Analysis of Hoare's FIND algorithm with Median‐of‐three partition

Abstract

Hoare's FIND algorithm can be used to select the jth element out of a file of n elements. It bears a remarkable similarity to Quicksort; in each pass of the algorithm, a pivot element is used to split the file into two subfiles, and recursively the algorithm proceeds with the subfile that contains the sought element. As in Quicksort, different strategies for selecting the pivot are reasonable. In this paper, we consider the Median-ofthree version, where the pivot element is chosen as the median of a random sample of three elements. Establishing some hypergeometric differential equations, we find explicit formulae for both the average number of passes and comparisons. We compare these results with the corresponding ones for the basic partition strategy.

Optimal Sampling Strategies in Quicksort and Quickselect

“…In section 3 we analyze the variants with fixed-size sampling. We rederive known results such as those of van Emden [26] and those of quickselect with median-of-three [10]. As far as we know, our results regarding the total cost of quicksort and of quickselect with fixed s > 3 are original.…”

“…Van Emden [26] showed that the average number of comparisons to sort an array of size n is q(k) • n ln n + O(n), where q(k) steadily decreases from q(0) = 2 to q(∞) = 1/ ln 2. The median-of-three strategy also improves the performance of quickselect [10].…”

Optimal Sampling Strategies in Quicksort and Quickselect

“…In section 3 we analyze the variants with fixed-size sampling. We rederive known results such as those of van Emden [26] and those of quickselect with median-of-three [10]. As far as we know, our results regarding the total cost of quicksort and of quickselect with fixed s > 3 are original.…”

“…Van Emden [26] showed that the average number of comparisons to sort an array of size n is q(k) • n ln n + O(n), where q(k) steadily decreases from q(0) = 2 to q(∞) = 1/ ln 2. The median-of-three strategy also improves the performance of quickselect [10].…”

A Gaussian limit process for optimal FIND algorithms

“…where the convergence in (25) with f = Z also holds with respect to all moments. For the mean, we have…”

Using nonlinear difference equations to study Quicksort algorithms

“……”

Optimal Sampling Strategies in Quicksort and Quickselect

“…In section 3 we analyze the variants with fixed-size sampling. We rederive known results such as those of van Emden [26] and those of quickselect with median-of-three [10]. As far as we know, our results regarding the total cost of quicksort and of quickselect with fixed s > 3 are original.…”

“…Van Emden [26] showed that the average number of comparisons to sort an array of size n is q(k) • n ln n + O(n), where q(k) steadily decreases from q(0) = 2 to q(∞) = 1/ ln 2. The median-of-three strategy also improves the performance of quickselect [10].…”

A Gaussian limit process for optimal FIND algorithms

“…where the convergence in (25) with f = Z also holds with respect to all moments. For the mean, we have…”

Using nonlinear difference equations to study Quicksort algorithms

“……”

Optimal Sampling Strategies in Quicksort and Quickselect

“…In section 3 we analyze the variants with fixed-size sampling. We rederive known results such as those of van Emden [26] and those of quickselect with median-of-three [10]. As far as we know, our results regarding the total cost of quicksort and of quickselect with fixed s > 3 are original.…”

“…Van Emden [26] showed that the average number of comparisons to sort an array of size n is q(k) • n ln n + O(n), where q(k) steadily decreases from q(0) = 2 to q(∞) = 1/ ln 2. The median-of-three strategy also improves the performance of quickselect [10].…”

A Gaussian limit process for optimal FIND algorithms

“…where the convergence in (25) with f = Z also holds with respect to all moments. For the mean, we have…”

Using nonlinear difference equations to study Quicksort algorithms

“……”