Real Two Dimensional Scaled Matching

AbstractScaled Matching refers to the problem of finding all locations in the text where the pattern, proportionally enlarged according to an arbitrary real-sized scale, appears. Scaled matching is an important problem that was originally inspired by Computer Vision. Finding a combinatorial definition that captures the concept of real scaling in discrete images has been a challenge in the pattern matching field. No definition existed that captured the concept of real scaling in discrete images, without assuming an underlying continuous signal, as done in the image processing field. We present a combinatorial definition for real scaled matching that scales images in a pleasing natural manner. We also present efficient algorithms for real scaled matching. The running times of our algorithms are as follows. For T, a two-dimensional n×n text array, and P, an m×m pattern array, we find in T all occurrences of P scaled to any real value in time O(nm3+n2mlog m).

[1]  Amihood Amir,et al.  Two-Dimensional Dictionary Matching , 1992, Inf. Process. Lett..

[2]  Kamala Krithivasan,et al.  Efficient two-dimensional pattern matching in the presence of errors , 1987, Inf. Sci..

[3]  Peter Weiner,et al.  Linear Pattern Matching Algorithms , 1973, SWAT.

[4]  M. Crochemore,et al.  On-line construction of suffix trees , 2002 .

[5]  Gad M. Landau,et al.  Two-dimensional pattern matching with rotations , 2004, Theor. Comput. Sci..

[6]  Uzi Vishkin,et al.  On Finding Lowest Common Ancestors: Simplification and Parallelization , 1988, AWOC.

[7]  Roberto Grossi,et al.  On the Construction of Classes of Suffix Trees for Square Matrices: Algorithms and Applications , 1995, ICALP.

[8]  Uzi Vishkin,et al.  Finding Level-Ancestors in Trees , 1994, J. Comput. Syst. Sci..

[9]  Robert E. Tarjan,et al.  Fast Algorithms for Finding Nearest Common Ancestors , 1984, SIAM J. Comput..

[10]  Gad M. Landau,et al.  Efficient String Matching with k Mismatches , 2018, Theor. Comput. Sci..

[11]  Gad M. Landau,et al.  Fast parallel and serial multidimensional approximate array matching , 1990 .

[12]  Donald E. Knuth,et al.  Fast Pattern Matching in Strings , 1977, SIAM J. Comput..

[13]  Edward M. McCreight,et al.  A Space-Economical Suffix Tree Construction Algorithm , 1976, JACM.

[14]  Amihood Amir,et al.  Efficient 2-dimensional approximate matching of non-rectangular figures , 1991, SODA '91.

[15]  Gad M. Landau,et al.  Efficient pattern matching with scaling , 1990, SODA '90.

[16]  S. Muthukrishnan,et al.  On the sorting-complexity of suffix tree construction , 2000, JACM.

[17]  Alejandro A. Schäffer,et al.  Multiple matching of rectangular patterns , 1993, STOC '93.

[18]  Peter Sanders,et al.  Simple Linear Work Suffix Array Construction , 2003, ICALP.

[19]  Esko Ukkonen,et al.  A Rotation Invariant Filter for Two-Dimensional String Matching , 1998, CPM.

[20]  Matt Brown,et al.  Invited talk , 2007 .

[21]  Amihood Amir,et al.  Alphabet Independent and Dictionary Scaled Matching , 1996, CPM.

[22]  Eugene W. Myers,et al.  Suffix arrays: a new method for on-line string searches , 1993, SODA '90.

[23]  Gad M. Landau,et al.  Pattern matching in a digitized image , 1992, SODA '92.

[24]  Moshe Lewenstein,et al.  Real scaled matching , 2000, SODA '00.

[25]  Robert S. Boyer,et al.  A fast string searching algorithm , 1977, CACM.