On Smoothed Analysis of Quicksort and Hoare's Find

We provide a smoothed analysis of Hoare’s find algorithm and we revisit the smoothed analysis of quicksort. Hoare’s find algorithm – often called quickselect – is an easy-to-implement algorithm for finding the k-th smallest element of a sequence. While the worst-case number of comparisons that Hoare’s find needs is Θ(n2 ), the average-case number is Θ(n). We analyze what happens between these two extremes by providing a smoothed analysis of the algorithm in terms of two different perturbation models: additive noise and partial permutations. In the first model, an adversary specifies a sequence of n numbers of [0, 1], and then each number is perturbed by adding a random number drawn from the interval [0, d]. We prove that Hoare’s find needs Θ( n d+1 p n/d+n) comparisons in expectation if the adversary may also specify the element that we would like to find. Furthermore, we show that Hoare’s find needs fewer comparisons for finding the median. In the second mo...

• Type: Conference
• Year: 2009
• Where: COCOON
• Authors: Mahmoud Fouz, Manfred Kufleitner, Bodo Manthey, Nima Zeini Jahromi
• Keywords: COCOON 2009, Combinatorics, Hoare’s find Algorithm, Hoare’s find Needs, Smoothed Analysis