## Improved algorithms for the k-maximum subarray problem for small k (2005)

Venue: | In Proceedings of the 11th Annual International Conference on Computing and Combinatorics, volume 3595 of LNCS |

Citations: | 15 - 5 self |

### BibTeX

@INPROCEEDINGS{Bae05improvedalgorithms,

author = {Sung E. Bae and Tadao Takaoka},

title = {Improved algorithms for the k-maximum subarray problem for small k},

booktitle = {In Proceedings of the 11th Annual International Conference on Computing and Combinatorics, volume 3595 of LNCS},

year = {2005},

pages = {621--631},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. The maximum subarray problem for a one- or two-dimensional array is to find the array portion that maiximizes the sum of array elements in it. The K-maximum subarray problem is to find the K subarrays with largest sums. We improve the time complexity for the one-dimensional case from O(min{K + n log 2 n, n √ K}) for 0 ≤ K ≤ n(n − 1)/2 to O(n log K + K 2) for K ≤ n. The latter is better when K ≤ √ n log n. If we simply extend this result to the two-dimensional case, we will have the complexity of O(n 3 log K + K 2 n 2).We improve this complexity to O(n 3) for K ≤ √ n. 1

### Citations

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(Show Context)
Citation Context ...ucture of min. We choose a 2-3 tree with level-link as a suitable data structure.s626 Sung E. Bae and Tadao Takaoka A 2-3 tree keeps all the leaf nodes sorted. A 2-3 tree with level-link described in =-=[7]-=- has all the internal nodes at the same depth connected, enabling finger searches. Finger search trees with constant update time are discussed in [6, 9], but they both require logarithmic time for pos... |

43 | A linear time algorithm for finding all maximal scoring subsequences
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Citation Context ...ess, we can find the second maximum sum, the third, etc. For a one-dimensional array, as each run takes O(n) time, we can find the Kmaximum subarray in O(Kn) time. This is however solved in O(n) time =-=[13]-=-. We can extend the O(Kn) time algorithm to two dimensions with O(Kn3 ) time. It remains to be seen if we can extend the O(n) time algorithm to two dimensions with O(n3 ) time. The sum of those maximu... |

43 |
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Citation Context ..., 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... |

34 |
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Citation Context ...omplexity of O(n 3 log K + K 2 n 2 ).We improve this complexity to O(n 3 ) for K ≤ √ n. 1 Introduction The maximum subarray problem was first described by Bentley in his literature Programming Pearls =-=[4, 5]-=- as an example to discuss the efficiency of computer programs. This problem determines an array portion that sums to the maximum value with respect to all possible array portions within the input arra... |

22 | Efficient algorithms for the maximum subarray problem by distance matrix multiplication
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(Show Context)
Citation Context ...e, 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]. For the twodimensional case, EREW PRAM solutions achieving... |

18 |
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Citation Context ...s 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]. For the twodi... |

17 |
Algorithms for the problem of k maximum sums and a vlsi algorithm for the k maximum subarrays problem
- Bae, Takaoka
- 2004
(Show Context)
Citation Context ...s for N = n/K as follows: ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎝ A1,1 ··· A1,N ··· AN,1 ··· AN,N ⎠ Matrix C can be computed by ⎝ B1,1 ··· B1,N ··· BN,1 ··· BN,N ⎠ = ⎝ C1,1 ··· C1,N ··· Cij =MIN N k=1{AikBkj}(i, j =1···,N) ···=-=(2)-=- CN,1 ··· CN,N where the product of submatrices is defined similarly to (1) and the MIN operation is defined on submatrices by taking the MIN operation component-wise. Since comparisons and additions ... |

17 |
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Citation Context ...the maximum subarray problem achieves T = m + n − 2 steps, which is O(n) time using O(n 2 ) sized hardware circuit. Finding K maximum sums is a natural extension. This problem is discussed in [2] and =-=[3]-=-. The former provides O(Kn) and O(Km 2 n) time solutions for the one- and two-dimensional cases in the course of development of a systolic array algorithm of O(n) time using O(n 2 ) size hardware for ... |

17 |
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Citation Context ...es sorted. A 2-3 tree with level-link described in [7] has all the internal nodes at the same depth connected, enabling finger searches. Finger search trees with constant update time are discussed in =-=[6, 9]-=-, but they both require logarithmic time for positioning and do not improve overall time complexity. Now we analyze each part. Part I. For Part I, generating the candidate list involves access to the ... |

