# Permutation Editing and Matching via Embeddings

## Abstract

If the genetic maps of two species are modelled as permutations of (homologous) genes, the number of chromosomal rearrangements in the form of deletions, block moves, inversions etc. to transform one such permutation to another can be used as a measure of their evolutionary distance. Motivated by such scenarios, we study problems of computing distances between permutations as well as matching permutations in sequences, and finding most similar permutation from a collection (“nearest neighbor”).

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We present the first known approximately distance preserving embeddings of these permutation distances into well-known spaces.

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Using these embeddings, we obtain several results, including the first known efficient solution for approximately solving nearest neighbor problems with permutations and the first known algorithms for finding permutation distances in the “data stream” model.

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We consider a novel class of problems called

*permutation matching*problems which are similar to string matching problems, except that the pattern is a permutation (rather than a string) and present linear or near-linear time algorithms for approximately solving permutation matching problems; in contrast, the corresponding string problems take significantly longer.

## Keywords

Edit Distance Neighbor Problem Identity Permutation Longe Common Subsequence Signed Permutation## Preview

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## References

- 1.V. Bafna and P. A. Pevzner. Genome rearrangements and sorting by reversals. In
*Proceedings of the 34th Annual Symposium on Foundations of Comptuer Science*, pages 148–157, Palo Alto, CA, 1993. IEEE Computer Society Press.Google Scholar - 2.Vineet Bafna and Pavel A. Pevzner. Sorting by transpositions.
*SIAM Journal on Discrete Mathematics*, 11(2):224–240, May 1998.MATHCrossRefMathSciNetGoogle Scholar - 3.A. Caprara. Sorting by reversals is difficult. In
*Proceedings of the First International Conference on Computational Molecular Biology*, pages 75–83, 1997.Google Scholar - 4.David A. Christie. A 3/2-approximation algorithm for sorting by reversals. In
*Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms*, pages 244–252, San Francisco, California, 25-27 January 1998.Google Scholar - 5.J. Feigenbaum, S. Kannan, M. Strauss, and M. Viswanathan. An approximate L1-difference algorithm for massive data streams. In
*IEEE Symposium on Foundations of Computer Science (FOCS)*, pages 501–511, 1999.Google Scholar - 6.Vincent Ferretti, Joseph H. Nadeau, and David Sankoff. Original synteny. In
*Combinatorial Pattern Matching, 7th Annual Symposium*, volume 1075 of*Lecture Notes in Computer Science*, pages 159–167. Springer, 1996.Google Scholar - 7.Leslie Ann Goldberg, Paul W. Goldberg, Mike Paterson, Pavel Pevzner, Süleyman Cenk Sahinalp, and Elizabeth Sweedyk. The complexity of gene placement. In
*Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms*, pages 386–395, N.Y., January 17-19 1999. ACM-SIAM.Google Scholar - 8.Qian-Ping Gu, Shietung Peng, and Hal Sudborough. A 2-approximation algorithm for genome rearrangements by reversals and transpositions.
*Theoretical Computer Science*, 210(2):327–339, 17 January 1999.MATHCrossRefMathSciNetGoogle Scholar - 9.Piotr Indyk and Rajeev Motwani. Approximate nearest neighbors: Towards removing the curse of dimensionality. In
*Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC-98)*, pages 604–613, 1998.Google Scholar - 10.Howard Karloff. Fast algorithms for approximately counting mismatches.
*Information Processing Letters*, 48(2):53–60, November 1993.MATHCrossRefMathSciNetGoogle Scholar - 11.J. Kececioglu and D. Sankoff. Exact and approximation algorithms for sorting by reversals, with application to genome rearrangement.
*Algorithmica*, 13(1/2):180–210, January 1995.MATHCrossRefMathSciNetGoogle Scholar - 12.E. Kushilevitz, R. Ostrovsky, and Y. Rabani. Effiient search for approximate nearest neighbor in high dimensional spaces. In
*Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC-98)*, pages 614–623, 1998.Google Scholar - 13.J. H. Nadeau and B. A. Taylor. Lengths of chromosome segments conserved since divergence of man and mouse.
*Proc. Nat’l Acad. Sci. USA*, 81:814–818, 1984.CrossRefGoogle Scholar - 14.D. Sankoff and J. Nadeau. Conserved synteny as a measure of genomic distance.
*DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science*, 71, 1996.Google Scholar