Finding Short Synchronizing Words for Prefix Codes

Authors Andrew Ryzhikov, Marek Szykula

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Author Details

Andrew Ryzhikov
  • Université Paris-Est, LIGM, Marne-la-Vallée, France
Marek Szykula
  • Institute of Computer Science, University of Wrocław, Wrocław, Poland

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Andrew Ryzhikov and Marek Szykula. Finding Short Synchronizing Words for Prefix Codes. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We study the problems of finding a shortest synchronizing word and its length for a given prefix code. This is done in two different settings: when the code is defined by an arbitrary decoder recognizing its star and when the code is defined by its literal decoder (whose size is polynomially equivalent to the total length of all words in the code). For the first case for every epsilon > 0 we prove n^(1 - epsilon)-inapproximability for recognizable binary maximal prefix codes, Theta(log n)-inapproximability for finite binary maximal prefix codes and n^(1/2 - epsilon)-inapproximability for finite binary prefix codes. By c-inapproximability here we mean the non-existence of a c-approximation polynomial time algorithm under the assumption P != NP, and by n the number of states of the decoder in the input. For the second case, we propose approximation and exact algorithms and conjecture that for finite maximal prefix codes the problem can be solved in polynomial time. We also study the related problems of finding a shortest mortal and a shortest avoiding word.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • synchronizing word
  • mortal word
  • avoiding word
  • Huffman decoder
  • inapproximability


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