14 | Finger Search Trees with Constant Insertion Time
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Citation Context ...es sorted. A 2-3 tree with level-link described in [7] has all the internal nodes at the same depth connected, enabling finger searches. Finger search trees with constant update time are discussed in =-=[6, 9]-=-, but they both require logarithmic time for positioning and do not improve overall time complexity. Now we analyze each part. Part I. For Part I, generating the candidate list involves access to the ... |

13 |
A slightly improved sub-cubic algorithm for the all pairs shortest paths problem with real edge lengths
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Citation Context ...ic complexity for the the two-di� log log n log n � log n mensional case for even smaller K ≤ O( log log n ), using the same frame work of divide-and-conquer and K-tuples. Recent developments for DMM =-=[16, 19]-=- can also be incorporated. Details are omitted here. If we find K-maximum subarray in a graphic image, those will heavily overlap. That is, we will find many array portions that only slightly differ i... |

11 |
Programming pearls: Perspective on performance
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Citation Context ...omplexity of O(n 3 log K + K 2 n 2 ).We improve this complexity to O(n 3 ) for K ≤ √ n. 1 Introduction The maximum subarray problem was first described by Bentley in his literature Programming Pearls =-=[4, 5]-=- as an example to discuss the efficiency of computer programs. This problem determines an array portion that sums to the maximum value with respect to all possible array portions within the input arra... |

11 |
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Citation Context ...ic complexity for the the two-di� log log n log n � log n mensional case for even smaller K ≤ O( log log n ), using the same frame work of divide-and-conquer and K-tuples. Recent developments for DMM =-=[16, 19]-=- can also be incorporated. Details are omitted here. If we find K-maximum subarray in a graphic image, those will heavily overlap. That is, we will find many array portions that only slightly differ i... |

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Citation Context ...timal PRAM and mesh algorithms for the one-dimensional case are described in [10]. For the twodimensional case, EREW PRAM solutions achieving O(log n) time with O(n 3 / log n) processors are given in =-=[11, 18]-=- and comparable result on interconnection networks is given in [12]. The EREW PRAM version of the subcubic algorithm in [15, 17] is given in [1], which also features a VLSI algorithm based on the tech... |

7 | Algorithms Sequential and Parallel: A Unified Approach . Second edition
- Miller
- 2005
(Show Context)
Citation Context ...i 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]-=-. For the twodimensional case, EREW PRAM solutions achieving O(log n) time with O(n 3 / log n) processors are given in [11, 18] and comparable result on interconnection networks is given in [12]. The ... |

6 | Parallel maximum sum algorithms on interconnection networks
- Qiu, Akl
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(Show Context)
Citation Context ...ed in [10]. For the twodimensional case, EREW PRAM solutions achieving O(log n) time with O(n 3 / log n) processors are given in [11, 18] and comparable result on interconnection networks is given in =-=[12]-=-. The EREW PRAM version of the subcubic algorithm in [15, 17] is given in [1], which also features a VLSI algorithm based on the technique introduced L. Wang (Ed.): COCOON 2005, LNCS 3595, pp. 621–631... |

5 |
Parallel approaches to the maximum subarray problem
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(Show Context)
Citation Context ...) time with O(n 3 / log n) processors are given in [11, 18] and comparable result on interconnection networks is given in [12]. The EREW PRAM version of the subcubic algorithm in [15, 17] is given in =-=[1]-=-, which also features a VLSI algorithm based on the technique introduced L. Wang (Ed.): COCOON 2005, LNCS 3595, pp. 621–631, 2005. c○ Springer-Verlag Berlin Heidelberg 2005 621s622 Sung E. Bae and Tad... |

4 |
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(Show Context)
Citation Context ...timal PRAM and mesh algorithms for the one-dimensional case are described in [10]. For the twodimensional case, EREW PRAM solutions achieving O(log n) time with O(n 3 / log n) processors are given in =-=[11, 18]-=- and comparable result on interconnection networks is given in [12]. The EREW PRAM version of the subcubic algorithm in [15, 17] is given in [1], which also features a VLSI algorithm based on the tech... |

2 | Algorithms for finding maxima-scoring segment sets - Csürös - 2004 |