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22
Efficient Algorithms for the Maximum Subarray Problem by Distance Matrix Multiplication
, 2002
"... We design an e#cient algorithm that maximizes the sum of array elements of a subarray of a twodimensional array. The solution can be used to find the most promising array portion that correlates two parameters involved in data, such as ages and income for the amount of sales per some period. The pr ..."
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Cited by 23 (6 self)
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We design an e#cient algorithm that maximizes the sum of array elements of a subarray of a twodimensional array. The solution can be used to find the most promising array portion that correlates two parameters involved in data, such as ages and income for the amount of sales per some period. The previous subcubic time algorithm is simplified, and the time complexity is improved for the worst case. We also give a more practical algorithm whose expected time is better than the worst case time.
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 17 (6 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
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 11 (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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Cited by 7 (0 self)
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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.
An O(n 3 log log n / log n) Time Algorithm for the AllPairs Shortest Path Problem
, 2004
"... We design a faster algorithm for the allpairs shortest path problem under the conventional RAM model, based on distance matrix multiplication (DMM). Specifically we improve the best known time complexity of O(n 3 (log log n) 2 / log n) to O(n 3 log log n / log n). As an application, we show the km ..."
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We design a faster algorithm for the allpairs shortest path problem under the conventional RAM model, based on distance matrix multiplication (DMM). Specifically we improve the best known time complexity of O(n 3 (log log n) 2 / log n) to O(n 3 log log n / log n). As an application, we show the kmaximum subarray problem can be solved in O(kn 3 log log n/ log n) time for small k. 1
Selecting Sums in Arrays
"... Abstract. In an array of n numbers each of the ` ´ ..."
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A Subcubic Time Algorithm for the kMaximum Subarray Problem
"... Abstract. We design a faster algorithm for the kmaximum subarray problem under the conventional RAM model, based on distance matrix multiplication (DMM). Specifically we achieve O(n 3 √ log log n/log n + k log n) for a general problem where overlapping is allowed for solution arrays. This complexi ..."
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Abstract. We design a faster algorithm for the kmaximum subarray problem under the conventional RAM model, based on distance matrix multiplication (DMM). Specifically we achieve O(n 3 √ log log n/log n + k log n) for a general problem where overlapping is allowed for solution arrays. This complexity is subcubic when k = o(n 3 / log n). The best known complexities of this problem are O(n 3 + k log n), which is cubic when k = O(n 3 /log n), and O(kn 3 √ log log n / log n), which is subcubic when k = o ( √ log n / log log n). 1
Algorithms for Aggregate Information Extraction from Sequences
"... In this thesis, we propose efficient algorithms for aggregate information extraction from sequences and multidimensional arrays. The algorithms proposed are applicable in several important areas, including large databases and DNA sequence segmentation. We first study the problem of efficiently compu ..."
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In this thesis, we propose efficient algorithms for aggregate information extraction from sequences and multidimensional arrays. The algorithms proposed are applicable in several important areas, including large databases and DNA sequence segmentation. We first study the problem of efficiently computing, for a given range, the rangesum in a multidimensional array as well as computing the k maximum values, called the topk values. We design two efficient data structures for these problems. For the rangesum problem, our structure supports fast update while preserving low complexity of rangesum query. The proposed topk structure provides fast query computation in linear time proportional to the sum of the sizes of a twodimensional query region. We also study the k maximum sum subsequences problem and develop several efficient algorithms. In this problem, the k subsegments of consecutive elements with largest sum are to be found. The segments can potentially
Data Structures: Sequence Problems, Range Queries and Fault Tolerance
, 2010
"... The focus of this dissertation is on algorithms, in particular data structures that give provably efficient solutions for sequence analysis problems, range queries, and fault tolerant computing. The work presented in this dissertation is divided into three parts. In Part I we consider algorithms for ..."
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The focus of this dissertation is on algorithms, in particular data structures that give provably efficient solutions for sequence analysis problems, range queries, and fault tolerant computing. The work presented in this dissertation is divided into three parts. In Part I we consider algorithms for a range of sequence analysis problems that have risen from applications in pattern matching, bioinformatics, and data mining. On a high level, each problem is defined by a function and some constraints and the job at hand is to locate subsequences that score high with this function and are not invalidated by the constraints. Many variants and similar problems have been proposed leading to several different approaches and algorithms. We consider problems where the function is the sum of the elements in the sequence and the constraints only bound the length of the subsequences considered. We give optimal algorithms for several variants of the problem based on a simple idea and classic algorithms and data structures. In Part II we consider range query data structures. This a category of