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In this paper, we study a new version from Dual-pivot Quicksort algorithm when we have some other number of pivots. Hence, we discuss the idea of picking pivots by random way and splitting the list simultaneously according to these. The modified version generalizes these results for multi process. We show that the average number of swaps done by Multi-pivot Quicksort process and we present a special case. Moreover, we obtain a relationship between the average number of swaps of Multi-pivot Quicksort and Stirling numbers of the first kind.

Quicksort studied in many books such as [

Yaroslavskiy’s algorithm replaced the new standard Quicksort algorithm in Oracle’s Java 7 runtime library. This algorithm uses two items as pivots to divide the items. If two pivots

We analyze the limiting distribution of the number of swaps needed by the duality process is proposed. It is known to be the unique fixed point of a certain distributional transformation

Later, many researchers has received the interest of the visualization of multi-pivot Quicksort in accordance with Yaroslavskiy proposed the duality pivot process which outperforms standard Quicksort by Java JVM. After that, this algorithm has been explained in terms of comparisons and swaps by Wild and Nebel [

A normal expansion of duality process would be to have some other number

There are

Each item of the

For the pervious process, there are

We assume that

As long as

where

The average number of swaps done by the multi algorithm applied to an list of

where

Let

first recursive call, where

the average number of swaps for ordering the sublist of

By collecting terms with a common factor

Multiplying both sides by

multiplying by

tion

We find that

Such that

The recurrence becomes as follows

In the right -hand side of Equation (5). The first sum becomes in this form because it may be easily explained by mathematical induction that the k-th order derivative of

is

The recurrence is converted to the following differential equation [

Multiplying by

This differential equation is a Cauchy-Euler equation [

By using the differential operator

and using the mathematical induction we find that at

and

We find

the relation holds at

at

So, it is easy to find the relation is satisfied for all values of

When we apply the operator

where

where

And we get

Setting

This differential equation can be written as

where

Then

By using the property of linearity of differential operator

if we apply

where

Therefore, to evaluate

Moreover

Combining both solutions,

such that

In terms of series;

The third sum of Equation (15) collects to the solution a stationary contribution. Furthermore, the root

The number of methods to permute a list of

We show the relation between the number of swaps done by the multi Quicksort process and Sirling number of the first kind. Form Equation (17) we assume that

The relation is converted to a k-th order differential equation

This differential equation is a Cauchy-Euler equation. We use the deferential operator

also, by induction

applying the operator

where

where

hence

by equality the coefficients we obtain

In this section we show the average number of swaps needed by the Quicksort is a particular case form the public case of the multi-pivot Quicksort [

where

Assume that if we need to sort list of

Subsequently, we consider as well that pivots are uniformly picked and noticing that we have to number the final swap with the pivot at the end of split operation [

So, we find the toll function given by

We find

We solve this recurrence relation by transforming into a differential equation. First multiply both sides by

Let

multiplying by

Multiplying by

We can solve this differential equation using basic principles

This differential equation is a Cauchy-Euler equation [

we use the differential operator

applying the operator

and applying the pervious technique we find the solution of the differential equation given by

where

Extracting the coefficients, the expected number of swaps for Multi-pivot Quicksort is

We study a new version from Dual-pivot Quicksort algorithm when we have some other number

We thank the Editor and the referee for their comments.

Ragab, M., El- Desouky, B.E.-S. and Nader, N. (2017) Ana- lysis of the Multi-Pivot Quicksort Process. Open Journal of Modelling and Simulation, 5, 47-58. http://dx.doi.org/10.4236/ojmsi.2017.51004