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String Powers in Trees

  • Tomasz Kociumaka
  • Jakub Radoszewski
  • Wojciech Rytter
  • Tomasz Waleń
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9133)

Abstract

We investigate the asymptotic growth of the maximal number \(\mathsf{powers }_{\alpha }(n)\) of different \(\alpha \)-powers (strings \(w\) with a period \(|w|/\alpha \)) in an edge-labeled unrooted tree of size \(n\). The number of different powers in trees behaves much unlike in strings. In a previous work (CPM, 2012) it was proved that the number of different squares in a tree is \(\mathsf{powers }_2(n) = \varTheta (n^{4/3})\). We extend this result and analyze other powers. We show that there are phase-transition thresholds:
  1. 1.

    \(\mathsf{powers }_{\alpha }(n)\;=\;\varTheta (n^2)\) for \(\alpha <2\);

     
  2. 2.

    \(\mathsf{powers }_{\alpha }(n)\;=\; \varTheta (n^{4/3})\) for \(2\le \alpha <3\);

     
  3. 3.

    \(\mathsf{powers }_{\alpha }(n)\;=\; {\mathcal {O}}(n \log n)\) for \(3\le \alpha <4\);

     
  4. 4.

    \(\mathsf{powers }_{\alpha }(n)\;=\; \varTheta (n)\) for \(4\le \alpha \).

     

The difficult case is the third point, which follows from the fact that the number of different cubes in a rooted tree is linear (in this case, only cubes passing through the root are counted).

Keywords

Rooted Tree Directed Tree Simple Path Asymptotic Bound Label Tree 
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

  • Tomasz Kociumaka
    • 1
  • Jakub Radoszewski
    • 1
  • Wojciech Rytter
    • 1
  • Tomasz Waleń
    • 1
  1. 1.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland

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