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Binary jumbled pattern matching on trees and treelike structures
 IN PROC. OF THE 21ST ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHM (ESA 2013
, 2013
"... Binary jumbled pattern matching asks to preprocess a binary string S in order to answer queries (i, j) which ask for a substring of S that is of length i and has exactly j 1bits. This problem naturally generalizes to vertexlabeled trees and graphs by replacing “substring ” with “connected subgr ..."
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Binary jumbled pattern matching asks to preprocess a binary string S in order to answer queries (i, j) which ask for a substring of S that is of length i and has exactly j 1bits. This problem naturally generalizes to vertexlabeled trees and graphs by replacing “substring ” with “connected subgraph”. In this paper, we give an O(n2 / log2 n)time solution for trees, matching the currently best bound for (the simpler problem of) strings. We also give an O(g2/3n4/3/(logn)4/3)time solution for strings that are compressed by a grammar of size g. This solution improves the known bounds when the string is compressible under many popular compression schemes. Finally, we prove that the problem is fixedparameter tractable with respect to the treewidth w of the graph, even for a constant number of different vertexlabels, thus improving the previous best nO(w) algorithm [ICALP’07].
Binary jumbled string matching for highly runlength compressible texts
 Inf. Process. Lett
"... The Binary Jumbled String Matching problem is defined as: Given a string s over {a, b} of length n and a query (x, y), with x, y nonnegative integers, decide whether s has a substring t with exactly x a’s and y b’s. Previous solutions created an index of size O(n) in a preprocessing step, which wa ..."
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The Binary Jumbled String Matching problem is defined as: Given a string s over {a, b} of length n and a query (x, y), with x, y nonnegative integers, decide whether s has a substring t with exactly x a’s and y b’s. Previous solutions created an index of size O(n) in a preprocessing step, which was then used to answer queries in constant time. The fastest algorithms for construction of this index have running time O(n2 / log n) [Burcsi et al., FUN 2010; Moosa and Rahman, IPL 2010], or O(n2 / log2 n) in the wordRAM model [Moosa and Rahman, JDA 2012]. We propose an index constructed directly from the runlength encoding of s. The construction time of our index is O(n+ ρ2 log ρ), where O(n) is the time for computing the runlength encoding of s and ρ is the length of this encoding—this is no worse than previous solutions if ρ = O(n / log n) and better if ρ = o(n / log n). Our index L can be queried in O(log ρ) time. While L  = O(min(n, ρ2)) in the worst case, preliminary investigations have indicated that L  may often be close to ρ. Furthermore, the algorithm for constructing the index is conceptually simple and easy to implement. In an attempt to shed light on the structure and size of our index, we characterize it in terms of the prefix normal forms of s introduced in [Fici and Lipták, DLT 2011].
On Hardness of Jumbled Indexing
"... 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 ..."
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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
Indexed Geometric Jumbled Pattern Matching
"... Abstract. We consider how to preprocess n colored points in the plane such that later, given a multiset of colors, we can quickly find an axisaligned rectangle containing a subset of the points with exactly those colors, if one exists. We first give an index that uses o(n4) space and o(n) query tim ..."
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Abstract. We consider how to preprocess n colored points in the plane such that later, given a multiset of colors, we can quickly find an axisaligned rectangle containing a subset of the points with exactly those colors, if one exists. We first give an index that uses o(n4) space and o(n) query time when there are O(1) distinct colors. We then restrict our attention to the case in which there are only two distinct colors. We give an index that uses O(n) bits and O(1) query time to detect whether there exists a matching rectangle. Finally, we give a O(n)space index that returns a matching rectangle, if one exists, in O(lg2 n / lg lgn) time. 1
On Combinatorial Generation of Prefix Normal Words
"... Abstract. A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. This class of words is important in the context of binary jumbled string matching. In this paper we present an efficient algorithm for exhaustively listing the prefix n ..."
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Abstract. A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. This class of words is important in the context of binary jumbled string matching. In this paper we present an efficient algorithm for exhaustively listing the prefix normal words with a fixed length. The algorithm is based on the fact that the language of prefix normal words is a bubble language, a class of binary languages with the property that, for any word w in the language, exchanging the first occurrence of 01 by 10 in w results in another word in the language. We prove that each prefix normal word is produced in O(n) amortized time, and conjecture, based on experimental evidence, that the true amortized running time is O(polylog(n)). 1
Normal, Abby Normal, Prefix Normal
, 2014
"... A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. This class of words is important in the context of binary jumbled pattern matching. In this paper we present results about the number pnw(n) of prefix normal words of length n, ..."
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A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. This class of words is important in the context of binary jumbled pattern matching. In this paper we present results about the number pnw(n) of prefix normal words of length n, showing that pnw(n) = Ω
On lower bounds for the Maximum Consecutive Subsums Problem and the (min,+)convolution
"... AbstractGiven a sequence of n numbers, the MAXIMUM CONSECUTIVE SUBSUMS PROBLEM (MCSP) asks for the maximum consecutive sum of lengths for each = 1, . . . , n. No algorithm is known for this problem which is significantly better than the naive quadratic solution. Nor a super linear lower bound is k ..."
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AbstractGiven a sequence of n numbers, the MAXIMUM CONSECUTIVE SUBSUMS PROBLEM (MCSP) asks for the maximum consecutive sum of lengths for each = 1, . . . , n. No algorithm is known for this problem which is significantly better than the naive quadratic solution. Nor a super linear lower bound is known. The best known bound for the MCSP is based on the the computation of the (min, +)convolution, another problem for which neither an O(n 2− ) upper bound is known nor a super linear lower bound. We show that the two problems are in fact computationally equivalent by providing linear reductions between them. Then, we concentrate on the problem of finding super linear lower bounds and provide empirical evidence for our conjecture that the solution of both problems requires Ω(n log n) time in the decision tree model.