How Many Squares Can a String Contain?

Fraenkel, Aviezri S.; Simpson, Jamie

doi:10.1006/jcta.1997.2843

Cited by 113 publications

(85 citation statements)

References 4 publications

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“…Every substring of a directed labeled tree corresponds to a directed path in the tree. The following fact is a simple generalization of the upper bound of 2n on the number of squares in a string of length n; see [9,12]. A proof of this fact was also implicitly presented in [15].…”

Section: Labeled Treesmentioning

confidence: 90%

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String Powers in Trees

et al. 2017

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In this paper we consider substrings of an unrooted edge-labeled tree, which are defined as the composite labels of simple paths. We study how the number of distinct repetitive substrings depends on their exponent α. An α-power is defined as a string U with an (integral, not necessarily shortest) period |U |/α. For example, squares are 2-powers and cubes are 3-powers. We investigate the asymptotic growth of the maximal number powers α (n) of distinct α-powers occurring as substrings of a tree with n nodes. The maximum number of such powers behaves much unlike in strings. In a previous work (CPM 2012. LNCS, vol 7354. Springer, Berlin, pp 27-40, 2012. It was proved that the number of different squares in a tree is powers 2 (n) = Θ(n 4/3 ).We extend this result and analyze powers of arbitrary rational exponent α ≥ 1. We identify two phase-transition thresholds:This is a full version of a paper presented at CPM 2015. LNCS, vol 9133. Springer, Berlin, pp 284-294, 2015. Compared to the earlier version, we improve our main technical contribution, i.e., the upper bound on the number of cubes in a tree, from O(n log n) to O(n). This lets us obtain a tight asymptotic characterization of the powers function.

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Section: Labeled Treesmentioning

confidence: 90%

“…A proof of this fact was also implicitly presented in [15]. Proof It suffices to note that there are at most two topmost occurrences of different squares starting at each node of the tree; see [9,12].…”

Section: Labeled Treesmentioning

confidence: 96%

String Powers in Trees

et al. 2017

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“…In this paper we present an improved algorithm for constructing MAST(S) with preprocessing time O(n log n), and prove that MAST(S) requires space O(n), which follows from a recent theorem of Fraenkel and Simpson [9].…”

Section: Introductionmentioning

confidence: 91%

“…To bound the size of a minimal augmented suffix tree we need the following theorem of Fraenkel and Simpson, who proved that a string can at most contain a linear number of distinct squares [9]. The proof given below is a slight simplification of [9, Theorem 1].…”

Section: Lemma 5 the Maximum Number Of Non-overlapping Occurrences Ofmentioning

confidence: 99%

Solving the String Statistics Problem in Time O(n log n)

Brodal¹,

Lyngsø²,

Östlin³

et al. 2002

BRICS

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The string statistics problem consists of preprocessing a string of length n such that given a query pattern of length m, the maximum number of non-overlapping occurrences of the query pattern in the string can be reported efficiently. Apostolico and Preparata introduced the minimal augmented suffix tree (MAST) as a data structure for the string statistics problem, and showed how to construct the MAST in time O(n log 2 n) and how it supports queries in time O(m) for constant sized alphabets. A subsequent theorem by Fraenkel and Simpson stating that a string has at most a linear number of distinct squares implies that the MAST requires space O(n). In this paper we improve the construction time for the MAST to O(n log n) by extending the algorithm of Apostolico and Preparata to exploit properties of efficient joining and splitting of search trees together with a refined analysis.

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“…On the other hand, one may try to maximize the number of square factors in a word. The theorem of Fraenkel and Simpson states that a word of length n contains always less than 2n distinct squares [6]. A very short proof for this and an improved upper bound 2n − Θ(log n) was given by Ilie in [9] and [10].…”

Section: Introductionmentioning

confidence: 99%

On the number of squares in partial words

Halava

Harju

Kärki

2010

RAIRO-Theor. Inf. Appl.

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The theorem of Fraenkel and Simpson states that the maximum number of distinct squares that a word w of length n can contain is less than 2n. This is based on the fact that no more than two squares can have their last occurrences starting at the same position. In this paper we show that the maximum number of the last occurrences of squares per position in a partial word containing one hole is 2k, where k is the size of the alphabet. Moreover, we prove that the number of distinct squares in a partial word with one hole and of length n is less than 4n, regardless of the size of the alphabet. For binary partial words, this upper bound can be reduced to 3n.

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How Many Squares Can a String Contain?

Cited by 113 publications

References 4 publications

String Powers in Trees

String Powers in Trees

Solving the String Statistics Problem in Time O(n log n)

On the number of squares in partial words

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