Efficiently approximating polygonal paths in three and higher dimensions

We present efficient algorithms for solving polygonal-path approximation problems in three and higher dimensions. Given an n-vertex polygonal curve P inRd , d‚ 3, we approximate P by another poly- gonal curve P0 of mn vertices in Rd such that the vertex sequence of P0 is an ordered subsequence of the vertices of P. The goal is either to minimize the size m of P0 for a given error tolerance " (called the min-# problem), or to minimize the deviation error " between P and P 0 for a given size m of P 0 (called the min-" problem). Our techniques enable us to develop efficient near-quadratic-time algorithms in three dimensions and subcubic-time algorithms in four dimensions for solving the min-# and min-" problems. We discuss extensions of our solutions to d-dimensional space, where d > 4, and for the L1 and L1 metrics.

[1]  David H. Douglas,et al.  ALGORITHMS FOR THE REDUCTION OF THE NUMBER OF POINTS REQUIRED TO REPRESENT A DIGITIZED LINE OR ITS CARICATURE , 1973 .

[2]  Kevin Q. Brown.  Geometric transforms for fast geometric algorithms , 1979 .

[3]  Nimrod Megiddo,et al.  Applying parallel computation algorithms in the design of serial algorithms , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[4]  B. Buttenfield.  TREATMENT OF THE CARTOGRAPHIC LINE , 1985 .

[5]  Hiroshi Imai,et al.  Computational-geometric methods for polygonal approximations of a curve , 1986, Comput. Vis. Graph. Image Process..

[6]  A. Melkman,et al.  On Polygonal Chain Approximation , 1988 .

[7]  M. Iri,et al.  Polygonal Approximations of a Curve — Formulations and Algorithms , 1988 .

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

[9]  Esther M. Arkin,et al.  An efficiently computable metric for comparing polygonal shapes , 1991, SODA '90.

[10]  B. Naylor,et al.  Piecewise linear approximations of digitized space curves with applications , 1991 .

[11]  Michael T. Goodrich.  Using approximation algorithms to design parallel algorithms that may ignore processor allocation , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[12]  Helmut Alt,et al.  Measuring the resemblance of polygonal curves , 1992, SCG '92.

[13]  Günter Rote,et al.  A New Metric Between Polygons and How to Compute it , 1992, ICALP.

[14]  Michael T. Goodrich,et al.  Dynamic ray shooting and shortest paths via balanced geodesic triangulations , 1993, SCG '93.

[15]  Leonidas J. Guibas,et al.  Approximating Polygons and Subdivisions with Minimum Link Paths , 1991, Int. J. Comput. Geom. Appl..

[16]  Otfried Cheong,et al.  On ray shooting in convex polytopes , 1993, Discret. Comput. Geom..

[17]  Godfried T. Toussaint,et al.  On Approximating Polygonal Curves in Two and Three Dimensions , 1994, CVGIP Graph. Model. Image Process..

[18]  W. S. Chan,et al.  Approximation of Polygonal Curves with Minimum Number of Line Segments or Minimum error , 1996, Int. J. Comput. Geom. Appl..

[19]  Kasturi R. Varadarajan.  Approximating monotone polygonal curves using the uniform metric , 1996, SCG '96.

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

[21]  David Eppstein,et al.  Faster construction of planar two-centers , 1997, SODA '97.

[22]  Micha Sharir.  A Near-Linear Algorithm for the Planar 2-Center Problem , 1997, Discret. Comput. Geom..

[23]  Jeff Erickson.  Space-time tradeoffs for emptiness queries (extended abstract) , 1997, SCG '97.

[24]  Jirí Matousek,et al.  Computing Many Faces in Arrangements of Lines and Segments , 1998, SIAM J. Comput..

[25]  Space-Efficient Algorithms for Approximating Polygonal Curves in Two Dimensional Space , 1998, COCOON.

[26]  Michael T. Goodrich,et al.  Dynamic trees and dynamic point location , 1991, STOC '91.