# Approximation algorithms for maximum two-dimensional pattern matching

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## Abstract

We introduce the following optimization version of the classical pattern matching problem (referred to as the *maximum pattern matching problem*). Given a two-dimensional rectangular text and a 2-dimensional rectangular pattern find the maximum number of non-overlapping occurrences of the the pattern in the text.

Unlike the classical 2-dimensional pattern matching problem, the maximum 2-dimensional pattern matching problem is NP-complete. We devise polynomial time approximation algorithms and approximation schemes for this problem. We also briefly discuss how the approximation algorithms can be extended to include a number of other variants of the problem.

## Keywords

Intersection Graph Chordal Graph Performance Guarantee Polynomial Time Approximation Algorithm Pattern Match Problem
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## References

- 1.A. Amir, and M. Farach, “Efficient 2-dimensional Approximate Matching of Non-Rectangular Figures,”
*Proc. 2nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)*, 1992, pp. 212–223.Google Scholar - 2.A. Amir, G. Benson, G. Benson and M. Farach. An Alphabet-independent Approach to Two-Dimensional Matching.
*Proc. 24th ACM Symposium on Theory of Computing*, 1992, pp. 59–68. Journal version to appear in*SIAM J. Computing*.Google Scholar - 3.T. P. Baker. A technique for extending rapid exact-match string matching to arrays of more than one dimension.
*SIAM J. Computing*, No. 7, 1978, pp. 533–541.Google Scholar - 4.B.S. Baker. “Approximation Algorithms for NP-complete Problems on Planar Graphs,”
*24th IEEE Symposium on Foundations of Computer Science (FOCS)*, 1983, pp 265–273. (Journal version in*J. ACM*, Vol. 41, No. 1, 1994, pp. 153–180.)Google Scholar - 5.B. Baker, “A Theory of Parameterized Pattern Matching: Algorithms and Applications,”
*Proc. 25th ACM Symposium on Theory of Computing*, 1993, pp. 71–80. Journal version to appear in*Journal of Computer and System Sciences (JCSS)*.Google Scholar - 6.R. S. Bird. “Two-dimensional pattern matching,”
*Information Processing Letters*No. 6, 1977, pp. 168–170.Google Scholar - 7.H. L. Bodlaender, “Dynamic programming on graphs of bounded treewidth,”
*Proc. 15th International Colloquium on Automata Languages and Programming (ICALP)*, LNCS Vol. 317, 1988, pp. 105–118.Google Scholar - 8.“CANDID Project,” Los Alamos National Laboratory, 1993.Google Scholar
- 9.M. Crochemore and W. Rytter.
*Text Algorithms*, Oxford University Press, New York, 1994.Google Scholar - 10.A. Czumaj, Z. Galil, L. Gasieniec, K. Park and W. Plandowski, “Work-Time-Optimal Parallel Algorithms for String Problems,”
*27th ACM Symposium on Theory Of Computing (STOC)*, pp. 713–722, 1995.Google Scholar - 11.D. Eppstein, G.L. Miller, S.H. Teng. “A Deterministic Linear Time Algorithm for Geometric Separators and its Application,”
*9th ACM Symposium on Computational Geometry*, pp 99–108, 1993.Google Scholar - 12.M. Farach and M. Thorup, “String Matching in Lempel-Ziv Compressed Strings,”
*27th ACM Symposium on Theory Of Computing (STOC)*, pp. 703–712, 1995.Google Scholar - 13.T. Feder and D. Greene. “Optimal Algorithms for Approximate Clustering”,
*20th ACM Symposium on Theory Of Computing (STOC)*, pp. 434–444, 1988.Google Scholar - 14.R.J. Fowler, M.S. Paterson and S.L. Tanimoto. “Optimal Packing and Covering in the Plane are NP-Complete,”
*Information Processing Letters*, Vol 12, No.3, June 1981, pp. 133–137.Google Scholar - 15.G.H. Gonnet and R. Baeza Yates,
*Handbook of Algorithms and Data Structures*, Addison-Wesley, Reading, MA, 1991.Google Scholar - 16.F. Gavril, “The intersection graphs of subtrees in trees are exactly the chordal graphs,”
*J. Combin. Theory B*, 16, 1974, pp. 47–56.Google Scholar - 17.F. Gavril, “Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph,”
*SIAM J. Computing*, 1, 1972, pp. 180–187.Google Scholar - 18.M.M. Halldórsson, “Approximating Discrete Collections via Local Improvement,”
*Proc. 6th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)*, 1995, pp. 160–169Google Scholar - 19.F. Harary,
*Graph Theory*, Addison-Wesley, Reading, Massachusetts, 1979.Google Scholar - 20.D.S. Hochbaum and W. Maass, “Approximation Schemes for Covering and Packing Problems in Image Processing and VLSI,”
*J. ACM*, Vol 32, No. 1, 1985, pp 130–136.Google Scholar - 21.D.S. Hochbaum and W. Maass, “Fast Approximation Algorithms for a Nonconvex Covering Problem,”
*Journal of Algorithms*, Vol. 8, 1987, pp. 305–323.Google Scholar - 22.R. Idury and A. Schäffer, “Multiple Matching of Rectangular Figures,”
*Proc. 25th ACM Symposium on Theory of Computing*, 1993, pp. 81–89.Google Scholar - 23.P.M. Kelly, T.M. Cannon, and D.R. Hush, “Query by image example: the CANDID approach,”
*SPIE Vol. 2420 Storage and Retrieval for Image and Video Databases III*, pages 238–248, 1995.Google Scholar - 24.P.M. Kelly and T.M. Cannon, “CANDID: Comparison Algorithm for Navigating Digital Image Databases,”
*In Proc. of the 7th International Working Conference on Scientific and Statistical Database Management, pages 252–258. Charlottesville, VA*, Sept., 1994.Google Scholar - 25.S. R. Kosaraju, “Faster Algorithms for the Construction of Parameterized Suffix Trees,”
*36th IEEE Symposium on Foundations of Computer Science (FOCS)*, 1995, pp 631–637.Google Scholar - 26.R. J. Lipton and R. E. Tarjan, “Applications of a planar separator theorem,”
*SIAM J. Computing*, 9, 1980, pp. 615–627.Google Scholar - 27.G.L. Miller, S.-H. Teng and S.A. Vavasis, “A Unified Geometric Approach to Graph Separators,”
*Proc. of the 32nd IEEE Symposium on Foundations of Computer Science*, 1991, pp. 538–547.Google Scholar - 28.F. P. Preparata and M. I. Shamos,
*Computational Geometry*, Springer Verlag, New York, 1985.Google Scholar - 29.N. Robertson and P. Seymour, “Graph Minors II. Algorithmic aspects of treewidth,”
*J. Algorithms*, No. 7 (1986), 309–322.Google Scholar - 30.D. J. Rose, R. E. Tarjan and G. S. Lueker, “Algorithmic aspects of vertex elimination on graphs,”
*SIAM J. Computing*, 5, 1976, pp. 266–283.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1996