Randomized algorithm for the sum selection problem

Lin, Tien-Ching; Lee, D. T.

doi:10.1016/j.tcs.2007.02.027

Cited by 12 publications

(6 citation statements)

References 25 publications

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“…-The Minkowski Sum Finding problem with any fixed number of constraints can be solved in optimal O(n log n) time if the objective function f (x, y) is linear or of the form by ax . As a byproduct, we obtain improved algorithms for the Length-Constrained Sum Selection problem [14] and the Density Finding problem [12]. Recently, Lin and Lee [14] proposed an expected O(n log(u − l + 1))-time randomized algorithm for the Length-Constrained Sum Selection problem, where n is the size of the input instance and l, u ∈ N are two given parameters with 1 ≤ l < u ≤ n. In this paper, we obtain a worst-case O(n log(u − l + 1))-time deterministic algorithm for the Length-Constrained Sum Selection problem (see Appendix A).…”

Section: Introductionmentioning

confidence: 99%

“…The Minkowski Sum Optimization problem is equivalent to the Minkowski Sum Selection problem with k = 1. A variety of selection problems, including the Sum Selection problem [3,13], the Length-Constrained Sum Selection problem [14], and the Slope Selection problem [7,18], are linear-time reducible to the Minkowski Sum Selection problem with a linear objective function or an objective function of the form f (x, y) = by ax . It is desirable that relevant selection problems from diverse fields are integrated into a single one, so we don't have to consider them separately.…”

Section: Introductionmentioning

confidence: 99%

“…As a byproduct, we obtain improved algorithms for the Length-Constrained Sum Selection problem [14] and the Density Finding problem [12]. Recently, Lin and Lee [14] proposed an expected O(n log(u − l + 1))-time randomized algorithm for the Length-Constrained Sum Selection problem, where n is the size of the input instance and l, u ∈ N are two given parameters with 1 ≤ l < u ≤ n. In this paper, we obtain a worst-case O(n log(u − l + 1))-time deterministic algorithm for the Length-Constrained Sum Selection problem (see Appendix A). Lee, Lin, and Lu [12] showed the Density Finding problem has a lower bound of Ω(n log n) and proposed an O(n log 2 m)-time algorithm for it, where n is the size of the input instance and m is a parameter whose value may be as large as n. In this paper, we give an optimal O(n log n)-time algorithm for the Density Finding problem (see Appendix B).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Minkowski Sum Selection and Finding

Luo

Liu

Chen

et al. 2011

Int. J. Comput. Geom. Appl.

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Let P, Q ⊆ R 2 be two n-point multisets and Ar ≥ b be a set of λ inequalities on x and y, where A ∈ R λ×2 , r = [ x y ], and b ∈ R λ . Define the constrained Minkowski sum (P ⊕ Q) Ar≥b as the multiset {(p + q)|p ∈ P, q ∈ Q, A(p + q) ≥ b}. Given P , Q, Ar ≥ b, an objective function f : R 2 → R, and a positive integer k, the Minkowski Sum Selection problem is to find the k th largest objective value among all objective values of points in (P ⊕Q) Ar≥b . Given P , Q, Ar ≥ b, an objective function f : R 2 → R, and a real number δ, the Minkowski Sum Finding problem is to find a point (x * , y * ) in (P ⊕ Q) Ar≥b such that |f (x * , y * ) − δ| is minimized. For the Minkowski Sum Selection problem with linear objective functions, we obtain the following results: (1) optimal O(n log n) time algorithms for λ = 1; (2) O(n log 2 n) time deterministic algorithms and expected O(n log n) time randomized algorithms for any fixed λ > 1. For the Minkowski Sum Finding problem with linear objective functions or objective functions of the form f (x, y) = by ax , we construct optimal O(n log n) time algorithms for any fixed λ ≥ 1. As a byproduct, we obtain improved algorithms for the Length-Constrained Sum Selection problem and the Density Finding problem.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Minkowski Sum Selection and Finding

Luo

Liu

Chen

et al. 2011

Int. J. Comput. Geom. Appl.

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“…Several other generalizations of this classical problem have been investigated as well. For example when one is interested in not only the largest, but also k continuous largest subsequences (for some parameter k ) [2,3,5,14]. Other generalizations arising from bioinformatics is to look for an interesting segment (or segments) with constrained length [8,9,15].…”

Section: Introductionmentioning

confidence: 99%

Computing Maximum-Scoring Segments in Almost Linear Time

Bengtsson

Chen

2006

Lecture Notes in Computer Science

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Given a sequence, the problem studied in this paper is to find a set of k disjoint continuous subsequences such that the total sum of all elements in the set is maximized. This problem arises naturally in the analysis of DNA sequences. The previous best known algorithm requires Θ(n log n) time in the worst case. For a given sequence of length n, we present an almost linear-time algorithm for this problem. Our algorithm uses a disjoint-set data structure and requires O(nα(n, n)) time in the worst case, where α(n, n) is the inverse Ackermann function.

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“…In this paper, we propose an O(n + k log(min{n, k}))-time algorithm which is superior to all the previous methods when k is o(n log n). It should be noted that recently Lin and Lee [14] gave a randomized O(n log n + k)-time algorithm, which is a better choice when k is sufficiently large.…”

Section: Introductionmentioning

confidence: 99%

Improved Algorithms for the k Maximum-Sums Problems

Cheng¹,

Chen²,

Tien³

et al. 2005

Algorithms and Computation

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Given a sequence of n real numbers and an integer k, 1 k 1 2 n(n − 1), the k maximum-sum segments problem is to locate the k segments whose sums are the k largest among all possible segment sums. Recently, Bengtsson and Chen gave an O(min{k + n log 2 n, n √ k})-time algorithm for this problem. Bae and Takaoka later proposed a more efficient algorithm for small k. In this paper, we propose an O(n + k log(min{n, k}))-time algorithm for the same problem, which is superior to both of them when k is o(n log n). We also give the first optimal algorithm for delivering the k maximum-sum segments in non-decreasing order if k n. Then we develop an O(n 2d−1 +k log min{n, k})-time algorithm for the d-dimensional version of the problem, where d > 1 and each dimension, without loss of generality, is of the same size n. This improves the best previously known O(n 2d−1 C)-time algorithm, also by Bengtsson and Chen, where C = min{k + n log 2 n, n √ k}. It should be pointed out that, given a two-dimensional array of size m×n, our algorithm for finding the k maximum-sum subarrays is the first one achieving cubic time provided that k is O(m 2 n/ log n).

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Randomized algorithm for the sum selection problem

Cited by 12 publications

References 25 publications

Minkowski Sum Selection and Finding

Minkowski Sum Selection and Finding

Computing Maximum-Scoring Segments in Almost Linear Time

Improved Algorithms for the k Maximum-Sums Problems

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