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Refined Quicksort asymptotics. (English) Zbl 1327.68086
Summary: The complexity of the Quicksort algorithm is usually measured by the number of key comparisons used during its execution. When operating on a list of \(n\) data, permuted uniformly at random, the appropriately normalized complexity \(Y_n\) is known to converge almost surely to a non-degenerate random limit \(Y\). This assumes a natural embedding of all \(Y_n\) on one probability space, e.g., via random binary search trees. In this note a central limit theorem for the error term in the latter almost sure convergence is shown:
\[ \sqrt{\frac{n}{2\log n}}(Y_n-Y)\overset {d} \longrightarrow\mathcal{N} \quad {(n}\infty) \]
where \(\mathcal{N}\) denotes a standard normal random variable.

MSC:
68P10 Searching and sorting
60F05 Central limit and other weak theorems
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
Software:
Quicksort
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References:
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