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Lecture 11 Sorting

Parallel ComputingFall 2008

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Sorting Algorithm Rearranging a list of numbers

into increasing (strictly nondecreasing) order.

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Potential Speedup O(nlogn) optimal for any sequential sorting algorithm

without using special properties of the numbers. Best we can expect based upon a sequential sorting

algorithm using n processors is

Optimal parallel time complexity = O(n logn)/n= O(logn)

Has been obtained but the constant hidden in the order notation extremely large.

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Compare-and-Exchange Sorting Algorithms: Compare and Exchange Form the basis of several, if not most, classical

sequential sorting algorithms. Two numbers, say A and B, are compared. If A > B, A

and B are exchanged, i.e.:if (A > B) {

temp = A;A = B;B = temp;

}

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Message Passing Method For P1 to send A to P2 and P2 to send B to P1. Then both processes perform compare operations. P1 keeps the larger of A and B and P2 keeps the smaller of A and B:

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Merging Two Sublists

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Bubble Sort First, largest number moved to the end

of list by a series ofcompares and exchanges, starting at the opposite end.

Actions repeated with subsequent numbers, stopping just before the previously positioned number.

In this way, the larger numbers move (“bubble”) toward one end,

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Bubble Sort

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Time Complexity Number of compare and exchange

operations

Indicates a time complexity of O(n^2) given that a single compare-and-exchange operation has a constant complexity, O(1).

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Parallel Bubble SortIteration could start before previous iteration finished

if doesnot overtake previous bubbling action:

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Odd-Even (Transposition) Sort

Variation of bubble sort.Operates in two alternating phases, even

phase and odd phase.Even phase

Even-numbered processes exchange numbers with their right neighbor.

Odd phaseOdd-numbered processes exchange numbers

with their right neighbor.

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Sequential Odd-Even Transposition Sort

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Parallel Odd-Even Transposition SortSorting eight numbers

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Parallel Odd-Even Transposition Sort

Consider the one item per processor case. There are n iterations; in each iteration,

each processor does one compare-exchange –which can all be done in parallel.

The parallel run time of this formulation is Θ(n).

This is cost optimal with respect to the base serial algorithm but not the optimal serial algorithm.

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Parallel Odd-Even Transposition Sort

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Parallel Odd-Even Transposition Sort Consider a block of n/p elements per processor. The first step is a local sort. In each subsequent step, the compare

exchange operation is replaced by the compare split operation.

There are p phases with each phase performing Θ(n/p) compares and Θ(n/p) communication.

The parallel run time of the formulation is

The parallel formulation is cost-optimal for p= O(logn).

( log ) ( ) ( )pn nT n np p

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Quicksort Very popular sequential sorting algorithm that

performs well with average sequential time complexity of O(nlogn).

First list divided into two sublists. All numbers in one sublist arranged to be smaller than all numbers in other sublist.

Achieved by first selecting one number, called a pivot, against which every other number is compared. If the number is less than the pivot, it is placed in one sublist. Otherwise, it is placed in the other sublist.

Pivot could be any number in the list, but often first number in list chosen. Pivot itself could be placed in one sublist, or the pivot could be separated and placed in its final position

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Quicksort

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Quicksort

Example of the quicksort algorithm sorting a sequence of size n= 8.

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Parallel Quicksort Lets start with recursive decomposition -the list is

partitioned by a single process and then each of the subproblems is handled by a different processor.

The time for this algorithm is lower-bounded by Ω(n)!

–Not cost optimal as the process-time product is Ω(n^2).

Can we parallelize the partitioning step -in particular, if we can use n processors to partition a list of length n around a pivot in O(1)time, we have a winner.

This is difficult to do on real machines, though.

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Parallel Quicksort Using tree allocation of processes

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Parallel Quicksort With the pivot being withheld in processes:

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Analysis Fundamental problem with all tree

constructions –initial division done by a single processor, which will seriously limit speed.

Tree in quicksort will not, in general, be perfectly balanced

Pivot selection very important to make quicksort operate fast.

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Parallelizing Quicksort: PRAM Formulation We assume a CRCW (concurrent read, concurrent write)

PRAM with concurrent writes resulting in an arbitrary write succeeding.

The formulation works by creating pools of processors. Every processor is assigned to the same pool initially and has one element.

Each processor attempts to write its element to a common location (for the pool).

Each processor tries to read back the location. If the value read back is greater than the processor's value, it assigns itself to the 'left' pool, else, it assigns itself to the 'right' pool.

Each pool performs this operation recursively. Note that the algorithm generates a tree of pivots. The

depth of the tree is the expected parallel runtime. The average value is O(logn).

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Parallelizing Quicksort: PRAM Formulation

A binary tree generated by the execution of the quicksortalgorithm. Each level of the tree represents a different array-partitioning iteration. If pivot selection is optimal, then the height of the tree is Θ(log n), which is also the number of iterations.

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Parallelizing Quicksort: PRAM Formulation

The execution of the PRAM algorithm on the array shown in (a).

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End

Thank you!