Abstract
We consider the complexity for computing the approximate sum a 1 + a 2 + ⋯ + a n of a sorted list of numbers a 1 ≤ a 2 ≤ ⋯ ≤ a n . We show an algorithm that computes an (1 + ε)-approximation for the sum of a sorted list of nonnegative numbers in an \(O({1\over \epsilon}\min(\log n, {\log ({x_{max}\over x_{min}})})\cdot (\log{1\over \epsilon}+\log\log n))\) time, where x max and x min are the largest and the least positive elements of the input list, respectively. We prove a lower bound \(\Omega(\min(\log n,\log ({x_{max}\over x_{min}}))\) time for every O(1)-approximation algorithm for the sum of a sorted list of nonnegative elements. We also show that there is no sublinear time approximation algorithm for the sum of a sorted list that contains at least one negative number.
This research is supported in part by NSF HRD-1137764 and NSF Early Career Award 0845376.
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Fu, B. (2013). On the Complexity of Approximate Sum of Sorted List. In: Fellows, M., Tan, X., Zhu, B. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 7924. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38756-2_29
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