '87.

[10]  Yajun Wang,et al.  Constant-Work-Space Algorithms for Shortest Paths in Trees and Simple Polygons , 2011, J. Graph Algorithms Appl..

[11]  Omer Reingold,et al.  Undirected connectivity in log-space , 2008, JACM.

[12]  Pat Morin,et al.  Space-efficient planar convex hull algorithms , 2004, Theor. Comput. Sci..

[13]  Alain Fournier,et al.  Triangulating Simple Polygons and Equivalent Problems , 1984, TOGS.

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

[15]  Sanjeev Khanna,et al.  Space-efficient online computation of quantile summaries , 2001, SIGMOD '01.

[16]  Hazel Everett,et al.  Slicing an ear using prune-and-search , 1993, Pattern Recognit. Lett..

[17]  Timothy M. Chan,et al.  Optimal in-place and cache-oblivious algorithms for 3-d convex hulls and 2-d segment intersection , 2010, Comput. Geom..

[18]  Kunihiko Sadakane,et al.  Space–Time Trade-offs for Stack-Based Algorithms , 2012, Algorithmica.

[19]  J. Ian Munro,et al.  Selection and sorting with limited storage , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[20]  Greg N. Frederickson,et al.  Upper Bounds for Time-Space Trade-Offs in Sorting and Selection , 1987, J. Comput. Syst. Sci..

[21]  Günter Rote,et al.  Constant-Working-Space Algorithms for Geometric Problems , 2009, CCCG.

[22]  Venkatesh Raman,et al.  Improved Upper Bounds for Time-Space Tradeoffs for Selection with Limited Storage , 1998, SWAT.

[23]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[24]  Günter Rote,et al.  Reprint of: Memory-constrained algorithms for simple polygons , 2014, Comput. Geom..

[25]  Timothy M. Chan,et al.  Space-efficient algorithms for computing the convex hull of a simple polygonal line in linear time , 2004, Comput. Geom..

[26]  Guy Joseph Jacobson,et al.  Succinct static data structures , 1988 .

[27]  Venkatesh Raman,et al.  Selection from Read-Only Memory and Sorting with Minimum Data Movement , 1996, Theor. Comput. Sci..

[28]  Luis Barba,et al.  Computing a visibility polygon using few variables , 2011, Comput. Geom..

[29]  Timothy M. Chan,et al.  Multi-Pass Geometric Algorithms , 2005, Discret. Comput. Geom..

[30]  Timothy M. Chan.  Comparison-based time-space lower bounds for selection , 2009, TALG.

[31]  Bernard Chazelle,et al.  Triangulation and shape-complexity , 1984, TOGS.