[Solved]: Which measure of sortedness explains the phase transition in Quicksort's runtime?

Problem Detail: I’m currently creating a program to analyse the pathological cases of Quicksort. Namely, the transition of complexity from $O(n^2)$ to $O(n log n)$ as a data set gets less ordered. Since Quicksort is a value-based algorithm, the choice of pivot determines its efficiency. We usually (Quicksort 3, and choosing a random element aside) select the first or last element as the pivot in an array. Therefore, since a good pivot is one that splits the array in half symmetrically, a more disordered array will make selecting a good pivot easier; or at least reduce the likely hood of running into the pathological case. My program gradually randomises this array by swapping some elements each time. When I started to plot the graph of this using gnuplot I notice the trend I expected; a exponential distribution, branching off to a uniform distribution. My question is, how do I quantify a level of disorder in a array? Is their a better way to gradually make an array of values more disordered?

Asked By : user103853

Answered By : rphv

Inversions are one way to measure “disorder” in a list:

Let $A[1..n]$ be an array of $n$ distinct numbers. If $i < j$ and $A[i] < A[j]$ then the pair $(i,j)$ is an inversion of $A$.

However, it’s not the only such measure. In general, this concept is formalized in the notion of presortedness – roughly:

An integer function on a permutation $sigma$ of a totally ordered set that reflects how much $sigma$ differs from the total order.

The survey papers by Mannila [1] and Estivill-Castro & Wood [2] might be good places to start. The question How to measure “sortedness” is related. [1] Heikki Mannila. “Measures of Presortedness and Optimal Sorting Algorithms.” IEEE Transactions on Computers 34.4 (1985): 318-325. [2] Estivill-Castro, Vladmir, and Derick Wood. “A survey of adaptive sorting algorithms.” ACM Computing Surveys (CSUR) 24.4 (1992): 441-476.

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