International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 74-85 | Cite as

Computing Minimum Length Representations of Sets of Words of Uniform Length

  • Francine Blanchet-SadriEmail author
  • Andrew Lohr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)


Motivated by text compression, the problem of representing sets of words of uniform length by partial words, i.e., sequences that may have some wildcard characters or holes, was recently considered and shown to be in \(\mathcal {P}\). Polynomial-time algorithms that construct representations were described using graph theoretical approaches. As more holes are allowed, representations shrink, and if representation is given, the set can be reconstructed. We further study this problem by determining, for a binary alphabet, the largest possible value of the size of a set of partial words that is important in deciding the representability of a given set S of words of uniform length. This largest value, surprisingly, is \(\varSigma _{i=0}^{|S|-1} 2^{\chi (i)}\) where \(\chi (i)\) is the number of ones in the binary representation of i, a well-studied digital sum, and it is achieved when the cardinality of S is a power of two. We show that circular representability is in \(\mathcal {P}\) and that unlike non-circular representability, it is easy to decide. We also consider the problem of computing minimum length representation (circular) total words, those without holes, and reduce it to a cost/flow network problem.


Graph Isomorphism Hole Character Representation Word Uniform Length Total Word 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of North CarolinaGreensboroUSA
  2. 2.Department of MathematicsRutgers UniversityPiscatawayUSA

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