Efficient Algorithms for Approximating Polygonal Chains

Abstract. We consider the problem of approximating a polygonal chain C by another polygonal chain C' whose vertices are constrained to be a subset of the set of vertices of C . The goal is to minimize the number of vertices needed in the approximation C' . Based on a framework introduced by Imai and Iri [25], we define an error criterion for measuring the quality of an approximation. We consider two problems. (1) Given a polygonal chain C and a parameter ɛ \geq 0 , compute an approximation of C , among all approximations whose error is at most ɛ , that has the smallest number of vertices. We present an O(n4/3 + δ) -time algorithm to solve this problem, for any δ > 0; the constant of proportionality in the running time depends on δ . (2) Given a polygonal chain C and an integer k , compute an approximation of C with at most k vertices whose error is the smallest among all approximations with at most k vertices. We present a simple randomized algorithm, with expected running time O(n4/3 + δ) , to solve this problem.

[1]  Micha Sharir,et al.  An Expander-Based Approach to Geometric Optimization , 1997, SIAM J. Comput..

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

[3]  John Hershberger,et al.  Cartographic Line Simplification and Polygon CSG Formulae and in O(n log* n) Time , 1997, WADS.

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

[5]  Yoshisuke Kurozumi,et al.  Polygonal approximation by the minimax method , 1982, Computer Graphics and Image Processing.

[6]  W. S. Chan,et al.  Approximation of Polygonal Curves with Minimum Number of Line Segments , 1992, ISAAC.

[7]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

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

[9]  Robert B McMaster,et al.  Automated Line Generalization , 1987 .

[10]  John D. Hobby,et al.  Polygonal approximations that minimize the number of inflections , 1993, SODA '93.

[11]  Rajeev Motwani,et al.  Clique Partitions, Graph Compression and Speeding-Up Algorithms , 1995, J. Comput. Syst. Sci..

[12]  Avraham A. Melkman,et al.  On-Line Construction of the Convex Hull of a Simple Polyline , 1987, Inf. Process. Lett..

[13]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[15]  Samuel D. Conte,et al.  Elementary Numerical Analysis: An Algorithmic Approach , 1975 .

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

[17]  Rajeev Motwani,et al.  Clique partitions, graph compression and speeding-up algorithms , 1991, STOC '91.

[18]  Jack Snoeyink,et al.  On the bit complexity of minimum link paths: superquadratic algorithms for problems solvable in linear time , 1996, SCG '96.

[19]  J. Hershberger,et al.  Speeding Up the Douglas-Peucker Line-Simplification Algorithm , 1992 .

[20]  Michael T. Goodrich.  Efficient piecewise-linear function approximation using the uniform metric , 1995, Discret. Comput. Geom..

[21]  Leonidas J. Guibas,et al.  Approximating Polygons and Subdivisions with Minimum Link Paths , 1991, ISA.

[22]  Noga Alon,et al.  Can visibility graphs Be represented compactly? , 1993, SCG '93.

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

[24]  Stefan Schirra,et al.  A new approach to subdivision simplification , 1995 .

[25]  Stan Openshaw,et al.  Algorithms for automated line generalization1 based on a natural principle of objective generalization , 1992, Int. J. Geogr. Inf. Sci..

[26]  Paul Dierckx,et al.  Curve and surface fitting with splines , 1994, Monographs on numerical analysis.

[27]  Alok Aggarwal,et al.  Finding Minimal Convex Nested Polygons , 1989, Inf. Comput..

[28]  John Hershberger,et al.  Computing Minimum Length Paths of a Given Homotopy Class (Extended Abstract) , 1991, WADS.

[29]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

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

[31]  Micha Sharir,et al.  Applications of a new space-partitioning technique , 1993, Discret. Comput. Geom..

[32]  Tetsuo Asano,et al.  Number Theory Helps Line Detection in Digital Images , 1993, ISAAC.

[33]  Jirí Matousek,et al.  Randomized Optimal Algorithm for Slope Selection , 1991, Inf. Process. Lett..

[34]  Noga Alon,et al.  Can visibility graphs Be represented compactly? , 1994, Discret. Comput. Geom..

[35]  S. Louis Hakimi,et al.  Fitting polygonal functions to a set of points in the plane , 1991, CVGIP Graph. Model. Image Process..