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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
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
Queries and Fault Tolerance
"... 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
Contents lists available at ScienceDirect Information Processing Letters
"... www.elsevier.com/locate/ipl Optimal algorithms for the averageconstrained maximumsum segment problem ..."
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www.elsevier.com/locate/ipl Optimal algorithms for the averageconstrained maximumsum segment problem