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Abstract: Given a sequence of real numbers and a positive integer , the Sum Selection Problem is to find the segment such that the rank of the sum is over all segments. We present a deterministic algorithm for this problem that runs in time. The previously best known result for this problem is a randomized algorithm that runs in expected time. Applying this algorithm we can obtain a deterministic algorithm for the k Maximum Sums Problem, i.e., the problem of enumerating the largest sum segments, that runs in time. The previously best known randomized and deterministic algorithms for the k Maximum Sums Problem run respectively in expected time and in worst case time. [Copyright &y& Elsevier]

Abstract: To any sequence of real numbers , we can associate another sequence , which Knuth calls its binomial transform. This transform is defined through the rule We study the properties of this transform, obtaining rules for its manipulation and a table of transforms, that allow us to invert many transforms by inspection. We use these methods to perform a detailed analysis of skip lists, a probabilistic data structure introduced by Pugh as an alternative to balanced trees. In particular, we obtain the mean and variance for the cost of searching for the first or the last element in the list (confirming results obtained previously by other methods), and also for the cost of searching for a random element (whose variance was not known). We obtain exact solutions, although not always in closed form. From them we are able to find the corresponding asymptotic expressions. [Copyright &y& Elsevier]