A Parallel Algorithm for the Generation of Costas Sequences

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A Parallel Algorithm for the Generation of
Costas Sequences

• E. Orozco - D. Bollman
• MPI Seminar
• April 17, 2009
• Department of Computer Science
• UPR-Rio Piedras

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References

• O. Moreno, J. Ramirez, D. Bollman, and E. Orozco
  Faster Backtracking Algorithms for the Generation
  of Symmetry-Invariant Permutations. Journal of
  Applied Mathematics 26 (2002) 277-287.
• D. Bollman and E. Orozco A Faster Algorithm for
  the n-Queens Problem. Congressus Numerantium,
  Vol. 148 (2001), 193-200.
• E. Horowitz, Sahni and Rajasekaran Computer
  Algorithms/C. Computer Science Press, 1996.

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Application of Costas Sequences

• Costas arrays are permutation matrices that
  provide a frequency indexing sequence that
  permits at most one tone in cross-correlation of
  certain waveforms.

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MPI Seminar Orozco-Bollman
MPI Seminar Orozco-Bollman Dec 09, 2004
Examples

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Not a solution to 5-Costas
A solution to 5-Costas
(4, 1, 2, 5, 3)
(3, 5, 1, 2, 4)
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MPI Seminar Orozco-Bollman Dec 09, 2004
Symmetries
(4, 1, 2, 5, 3)
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MPI Seminar Orozco-Bollman Dec 09, 2004
Symmetries
(4, 1, 2, 5, 3)
(3, 5, 2, 1, 4)
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MPI Seminar Orozco-Bollman Dec 09, 2004
Symmetries
(4, 1, 2, 5, 3)
(3, 5, 2, 1, 4)
(2, 5, 4, 1, 3)
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MPI Seminar Orozco-Bollman Dec 09, 2004
Symmetries
(4, 1, 2, 5, 3)
(3, 5, 2, 1, 4)
(3, 1, 4, 5, 2)
(2, 5, 4, 1, 3)
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Costas Property Difference Triangle
Let
be a permutation of the integers from 1 to n.
Then X is a Costas sequence if and only if
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Examples Costas Property Difference Triangle
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Examples Costas Property Difference Triangle

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Backtracking Technique Example n 4
1
12
124
1243
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Backtracking Technique Example n 4
1
13
12
132
124
1243
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Backtracking Technique Example n 4
1
13
12
132
124
134
1243
1342
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Backtracking Technique Example n 4
1
13
12
14
132
124
143
134
142
1243
1342
1423
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Backtracking Technique Example n 4
1
2
13
12
14
23
21
24
132
124
143
231
214
134
213
241
243
142
2314
2431
1243
1342
1423
2134
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Backtracking Technique Example n 4
1
2
3
13
12
14
23
21
24
32
31
34
132
124
143
231
214
314
312
134
213
241
243
324
342
142
2314
2431
1243
1342
1423
2134
3124
3241
3421
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Backtracking Technique Example n 4
1
2
3
4
13
12
14
23
21
24
32
31
34
42
41
43
132
124
143
231
214
314
312
412
421
413
134
213
241
243
324
342
423
431
142
2314
2431
1243
1342
1423
2134
3124
3241
3421
4132
4213
4312
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Master-Slave Technique
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Main code
if ( myid Master ) Master_Code( ) else
Worker_Code( )
MPI_Send (void buf, int count, MPI_Datatype
datatype, int dest, int tag, MPI_Comm
comm) MPI_Recv (void buf, int count,
MPI_Datatype datatype, int source, int tag,
MPI_Comm comm, MPI_Status status)
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Xi
Xi1

Xid




Master sends a subsequence with d terms having
the Costas property
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Worker receives the subsequence
Xi
Xi1

Xid
Y1

Yk




Worker tries to append as many terms to the right
as possible so that the enlarged subsequence
preserves the Costas property
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Xi
Xi1

Xid
Y1

Yk




Zt

Z1
Worker tries to append more terms to the left,
so that the enlarged subsequence preserves the
Costas property
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Xi
Xi1

Xid
Y1

Yk
Zt

Z1
Worker generates ¼ of ALL Costas solutions having
(Xi,Xi1,,Xid) as a subsequence.
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Masters Code
CostasMaster ( ) send each worker an
initial message //just the first chunk of
work while (there is a job to do)
receive answer from any worker Wi send
Wi another message to work on compute
if (there is no more job to do)
receive answer from any worker Wi send
Wi a STOP signal
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Workers Code
CostasWorker ( ) do receive message
from Master if (message ! STOP) compute
send answer to Master receive new
message from Master while (message !
STOP)
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Lets make some runs in your Gauss cluster!
Compile mpicc CostasPar.c Execute with 4
processes mpirun np 4 a.out
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