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 Weight-Constrained 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 Weight-Constrained 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 Length-Constrained k Maximum-Sum Segments problem is to find the k maximum-sum segments among all segments of A with lengths in the interval [L, U]. We show that the Length-Constrained k Maximum-Sum Segments problem can be solved in O(n + k) time. ∗Corresponding

Abstract. We design a faster algorithm for the k-maximum sub-array 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 sub-cubic 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 sub-cubic when k = o ( √ log n / log log n). 1

Abstract. In an array of n numbers each of the ` ´

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