Efficient Algorithms for the Maximum Subarray Problem by Distance Matrix Multiplication

Takaoka, Tadao

doi:10.1016/s1571-0661(04)00313-5

Cited by 57 publications

(52 citation statements)

References 7 publications

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“…Tamaki and Tokuyama present an Oðn 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi log log n=log n p Þ time, which is subcubic, for the two dimensional maximum subarray problem based on divideand-conquer approach [4] utilizing Takaoka's DMM algorithm [3]. The simplified solution with the same complexity is given in [5]. We modify this algorithm to compute the K-maximum subarray problem for two dimensions.…”

Section: Divide-and-conquermentioning

confidence: 99%

“…Subcubic time algorithm for the maximum subarray problem We review the divide-and-conquer approach given in [5]. Let a two-dimensional array a[1..m, 1..n] of real numbers be given as input data.…”

Section: Generalization Of Dmmmentioning

confidence: 99%

“…The subcubic time solution based on Takaoka's subcubic distance matrix multiplication (DMM) algorithm [3] is given by Tamaki and Tokuyama [4], which is further simplified by Takaoka [5]. In the context of parallel computations, time and cost optimal PRAM and mesh algorithms for the one-dimensional case are described in [6].…”

Section: Introductionmentioning

confidence: 99%

“…For the two-dimensional case, EREW PRAM solutions achieving O(log n) time with O(n 3 /log n) processors are given in [7,8] and comparable result on interconnection networks is given in [9]. The EREW PRAM version of the subcubic algorithm in [4,5] is given in [10], which also features a VLSI algorithm based on the technique introduced in Bentley's paper. This VLSI algorithm for the maximum subarray problem achieves T ¼ m + n À 2 steps, which is O(n) time using O(n 2 ) sized hardware circuit.…”

Section: Introductionmentioning

confidence: 99%

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Improved Algorithms for the K-Maximum Subarray Problem

Bae

Takaoka

2005

The Computer Journal

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The maximum subarray problem is to find the contiguous array elements having the largest possible sum. We extend this problem to find K maximum subarrays. For general K maximum subarrays where overlapping is allowed, Bengtsson and Chen presented OðminfK + n log 2 n‚ n ffiffiffi ffi K p gÞ time algorithm for one-dimensional case, which finds unsorted subarrays. Our algorithm finds K maximum subarrays in sorted order with improved complexity of O ((n + K ) log K ). For the twodimensional case, we introduce two techniques that establish O(n 3 ) and subcubic time.

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Section: Divide-and-conquermentioning

confidence: 99%

Section: Generalization Of Dmmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Improved Algorithms for the K-Maximum Subarray Problem

Bae

Takaoka

2005

The Computer Journal

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“…A simple extension of this solution can solve the two-dimensional problem in O(m 2 n) time for an m × n array (m ≤ n), which is cubic when m = n [4,5]. In this paper, if only n appears in complexities for the two-dimensional case, we assume m = n. The sub-cubic time solution based on Takaoka's sub-cubic distance matrix multiplication algorithm [14] is given by Tamaki and Tokuyama [17], which is further simplified by Takaoka [15]. In the context of parallel computations, time and cost optimal PRAM and mesh algorithms for the one-dimensional case are described in [10].…”

Section: Introductionmentioning

confidence: 99%

Improved Algorithms for the K-Maximum Subarray Problem for Small K

Bae

Takaoka

2005

Lecture Notes in Computer Science

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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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Efficient Algorithms for the Maximum Subarray Problem by Distance Matrix Multiplication

Cited by 57 publications

References 7 publications

Improved Algorithms for the K-Maximum Subarray Problem

Improved Algorithms for the K-Maximum Subarray Problem

Improved Algorithms for the K-Maximum Subarray Problem for Small K

Improved Algorithms for the k Maximum-Sums Problems

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