Copy of Parallel Graph Algorithms

8/14/2019 Copy of Parallel Graph Algorithms

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Parallel Graph Algorithms

Anshuman Chakraborty

A technical Seminar onParallel Graph Algorithms


CS200118214under the guidance of

Anisur Rahman

By


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What is Parallel Processing? Parallel processing is processing a

task with several processing units, or

many processors Most powerful computers containtwo or more processing units thatshare among themselves the jobssubmitted for processing


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Why Parallel processing? Several operations are performedsimultaneously, so the time taken by a

computation can be reduced. A problem to be solved is broken into a

no.of subproblems. These subproblems arenow solved simultaneously, each ondifferent processors. The results are then

combined to produce an answer to theoriginal problem. To achieve speedup of processing


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Speedup of Computation Sppedup is measured in contrast with sequential

version of the same algorithm

speedup =ws/wp where ws=worst case running

time of fastest known sequential algorithm for the problem and wp= worst case running time of the parallel algorithm for the same problem.

Pipelines have been extensively used in processorsto increase the performance.


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Computational Model The stream of instruction tells the computer what to do at eachstep and are divided into four types:1. Single Instruction stream, single Data stream (SISD)2. Multiple Instruction stream, Single Data stream(MISD)3. Single Instruction stream, Multiple Data stream(SIMD)

4. Multiple Instruction stream, Multiple Data stream(MIMD )


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Introducing Graph

A graph consists of afinite set of nodes and

a finite set of edgesconnecting pairs of these nodes

One graph can berepresented either byan adjacency matrix or adjacency list


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The connectivity matrix contd

Procedure: CUBE CONNECTIVITY (A, C)

Step 1 : {The diagonal elements of the adjacency matrix are made equal to 1}

for j = 0 to n -1 do in parallel

A(0, j, j) 1end for.

Step 2 : {The A registers are copied into the B registers}

for j = 0 to n-1 do in parallel

for k = 0 to n-1 do in parallel

B(0, j, k)



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The connectivity matrix contdStep 3 : {The connectivity matrix is obtained through repeated Boolean

multiplication)

for i = 1 to [log(n - 1)] do[3.1] CUBE MATRIX MULTIPLICATION (A, B, C)

[3.2] for j=0 to n-1 do in parallelfor k = 0 to n -1 do in parallel

(i)A(0, j, k) C(0, j, k)(ii) B(0, j ,k) C(0, j, k)

end for

end for end for It takesO(log n) time. This step is iterated (log n) times. It follows that the

overall running time of this procedure is t (n) = O (log 2 n).



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All pair shortest path Let d

k ij denote the length of the shortest path from v i to v j that goes through at most k-1 intermediate

vertices.

Thus d1

ij =w ij ,(the weight of the edge from v i to v j.

d1

ij = . Also d 1ii=0. G has no cycles of negative line, there is no advantage in visiting any vertex more thanonce in a shortest path from v i to v j

In order to compute for k > 1 we can use the fact that

dk

ij = min l{d k/2 il+d k/2 lj }and d ij =d n-1 ij

The procedure runs on a cube-connected SIMD computer with N = n3 processors, each with three registers

A, B, and C. As before, the processors can be regarded as being arranged in an (n x n x n) array pattern.



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All Pair Shortest Path algo

Procedure CUBE SHORTESTPATHS (A, C)

Step 1: (Construct the matrix D 1 andstore it in registers A and B)

for j = 0 to n-1 do in parallel

for k = 0 to n-1 do in parallel

if j!=k and A(0, j, k) = 0

then A(0, j, k) = Inf

end if

[1.2] B(0, j, k) = A(0, j, k)

end for

end for

Slep 2: (Construct the matrices D 2, D 4,.., D n-1 through repeated matrixmultiplication}

for i = l to log(n-1) do

[2.1] CUBE MATRIXMULTIPLICATION (A, B, C)[2.2] for j = 0 to n-1 do in parallel for k = 0 lo n-1 do in parallel [1]A(0, j, k )=C(0, j, k)

[2]B(0, j, k) =C(0, j, k)

end for end for

end for.



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All Pair Shortest Path(complexity)

Steps 1 and 2 take constant time. There are [log(n-1)] iterations of step 2.1 each requiring O(logn)time.

The overall running time of procedure CUBESHORTEST PATHS is therefore t(n) = O(log 2n).Since p(n)=n 3 ,c(n) = O(n 3log 2n)

The sequential algorithm takes O(n 3) time

The speed up = 2 k.k 2 where k=log 2n.



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Example



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Computation of All pair Shortest Path


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Conclusion Graph Theory and algorithms has a very

wide area of application in practical problems of Networking

So it is very good if a speed up is achievedin few graph algorithms like finding MSTor finding Transitive Closure

It is a great experience in doing a Technicalseminar related to Parallel Processing


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Thank You!!!

Copy of Parallel Graph Algorithms

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