A linear time and space algorithm for detecting path intersection I

The Freeman chain code is a common and useful way for representing discrete paths by means of words such that each letter encodes a step in a given direction. In the discrete plane Z^2 such a coding is widely used for representing connected discrete sets by their contour which forms a closed and intersection free path. In this paper, we use a multidimensional radix tree like data structure for storing paths in the discreted-dimensional space Z^d. It allows to design a simple and efficient algorithm for detecting path intersection. Even though an extra initialization is required, the time and space complexities remain linear for any fixed dimension d. Several problems that are solved by adapting our algorithm are also discussed.

[1]  Donald E. Knuth,et al.  The art of computer programming, volume 3: (2nd ed.) sorting and searching , 1998 .

[2]  Jacques-Olivier Lachaud,et al.  Lyndon + Christoffel = digitally convex , 2009, Pattern Recognit..

[3]  M. Lothaire,et al.  Applied Combinatorics on Words , 2005 .

[4]  Donald E. Knuth,et al.  The Art of Computer Programming: Volume 3: Sorting and Searching , 1998 .

[5]  Gilbert Labelle,et al.  Properties of the Contour Path of Discrete Sets , 2006, Int. J. Found. Comput. Sci..

[6]  Herbert Freeman,et al.  On the Encoding of Arbitrary Geometric Configurations , 1961, IRE Trans. Electron. Comput..

[7]  Srecko Brlek,et al.  On the tiling by translation problem , 2009, Discret. Appl. Math..

[8]  Isabelle Debled-Rennesson,et al.  A Linear Algorithm for Segmentation of Digital Curves , 1995, Int. J. Pattern Recognit. Artif. Intell..

[9]  Srecko Brlek,et al.  A Linear Time and Space Algorithm for Detecting Path Intersection , 2009, DGCI.

[10]  Laure Tougne,et al.  What Does Digital Straightness Tell about Digital Convexity? , 2009, IWCIA.

[11]  Isabelle Debled-Rennesson,et al.  Detection of the discrete convexity of polyominoes , 2003, Discret. Appl. Math..

[12]  Jon Louis Bentley,et al.  Quad trees a data structure for retrieval on composite keys , 1974, Acta Informatica.

[13]  M. V. Wilkes,et al.  The Art of Computer Programming, Volume 3, Sorting and Searching , 1974 .

[14]  Ulrich Eckhardt,et al.  Polygonal Representations of Digital Sets , 2003, Algorithmica.

[15]  Isabelle Debled-Rennesson,et al.  Detection of the discrete convexity of polyominoes , 2000, Discret. Appl. Math..

[16]  Jacques-Olivier Lachaud,et al.  Combinatorial View of Digital Convexity , 2008, DGCI.

[17]  M. Lothaire.  Applied Combinatorics on Words (Encyclopedia of Mathematics and its Applications) , 2005 .

[18]  Gilbert Labelle,et al.  A Note on a Result of Daurat and Nivat , 2005, Developments in Language Theory.

[19]  Xavier Provençal.  Combinatoire des mots, géométrie discrète et pavages , 2008 .

[20]  Azriel Rosenfeld,et al.  Picture Processing and Psychopictorics , 1970 .

[21]  Jacques-Olivier Lachaud,et al.  Two linear-time algorithms for computing the minimum length polygon of a digital contour , 2009, Discret. Appl. Math..

[22]  A. Smeulders,et al.  Discrete straight line segments: parameters, primitives and properties , 1991 .

[23]  Isabelle Debled-Rennesson,et al.  A Linear Algorithm for Polygonal Representations of Digital Sets , 2006, IWCIA.