CPM 2015: Combinatorial Pattern Matching pp 284-294

# String Powers in Trees

• Tomasz Kociumaka
• 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).

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© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Tomasz Kociumaka
• 1