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21
Improved algorithms for the kmaximum subarray problem for small k
 In Proceedings of the 11th Annual International Conference on Computing and Combinatorics, volume 3595 of LNCS
, 2005
"... Abstract. The maximum subarray problem for a one or twodimensional array is to find the array portion that maiximizes the sum of array elements in it. The Kmaximum subarray problem is to find the K subarrays with largest sums. We improve the time complexity for the onedimensional case from O(min ..."
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Cited by 19 (7 self)
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Abstract. The maximum subarray problem for a one or twodimensional array is to find the array portion that maiximizes the sum of array elements in it. The Kmaximum subarray problem is to find the K subarrays with largest sums. We improve the time complexity for the onedimensional 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 twodimensional 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
Improved algorithms for the k maximumsums problems
, 2006
"... Given a sequence of n real numbers and an integer k, 1�k � 1 2 n(n − 1), the k maximumsum 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 log2 n, n √ k})time algorithm for this problem. Bae ..."
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Cited by 13 (2 self)
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Given a sequence of n real numbers and an integer k, 1�k � 1 2 n(n − 1), the k maximumsum 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 log2 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 maximumsum segments in nondecreasing order if k �n. Then we develop an O(n2d−1 +k log min{n, k})time algorithm for the ddimensional 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(n2d−1C)time algorithm, also by Bengtsson and Chen, where C = min{k + n log2 n, n √ k}. It should be pointed out that, given a twodimensional array of size m×n, our algorithm for finding the k maximumsum subarrays is the first one achieving cubic time provided that k is O(m2n / log n).
A Linear Time Algorithm for the k Maximal Sums Problem
"... Abstract. Finding the subvector with the largest sum in a sequence of n numbers is known as the maximum sum problem. Finding the k subvectors with the largest sums is a natural extension of this, and is known as the k maximal sums problem. In this paper we design an optimal O(n+k) time algorithm f ..."
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Cited by 12 (2 self)
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Abstract. Finding the subvector with the largest sum in a sequence of n numbers is known as the maximum sum problem. Finding the k subvectors with the largest sums is a natural extension of this, and is known as the k maximal sums problem. In this paper we design an optimal O(n+k) time algorithm for the k maximal sums problem. We use this algorithm to obtain algorithms solving the twodimensional k maximal sums problem in O(m 2 ·n+k) time, where the input is an m ×n matrix with m ≤ n. We generalize this algorithm to solve the ddimensional problem in O(n 2d−1 +k) time. The space usage of all the algorithms can be reduced to O(n d−1 + k). This leads to the first algorithm for the k maximal sums problem in one dimension using O(n + k) time and O(k) space. 1
Efficient algorithms for the sum selection problem and k maximum sums problem
 In Proceedings of the 17th International Symposium on Algorithms and Computations
, 2006
"... Abstract. Given a sequence of n real numbers A = a1, a2,..., an and a positive integer k, the Sum Selection Problem is to find the segment A(i, j) = ai, ai+1,..., aj such that the rank of the sum s(i, j) = j t=i at is k over all n(n−1)2 segments. We present a deterministic algorithm for this proble ..."
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Abstract. Given a sequence of n real numbers A = a1, a2,..., an and a positive integer k, the Sum Selection Problem is to find the segment A(i, j) = ai, ai+1,..., aj such that the rank of the sum s(i, j) = j t=i at is k over all n(n−1)2 segments. We present a deterministic algorithm for this problem that runs in O(n log n) time. The previously best known randomized algorithm for this problem runs in expected O(n log n) time. Applying this algorithm we can obtain a deterministic algorithm for the k Maximum Sums Problem, i.e., the problem of enumerating the k largest sum segments, that runs in O(n log n + k) time. The previously best known randomized and deterministic algorithms for the k Maximum Sums Problem run respectively in expected O(n log n + k) and O(n log2 n + k) time in the worst case.
A note on ranking k maximum sums
, 2005
"... In this paper, we design a fast algorithm for ranking the k maximum sum subsequences. Given a sequence of real numbers 〈x1, x2, · · · , xn 〉 and an integer parameter k, the problem is to compute k subsequences of consecutive elements with the sums of their elements being the largest, second large ..."
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Cited by 5 (0 self)
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In this paper, we design a fast algorithm for ranking the k maximum sum subsequences. Given a sequence of real numbers 〈x1, x2, · · · , xn 〉 and an integer parameter k, the problem is to compute k subsequences of consecutive elements with the sums of their elements being the largest, second largest,..., and the k th largest among all possible range sums. For any value of k, 1 ≤ k ≤ n(n + 1)/2, our algorithm takes O(n + k log n) time in the worst case to rank all such subsequences. Our algorithm is optimal for k ≤ n.
Algorithm for k disjoint maximum subarrays
 In Proc. of the International Conference on Computational Science (ICCS 2006), Part I (2006
"... Abstract. The maximum subarray problem is to find the array portion that maximizes the sum of array elements in it. For K disjoint maximum subarrays, Ruzzo and Tompa gave an O(n) time solution for onedimension. This solution is, however, difficult to extend to twodimensions. While a trivial solut ..."
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Cited by 4 (4 self)
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Abstract. The maximum subarray problem is to find the array portion that maximizes the sum of array elements in it. For K disjoint maximum subarrays, Ruzzo and Tompa gave an O(n) time solution for onedimension. This solution is, however, difficult to extend to twodimensions. While a trivial solution of O(Kn3) time is easily obtainable for twodimensions, little study has been undertaken to better this. We first propose an O(n + K log K) time solution for onedimension. This is equivalent to Ruzzo and Tompa’s when order is considered. Based on this, we achieve O(n3 +Kn2 log n) time for twodimensions. This is cubic time when K ≤ n / log n. 1
Computing maximumscoring segments in almost linear time
 IN PROCEEDINGS OF THE 12TH ANNUAL INTERNATIONAL COMPUTING AND COMBINATORICS CONFERENCE, VOLUME 4112 OF LNCS
, 2006
"... 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 th ..."
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Cited by 3 (1 self)
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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 lineartime algorithm for this problem. Our algorithm uses a disjointset data structure and requires O(nα(n, n)) time in the worst case, where α(n, n) is the inverse Ackermann function.
Selecting Sums in Arrays
"... Abstract. In an array of n numbers each of the ` ´ ..."
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Algorithms for Finding the WeightConstrained k Longest Paths in a Tree and the LengthConstrained k MaximumSum Segments of a Sequence
, 2008
"... In this work, we obtain the following new results: – Given a tree T = (V, E) with a length function ℓ: E → R and a weight function w: E → R, a positive integer k, and an interval [L, U], the WeightConstrained k Longest Paths problem is to find the k longest paths among all paths in T with weights i ..."
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In this work, we obtain the following new results: – Given a tree T = (V, E) with a length function ℓ: E → R and a weight function w: E → R, a positive integer k, and an interval [L, U], the WeightConstrained k Longest Paths problem is to find the k longest paths among all paths in T with weights in the interval [L, U]. We show that the WeightConstrained k Longest Paths problem has a lower bound Ω(V log V + k) in the algebraic computation tree model and give an O(V log V + k)time algorithm for it. – Given a sequence A = (a1, a2,..., an) of numbers and an interval [L, U], we define the sum and length of a segment A[i, j] to be ai + ai+1 + · · · + aj and j − i + 1, respectively. The LengthConstrained k MaximumSum Segments problem is to find the k maximumsum segments among all segments of A with lengths in the interval [L, U]. We show that the LengthConstrained k MaximumSum Segments problem can be solved in O(n + k) time. ∗Corresponding