Results 1  10
of
17
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 ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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
Contents lists available at ScienceDirect Information Processing Letters
"... www.elsevier.com/locate/ipl Optimal algorithms for the averageconstrained maximumsum segment problem ..."
Abstract
 Add to MetaCart
www.elsevier.com/locate/ipl Optimal algorithms for the averageconstrained maximumsum segment problem
doi:10.1093/comjnl/bxl007 Improved Algorithms for the KMaximum Subarray Problem
"... 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 log2n ‚ n ffiffiffiffi p KgÞ time algorith ..."
Abstract
 Add to MetaCart
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 log2n ‚ n ffiffiffiffi p KgÞ time algorithm for onedimensional 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.