Scaled and permuted string matching

Butman, Ayelet; Eres, Revital; Landau, Gad M.

doi:10.1016/j.ipl.2004.09.002

Cited by 28 publications

(19 citation statements)

References 12 publications

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“…Nevertheless, this query time is o(n) only for m = o(n ε ). Jumbled pattern matching in a run-length encoded text over arbitrary alphabet was considered in [9].…”

Section: Indexes For Larger Alphabetsmentioning

confidence: 99%

Efficient Indexes for Jumbled Pattern Matching with Constant-Sized Alphabet

2016

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We introduce efficient indexes for a problem in non-standard stringology: jumbled pattern matching. An index is a data structure constructed for a text of length n over an alphabet of size σ that can answer queries asking if the text contains a fragment which is jumbled (Abelian) equivalent to a pattern, specified by its so-called Parikh vector. We denote the length of the pattern by m. Moosa and Rahman (J Discrete Algorithms 10:5-9, 2012) gave an index for the case of binary alphabets with O n 2 (log n) 2 -time construction in the word-RAM model. Several earlier papers stated as an open problem the existence of an efficient solution for larger alphabets. In this paper we develop an index for any constant-sized alphabet. The construction involves a trade-off parameter, which in particular lets us achieve the following complexities: O(n 2−δ ) space and O(m (2σ −1)δ ) query time for any 0 < δ < 1, or O n 2 (log log n) 2 log n space and polylogarithmic, o(log 2σ −1 m), query time. The construction time in both cases is subquadratic: O n 2 (log log n) 2 log n in the word-RAM model (using bit-parallelism). Our construction algorithms are randomized (Las Vegas, running time w.h.p.), which is due to the usage of perfect hashing. On the other hand, all queries are answered deterministically. A preliminary version of this work appeared at ESA 2013 (Kociumaka et al. in Algorithms, ESA 2013. LNCS, vol 8125. Springer, Berlin, pp. 625-636, 2013 time construction of the index with O(n 2−δ ) space and O(m (2σ −1)δ ) query time, which was not present in the preliminary version. We also extend the index so that the position of the leftmost occurrence of the query pattern is provided at no additional cost in the complexity; this required rather nontrivial changes in the construction algorithm.

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“…Nevertheless, this query time is o(n) only for m = o(n ε ). Jumbled pattern matching in a run-length encoded text over arbitrary alphabet was considered in [9].…”

Section: Indexes For Larger Alphabetsmentioning

confidence: 99%

Efficient Indexes for Jumbled Pattern Matching with Constant-Sized Alphabet

2016

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“…In contrast, the exact pattern matching problem can only be solved in linear time via more complex techniques (e.g., see [29,11]). Jumbled pattern matching has also been studied along with other metrics (e.g., see [14,15,1]).…”

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confidence: 99%

On Hardness of Jumbled Indexing

Amir

Chan

Lewenstein

et al. 2014

Automata, Languages, and Programming

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Abstract. Jumbled indexing is the problem of indexing a text T for queries that ask whether there is a substring of T matching a pattern represented as a Parikh vector, i.e., the vector of frequency counts for each character. Jumbled indexing has garnered a lot of interest in the last four years; for a partial list see [2,6,13,16,17,20,22,24,26,30,35,36]. There is a naive algorithm that preprocesses all answers in O(n 2 |Σ|) time allowing quick queries afterwards, and there is another naive algorithm that requires no preprocessing but has O(n log |Σ|) query time. Despite a tremendous amount of effort there has been little improvement over these running times. In this paper we provide good reason for this. We show that, under a 3SUM-hardness assumption, jumbled indexing for alphabets of size ω(1) requires Ω(n 2−ǫ ) preprocessing time or Ω(n 1−δ ) query time for any ǫ, δ > 0. In fact, under a stronger 3SUM-hardness assumption, for any constant alphabet size r ≥ 3 there exist describable fixed constant ǫr and δr such that jumbled indexing requires Ω(n 2−ǫr ) preprocessing time or Ω(n 1−δr ) query time.

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“…Moreover, for any binary word w, there is a prefix normal word w of the same length such that for any length k, the maximal number of 1's in a factor of length k coincide for w and w . Such w is called the prefix normal form of w. The motivation behind the study of prefix normal words and prefix normal forms comes from a variation of the binary jumbled pattern matching (binary JPM), which asks whether, for a text of length n over a binary alphabet and two numbers x and y, there exists a substring with x 1's and y 0's [4]. While this problem has a simple efficient solution, its indexing variation, called indexing jumbled pattern matching (IJPM) is not trivial.…”

Section: Introductionmentioning

confidence: 99%

Leaf realization problem, caterpillar graphs and prefix normal words

Massé¹,

Carufel²,

Goupil³

et al. 2018

Theoretical Computer Science

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Given a simple graph G with n vertices and a natural number i ≤ n, let L G (i) be the maximum number of leaves that can be realized by an induced subtree T of G with i vertices. We introduce a problem that we call the leaf realization problem, which consists in deciding whether, for a given sequence of n+1 natural numbers ( 0 , 1 , . . . , n ), there exists a simple graph G with n vertices such that i = L G (i) for i = 0, 1, . . . , n. We present basic observations on the structure of these sequences for general graphs and trees. In the particular case where G is a caterpillar graph, we exhibit a bijection between the set of the discrete derivatives of the form (∆L G (i)) 1≤i≤n−3 and the set of prefix normal words.

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Scaled and permuted string matching

Cited by 28 publications

References 12 publications

Efficient Indexes for Jumbled Pattern Matching with Constant-Sized Alphabet

Efficient Indexes for Jumbled Pattern Matching with Constant-Sized Alphabet

On Hardness of Jumbled Indexing

Leaf realization problem, caterpillar graphs and prefix normal words

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