Bi Tonic Sort with examples

7/23/2019 Bi Tonic Sort with examples

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CS 584


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Sorting

One of the most common operationsOne of the most common operations

Definition:Definition:

Arrange an unordered collection ofArrange an unordered collection of

elements into a monotonically increasing orelements into a monotonically increasing or

decreasing order.decreasing order.

Two categories of sortingTwo categories of sorting internal (fits in memory)internal (fits in memory)

external (uses auxiliary storage)external (uses auxiliary storage)


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Sorting Algorithms

Comparison asedComparison ased

compare!exchangecompare!exchange

O(n log n)O(n log n)

"oncomparison ased"oncomparison ased

#ses $nown properties of the elements#ses $nown properties of the elements

O(n)O(n)! uc$et sort etc.! uc$et sort etc.


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Parallel Sorting Issues

%nput and Output se&uence storage%nput and Output se&uence storage

'here'here

ocal to one processor or distriutedocal to one processor or distriuted

ComparisonsComparisons

*ow compare elements on different nodes*ow compare elements on different nodes

+ of elements per processor+ of elements per processor One (compare!exchange !!, comm.)One (compare!exchange !!, comm.)

-ultiple (compare!split !!, comm.)-ultiple (compare!split !!, comm.)


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Compare-Exchange


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Compare-Split


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Sorting Networks

peciali/ed hardware for sortingpeciali/ed hardware for sorting

ased on comparatorased on comparator

x

y

xy

maxx!y"

minx!y"

minx!y"maxx!y"


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Sorting Network


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Parallel Sorting Algorithms

-erge ort-erge ort

0uic$ ort0uic$ ort

1itonic ort1itonic ort

Others 2Others 2


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#erge Sort

implest parallel sorting algorithmimplest parallel sorting algorithm

tepsteps

Distriute the elementsDistriute the elements

34eryody sort their own se&uence34eryody sort their own se&uence

-erge the lists-erge the lists

5rolem5rolem

*ow to merge the lists*ow to merge the lists


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$itonic Sort

6ey operation:6ey operation:

rearrange a itonic se&uence to orderedrearrange a itonic se&uence to ordered

1itonic e&uence1itonic e&uence

se&uence of elements 7ase&uence of elements 7a889 a9 a9 2 9 a9 2 9 an!n!,,

There exists i such that 7aThere exists i such that 7a889 2 9a9 2 9a ii, is, is

monotonically increasing and 7amonotonically increasing and 7a i;i;92 9 a92 9 an!n!, is, ismonotonically decreasing ormonotonically decreasing or

There exists a cyclic shift of indicies such that theThere exists a cyclic shift of indicies such that the

ao4e is satisfied.ao4e is satisfied.


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$itonic Se%uences

79 9 ?9 8,79 9 ?9 8,

@irst it increases then decreases@irst it increases then decreases

i Bi B

79 9


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&earranging a $itonic Se%uence

et s 7aet s 7a889 a9 a9 2 9 a9 2 9 an!n!,,

aanE

all elements after all elements after iiare from decreasingare from decreasing

e&uence se&uence s


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34ery element of s34ery element of s::is smaller thanis smaller than

e4ery element of se4ery element of s


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$itonic #erging Network


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'hat a(out unor)ere) lists*

To use the itonic merge for n items9 we mustTo use the itonic merge for n items9 we must

first ha4e a itonic se&uence of n items.first ha4e a itonic se&uence of n items.

Two elements form a itonic se&uenceTwo elements form a itonic se&uence Any unsorted se&uence is a concatenation ofAny unsorted se&uence is a concatenation of

itonic se&uences of si/e


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#apping onto a hypercu(e

One element per processorOne element per processor

tart with the sorting networ$ mapstart with the sorting networ$ maps

3ach wire represents a processor3ach wire represents a processor

-ap processors to wires to minimi/e-ap processors to wires to minimi/e

the distance tra4eled during exchangethe distance tra4eled during exchange


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$itonic #erging Network


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$itonic #erge on +ypercu(e


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$itonic SortProce)ure $itonicSort

,or i . to ) -/

,or 0 i )ownto .

i, 1i 2 /3st(it o, iproc 0th(it o, iproc

comp6exchange6max10! item3

else

comp6exchange6min10! item3

en)i,

en),oren),or

comp6exchange6max an) comp6exchange6min compare an)

exchange the item with the neigh(or on the 0th)imension


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$itonic Sort Stages


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Assignment

5ic$ ? random integers5ic$ ? random integers

Draw the 1itonic ort networ$Draw the 1itonic ort networ$

tep through the 1itonic sort networ$ totep through the 1itonic sort networ$ to

produce a sorted list of integers.produce a sorted list of integers.

3xplain how the if statement in the3xplain how the if statement in the

1itonic sort algorithm wor$s.1itonic sort algorithm wor$s.

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Bi Tonic Sort with examples