Efficient Algorithms for the Sum Selection Problem and K Maximum Sums Problem

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

doi:10.1007/11940128_47

Cited by 5 publications

(3 citation statements)

References 21 publications

(18 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Bengtsson and Chen [3] first studied the Sum Selection problem and gave an O(n log 2 n)-time algorithm for it. Recently, Lin and Lee provided an O(n log n)-time algorithm [13] for the Sum Selection problem and an expected O(n log(u − l + 1))time randomized algorithm [14] for the Length-Constrained Sum Selection problem. In the following, we show how to solve the Length-Constrained Sum Selection problem in worst-case O(n log(u − l + 1)) time.…”

Section: Appendix A: Applications To the Length-constrained Sum Selec...mentioning

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%

See 1 more Smart Citation

Minkowski Sum Selection and Finding

Luo

Liu

Chen

et al. 2011

Int. J. Comput. Geom. Appl.

View full text Add to dashboard Cite

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.

show abstract

Section: Appendix A: Applications To the Length-constrained Sum Selec...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Minkowski Sum Selection and Finding

Luo

Liu

Chen

et al. 2011

Int. J. Comput. Geom. Appl.

View full text Add to dashboard Cite

show abstract

“…Since then, there have been many variants proposed. The k MaximumSum Segments problem [3][4][5]10,12,18,19] is to locate the k segments whose sums are the k largest among all possible sums, and is solvable in O (n + k) time [10,21]. The Range Maximum-Sum Segment Query (RMSQ) problem is to preprocess the input sequence such that any range maximum-sum segment query can be answered quickly, where a range maximum-sum segment query specifies two intervals [i, j] and [k, l] and the goal is to find a segment A(x, y) with maximum sum subject to i x j and k y .…”

Section: Introductionmentioning

confidence: 99%