Abstract Probabilistic recurrence relations (PRRs) are a standard formalism for describing the runtime of randomized algorithm. Given PRR and time limit $$\kappa $$ κ , we consider tail probability $$\Pr [T \ge \kappa ]$$ Pr [ T ≥ ...

We present a proof of an efficient, in-place Quicksort implementation [1] using single-threaded objects (stobjs) in ACL2 [3, 4]. We demonstrate that the Quicksort implementation is equivalent to a simple insertion-sort function that is shown to produce an ordered permutation of its input. For ease of reasoning, the demonstration is carried out by verifying a series of ”intermediate” sorting fun...

In quicksort, due to branch mispredictions, a skewed pivotselection strategy can lead to a better performance than the exactmedian pivot-selection strategy, even if the exact median is given for free. In this paper we investigate the effect of branch mispredictions on the behaviour of mergesort. By decoupling element comparisons from branches, we can avoid most negative effects caused by branch...

The number of comparisons Xn used by Quicksort to sort an array of n distinct numbers has mean μn of order n log n and standard deviation of order n. Using different methods, Régnier and Rösler each showed that the normalized variate Yn := (Xn−μn)/n converges in distribution, say to Y ; the distribution of Y can be characterized as the unique fixed point with zero mean of a certain distribution...

QuickSort Hoare [1962] (A) Pick a pivot element from array (B) Split array into 3 subarrays: those smaller than pivot, those larger than pivot, and the pivot itself. (C) Recursively sort the subarrays, and concatenate them. Randomized QuickSort (A) Pick a pivot element uniformly at random from the array (B) Split array into 3 subarrays: those smaller than pivot, those larger than pivot, and the...

Binary search trees are a fundamental data structure and their height plays a key role in the analysis of divide-and-conquer algorithms like quicksort. Their worst-case height is linear; their average height, whose exact value is one of the best-studied problems in average-case complexity, is logarithmic. We analyze their smoothed height under additive noise: An adversary chooses a sequence of ...