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Computing the quartet distance between evolutionary trees in time O(n log n
 Algorithmica
, 2001
"... Abstract Evolutionary trees describing the relationship for a set of species are central in evolutionarybiology, and quantifying differences between evolutionary trees is therefore an important task. The quartet distance is a distance measure between trees previously proposed by Estabrook,McMorris a ..."
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Cited by 16 (5 self)
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Abstract Evolutionary trees describing the relationship for a set of species are central in evolutionarybiology, and quantifying differences between evolutionary trees is therefore an important task. The quartet distance is a distance measure between trees previously proposed by Estabrook,McMorris and Meacham. The quartet distance between two unrooted evolutionary trees is the number of quartet topology differences between the two trees, where a quartet topologyis the topological subtree induced by four species. In this paper, we present an algorithm for computing the quartet distance between two unrooted evolutionary trees of n species, whereall internal nodes have degree three, in time O(n log n). The previous best algorithm for theproblem uses time O(n2).
Solving the string statistics problem in time O(n log n)
 Proc. 29th International Colloquium on Automata, Languages, and Programming
, 2002
"... 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 nonoverlapping occurrences of the query pattern in the string can be reported efficiently... ..."
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Cited by 9 (0 self)
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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 nonoverlapping occurrences of the query pattern in the string can be reported efficiently...
Finger Search Trees
, 2005
"... One of the most studied problems in computer science is the problem of maintaining a sorted sequence of elements to facilitate efficient searches. The prominent solution to the problem is to organize the sorted sequence as a balanced search tree, enabling insertions, deletions and searches in logari ..."
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Cited by 5 (0 self)
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One of the most studied problems in computer science is the problem of maintaining a sorted sequence of elements to facilitate efficient searches. The prominent solution to the problem is to organize the sorted sequence as a balanced search tree, enabling insertions, deletions and searches in logarithmic time. Many different search trees have been developed and studied intensively in the literature. A discussion of balanced binary search trees can e.g. be found in [4]. This chapter is devoted to finger search trees which are search trees supporting fingers, i.e. pointers, to elements in the search trees and supporting efficient updates and searches in the vicinity of the fingers. If the sorted sequence is a static set of n elements then a simple and space efficient representation is a sorted array. Searches can be performed by binary search using 1+⌊log n⌋ comparisons (we throughout this chapter let log x denote log 2 max{2, x}). A finger search starting at a particular element of the array can be performed by an exponential search by inspecting elements at distance 2 i − 1 from the finger for increasing i followed by a binary search in a range of 2 ⌊log d ⌋ − 1 elements, where d is the rank difference in the sequence between the finger and the search element. In Figure 11.1 is shown an exponential search for the element 42 starting at 5. In the example d = 20. An exponential search requires
A Linear Time Algorithm for Seeds Computation
"... A seed in a word is a relaxed version of a period. We show a linear time algorithm computing a compact representation of all the seeds of a word, in particular, the shortest seed. Thus, we solve an open problem stated in the survey by Smyth (2000) and improve upon a previous over 15year old O(n log ..."
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A seed in a word is a relaxed version of a period. We show a linear time algorithm computing a compact representation of all the seeds of a word, in particular, the shortest seed. Thus, we solve an open problem stated in the survey by Smyth (2000) and improve upon a previous over 15year old O(n log n) algorithm by Iliopoulos, Moore and Park (1996). Our approach is based on combinatorial relations between seeds and a variant of the LZfactorization (used here for the first time in context of seeds). 1
Computing the Quartet Distance between Evolutionary Trees in Time O(n log n)
"... Evolutionary trees describing the relationship for a set of species are central in evolutionary biology, and quantifying differences between evolutionary trees is an important task. The quartet distance is a distance measure between trees previously proposed by Estabrook, McMorris and Meacham. The q ..."
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Evolutionary trees describing the relationship for a set of species are central in evolutionary biology, and quantifying differences between evolutionary trees is an important task. The quartet distance is a distance measure between trees previously proposed by Estabrook, McMorris and Meacham. The quartet distance between two unrooted evolutionary trees is the number of quartet topology differences between the two trees, where a quartet topology is the topological subtree induced by four species. In this paper, we present an algorithm for computing the quartet distance between two unrooted evolutionary trees of n species in time O(n log n). The previous best algorithm for the problem uses time O(n 2).
unknown title
"... 1 at this time we can only prove that the averagecase complexity is O(jxj log jxj), tests carried out by one of us (Holub) strongly suggest that the worstcase complexity should be the same as the averagecase time complexity; at the same time, the possibility exists that the averagecase time compl ..."
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1 at this time we can only prove that the averagecase complexity is O(jxj log jxj), tests carried out by one of us (Holub) strongly suggest that the worstcase complexity should be the same as the averagecase time complexity; at the same time, the possibility exists that the averagecase time complexity of the best DIST algorithms is actually O(jxj), at least for strings on large alphabets.
Efficient Seeds Computation Revisited
 CPM, PALERMO: ITALY
, 2011
"... The notion of the cover is a generalization of a period of a string, and there are linear time algorithms for finding the shortest cover. The seed is a more complicated generalization of periodicity, it is a cover of a superstring of a given string, and the shortest seed problem is of much higher al ..."
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The notion of the cover is a generalization of a period of a string, and there are linear time algorithms for finding the shortest cover. The seed is a more complicated generalization of periodicity, it is a cover of a superstring of a given string, and the shortest seed problem is of much higher algorithmic difficulty. The problem is not well understood, no linear time algorithm is known. In the paper we give linear time algorithms for some of its versions — computing shortest leftseed array, longest leftseed array and checking for seeds of a given length. The algorithm for the last problem is used to compute the seed array of a string (i.e., the shortest seeds for all the prefixes of the string) in O(n 2) time. We describe also a simpler alternative algorithm computing efficiently the shortest seeds. As a byproduct we obtain an O(n log(n/m)) time algorithm checking if the shortest seed has length at least m and finding the corresponding seed. We also correct some important details missing in the previously known shortestseed algorithm (Iliopoulos et al., 1996).