Optimal algorithms for the average-constrained maximum-sum segment problem

Cheng, Chih-Huai; Liu, Hsiao-Fei; Chao, Kun-Mao

doi:10.1016/j.ipl.2008.09.024

Cited by 4 publications

(3 citation statements)

References 27 publications

(29 reference statements)

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“…With motivations from Bioinformatics, a further direction of research concerns problems where one or more measures of subsequences are constrained. For example, algorithms have been devised that compute the greatest sum among all subsequences subject to a length lower bound [21], a length upper bound [9,22], both length bounds [9] or average bounds [23] (the average of a sequence A being sum(A)/|A|). Optimal algorithms have also been devised for the length constrained versions of the multiple maximal sum subsequences [24].…”

Section: Related Workmentioning

confidence: 99%

Linear time computation of the maximal linear and circular sums of multiple independent insertions into a sequence

Corrêa

Farias

2017

Theoretical Computer Science

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The maximal sum of a sequence A of n real numbers is the greatest sum of all elements of any linearly contiguous and possibly empty subsequence of A. It can be computed in O(n) time by means of Kadane's algorithm. Letting A (x→p) denote the sequence which results from inserting a real number x just after element A[p − 1], we show how the maximal sum of A (x→p) can be computed in O(1) worst-case time for any given x and p, provided that an O(n) time preprocessing step has already been executed on A. In particular, this implies that, given m pairs (x0, p0), . . . , (xm−1, pm−1), we can compute the maximal sums of sequences A (x 0 →p 0 ) , . . . , A (x m−1 →p m−1 ) optimally in O(n + m) time, improving on the straightforward and suboptimal strategy of applying Kadane's algorithm to each sequence A (x i →p i ) , which takes a total of Θ(nm) time. We also show that the same time bound is attainable when circular subsequences of A (x→p) are taken into account. Our algorithms are easy to implement in practice, and they were motivated by a buffer minimization problem on wireless mesh networks.

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Section: Related Workmentioning

confidence: 99%

Linear time computation of the maximal linear and circular sums of multiple independent insertions into a sequence

Corrêa

Farias

2017

Theoretical Computer Science

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“…, (h n , s n )) of number pairs, where s i > 0 for all i, and a confidence lower bound L c , find a confident interval I = [i, j] maximizing the hit-support hit(i, j). Bernholt et al [5]'s results imply an O(n log n)-time algorithm for this problem, and recently, Cheng et al [10] obtained an O(n)-time algorithm.…”

Section: Introductionmentioning

confidence: 97%

Algorithms for Locating Constrained Optimal Intervals

Liu,

Chen,

Chao

2008

Preprint

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In this work, we obtain the following new results.-Given a sequence D = ((h 1 , s 1 ), (h 2 , s 2 ) . . . , (h n , s n )) of number pairs, where s i > 0 for all i, and a number L h , we propose an O(n)-time algorithm for finding an index interval [i, j] that maximizes-Given a sequence D = ((h 1 , s 1 ), (h 2 , s 2 ) . . . , (h n , s n )) of number pairs, where s i = 1 for all i, and an integer L s with 1 ≤ L s ≤ n, we propose an O(n T (L 1/2 s ) L 1/2 s )-time algorithm for finding an index interval [i, j] that maximizes P j k=i h k q P j k=i s k subject to j k=i s k ≥ L s , where T (n ′ ) is the time required to solve the all-pairs shortest paths problem on a graph of n ′ nodes. By the latest result of Chan [8], T (n ′ ) = O(n ′3 (log log n ′ ) 3 (log n ′ ) 2 ), so our algorithm runs in subquadratic time O(nL s (log log Ls) 3 (log Ls) 2 ).

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“…A restricted version of Problem DCHP is studied in [4], which proposes an optimal O(n log n) time algorithm for the case where T is a path, i.e., a sequence. All of [3], [4] and [9] are motivated by the observation that constraining density with upper and lower bounds is necessary to locate good-quality subsequences of DNA sequences, which are further analyzed to be confirmed as CpG islands.…”

Section: Introductionmentioning

confidence: 99%

Optimal Algorithms for Finding Density-Constrained Longest and Heaviest Paths in a Tree

Kim

2010

IEICE Trans. Inf. & Syst.

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SUMMARYLet T be a tree with n nodes, in which each edge is associated with a length and a weight. The density-constrained longest (heaviest) path problem is to find a path of T with maximum path length (weight) whose path density is bounded by an upper bound and a lower bound. The path density is the path weight divided by the path length. We show that both problems can be solved in optimal O(n log n) time.

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Optimal algorithms for the average-constrained maximum-sum segment problem

Cited by 4 publications

References 27 publications

Linear time computation of the maximal linear and circular sums of multiple independent insertions into a sequence

Linear time computation of the maximal linear and circular sums of multiple independent insertions into a sequence

Algorithms for Locating Constrained Optimal Intervals

Optimal Algorithms for Finding Density-Constrained Longest and Heaviest Paths in a Tree

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