Sorted data is usually easier to compress than unsorted permutations of the same data. This motivates a simple compression scheme: specify the sorted permutation of the data along with a representation of the sorted data compressed recursively. The sorted permutation can be specified by recording the decisions made by quicksort. If the size of the data is known, then the quicksort decisions des...

We characterize all limit laws of the quicksort type random variables defined recursively by Xn d = XIn + X ∗ n−1−In + Tn when the “toll function” Tn varies and satisfies general conditions, where (Xn), (X ∗ n ), (In, Tn) are independent, Xn d = X∗ n , and In is uniformly distributed over {0, . . . , n − 1}. When the “toll function” Tn (cost needed to partition the original problem into smaller...

The article is a walk through a DATA step implementation of one of the most versatile, fast, and well-rounded sorting algorithms Quicksort. Although a need in array sorting frequently arises in practical SAS programming, Base SAS does not provide a function or call routine to serve the purpose. However, the SAS Language is flexible and powerful enough to implement just about any algorithm. In t...

Sample sort, a generalization of quicksort that partitions the input into many pieces, is known as the best practical comparison based sorting algorithm for distributed memory parallel computers. We show that sample sort is also useful on a single processor. The main algorithmic insight is that element comparisons can be decoupled from expensive conditional branching using predicated instructio...

We present a deterministic oblivious LIFO (Stack), FIFO, double-ended and double-ended priority queue as well as an oblivious mergesort and quicksort algorithm. Our techniques and ideas include concatenating queues end-to-end, size balancing of multiple arrays, several multi-level partitionings of an array. Our queues are the first to enable executions of pop and push operations without any cha...

This paper presents an application of the theory of sorting networks to facilitate the synthesis of optimized general-purpose sorting libraries. Standard sorting libraries are often based on combinations of the classic Quicksort algorithm with insertion sort applied as base case for small, fixed, numbers of inputs. Unrolling the code for the base case by ignoring loop conditions eliminates bran...

(A) Let Q(A) be number of comparisons done on input array A: (A) R ij : event that rank i element is compared with rank j element, for 1 ≤ i < j ≤ n. (B) X ij is the indicator random variable for R ij. That is, X ij = 1 if rank i is compared with rank j element, otherwise 0. (B) Q(A) = ∑ 1≤i<j≤n X ij. (C) By linearity of expectation,

I prove that the average number of comparisons for median-of-k Quicksort (with fat-pivot a.k.a. three-way partitioning) is asymptotically only a constant αk times worse than the lower bound for sorting random multisets with Ω(nε) duplicates of each value (for any ε > 0). The constant is αk = ln(2)/ ( Hk+1 −H(k+1)/2 ) , which converges to 1 as k →∞, so Quicksort is asymptotically optimal for inp...

The in-situ permutation algorithm due to MacLeod replaces (x1, · · · , xn) by (xp(1), · · · , xp(n)) where π = (p(1), · · · , p(n)) is a permutation of {1, 2, · · · , n} using at most O(1) space. Kirshenhofer, Prodinger and Tichy have shown that the major cost incurred in the algorithm satisfies a recurrence similar to sequence of the number of key comparisons needed by the Quicksort algorithm ...