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We consider versions of the FIND algorithm where the pivot element used isthe median of a subset chosen uniformly at random from the data. For the medianselection we assume that subsamples of size asymptotic to $c \cdot n^\alpha$are chosen, where $0<\alpha\le \frac{1}{2}$, $c>0$ and $n$ is the size of thedata set to be split. We consider the complexity of FIND as a process in therank to be selected and measured by the number of key comparisons required.After normalization we show weak convergence of the complexity to a centeredGaussian process as $n\to\infty$, which depends on $\alpha$. The proof relieson a contraction argument for probability distributions on c{\`a}dl{\`a}gfunctions. We also identify the covariance function of the Gaussian limitprocess and discuss path and tail properties.

A Gaussian limit process for optimal FIND algorithms