Abstract
We present an approach to the problem of maximum number of distinct squares in a string which underlines the importance of considering as key variables both the length n and n − d where d is the size of the alphabet. We conjecture that a string of length n and containing d distinct symbols has no more than n − d distinct squares, show the critical role played by strings satisfying n = 2d, and present some properties satisfied by strings of length bounded by a constant times the size of the alphabet.
Supported in part by grants from the Natural Sciences and Engineering Research Council of Canada for the first 2 authors, and by the Canada Research Chair Programme and Mathematics of Information Technology and Complex Systems grants for the first author, and by Queen Elizabeth II Graduate Scholarship in Science and Technology for the third author.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Baker, A., Deza, A., Franek, F.: On the structure of relatively short run-maximal strings, AdvOL Technical Report 2011/02, Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada
Deza, A., Franek, F.: A d-step analogue for runs on strings, AdvOL Technical Report 2010/02, Department of Computing and Software, McMaster University, Hamilton, Ontario, Canada
Fraenkel, A.S., Simpson, J.: How Many Squares Can a String Contain? Journal of Combinatorial Theory Series A 82(1), 112–120 (1998)
Ilie, L.: A simple proof that a word of length n has at most 2n distinct squares. Journal of Combinatorial Theory Series A 112(1), 163–164 (2005)
Ilie, L.: A note on the number of squares in a word. Theoretical Computer Science 380(3), 373–376 (2007)
Matschke, B., Santos, F., Weibel, C.: The width of 5-prismatoids and smaller non-Hirsch polytopes (2011), http://www.cs.dartmouth.edu/~weibel/hirsch.php
Santos, F.: A counterexample to the Hirsch conjecture, arXiv:1006.2814v1 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Deza, A., Franek, F., Jiang, M. (2011). A d-Step Approach for Distinct Squares in Strings. In: Giancarlo, R., Manzini, G. (eds) Combinatorial Pattern Matching. CPM 2011. Lecture Notes in Computer Science, vol 6661. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21458-5_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-21458-5_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21457-8
Online ISBN: 978-3-642-21458-5
eBook Packages: Computer ScienceComputer Science (R0)