Book Chapter10.1007/11602613_52
Randomized algorithm for the sum selection problem
Tien-Ching Lin,Der-Tsai Lee +1 more
- 19 Dec 2005
- pp 515-523
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TL;DR: An algorithm is obtained for the kMaximum Sums Problem, i.e., the problem of enumerating the k largest sum segments, that runs in expected O(n log n + k) time.
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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 $s(i, j) = \sum_{t = i}^{j}{a_{t}}$ is k over all ${n(n-1)} \over {2}$ segments. We will give a randomized algorithm for this problem that runs in expected O(n log n) time. Applying this algorithm we can obtain an algorithm for the kMaximum Sums Problem, i.e., the problem of enumerating the k largest sum segments, that runs in expected O(n log n + k) time. The previously best known algorithm for the kMaximum Sums Problem runs in O(n log2n + k) time in the worst case.
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Citations
Sequential and Parallel Algorithms for the Generalized Maximum Subarray Problem
Sung Eun Bae
- 01 Jan 2007
TL;DR: This thesis explores various techniques to speed up the computation, and several new algorithms for the maximum subarray problem, and investigates a speed-up option through parallel computation.
A linear time algorithm for the k maximal sums problem
Gerth Stølting Brodal,Allan Grønlund Jørgensen +1 more
- 26 Aug 2007
TL;DR: This paper designs an optimal O(n+k) time algorithm and uses this algorithm to obtain algorithms solving the two-dimensional k maximal sums problem in O(m2 ċ n+ k) time, where the input is an m × n matrix with m ≤ n.
Improved algorithmms for the k maximum-sums problems
TL;DR: This paper proposes 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), and gives the first optimal algorithm for delivering the k maximum-sum segments in non-decreasing order if k ≤ n.
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Algorithms for finding the weight-constrained k longest paths in a tree and the length-constrained k maximum-sum segments of a sequence
Hsiao-Fei Liu,Kun-Mao Chao +1 more
TL;DR: This work shows that the Length-ConstrainedkMaximum-Sum Segments problem can be solved in O(n+k) time and gives an O(VlogV+k)-time algorithm for it.
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Fast algorithms for finding disjoint subsequences with extremal densities
Anders Bergkvist,Peter Damaschke +1 more
TL;DR: Fast algorithms are derived for the problem of finding a prescribed number of intervals of maximum total length that contain at most some prescribedNumber of points from a given point set that leads to an apparently open problem of computational geometry flavour (where the algorithm is seeking a subquadratic algorithm) which might be interesting in itself.
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