Enumerating cycles in bipartite graph using matrix approach

HUAWEI TECHNOLOGIES CO., LTD.
47pt
www.huawei.com
Usatyuk Vasiliy, 2013
L [dog] Lcrypto.com
Enumerating cycles
in bipartite graphs
using matrix approach

Let consider arbitrary codes on the graph
parity-check matrix
Enumerate cycle 4
girth_four_num=0;
for i=1:rows-1
for j=1:cols-1
if H(i,j)==1
%save to array (i,j)
x1=i;y1=j;
for j1=j+1:cols
if H(i,j1)==1
x2=i;y2=j1;
for i1=i+1:rows
if H(i1,j1)==1
x3=i1;y3=j1;
if H(i1,j)==1 %the forth point;
x4=i1;y4=j;
girth_four_num=girth_four_num+1;
else
%delete all saved element location for symbols and checks;
end
end
end
end
end
end
end
end
,
11
11

















H
Poor solution, complexity too high
Fan J.,Yang X., "A Method of Counting the Number of Cycles in LDPC Codes," Signal
Processing, 2006 8th International Conference on , vol.3, no., pp.,, 16-20 2006

Let try to estimate number of cycle using
properties of adjacency matrix
Hmatrixcheck-parityofmatrixadjacency,
0
0
isA
H
H
A T 






22
equallengthwholepathsofnumbergivesA 
gequallengthwholepathsofnumbergivesAg

But powers of the adjacency matrix takes into account not only the trail(simple way,
edges that form the walk distinct), but also way those go through the same edge
many times.

lollipop walk
Cycles of length 2m are thus lollipop walks.
All cycles in bipartite graphs contain an even number
of edges and all lollipop walk paths have
  mnm,
naaa ,...,, 21
 m2,0
  mnm,
  .2mod0 mn
1v 3v 4v 6v 2v 5v
4v 3v 5v
2v 6v
1v
 2,4 lollipop walk where
while are distinct.
57 aa 
is n length walks where are distinct and
for some
11   mn aa
].,1[ nm
1a
2a 3a 4a 5a 6a
7a
1a 2a
3a
4a 5a
6a
7a
621 ,...,, aaa
 4,2 lollipop walk where
while are distinct.
37 aa 
621 ,...,, aaa
Halford, T.R.; Chugg, K.M., "An algorithm for counting short cycles in bipartite graphs,"
Information Theory, IEEE Transactions on , vol.52, no.1, pp.287,292, Jan. 2006

v c v
c
c
1a 2a 3a
4a
lollipop walk from to 6,2 .39 aa 1a
Subtract non-trail way
v
c
v
5a
6a
7a8a9a
 
cv
LH 6,1 Count lollipop walk and walks where 
sv
L 6,2 715131 ,, aaaaaa 
which necessary to substract to get  
sv
L 6,2

Let define number of paths length 2k from variable i to j variable
and from i check node i to j check nodes
ss
v
k VVP s
,2
cc
v
k VVsizeP c
,2
Let define number of paths length 2k+1 from variable i to j check node
and from i check node i to j symbol nodes
cs
v
k VVP s
 ,12
sc
v
k VVsizeP c
 ,12
Let define number lollipop walks from check i to j check
and from i check node to j symbol nodes
cc
v
kkk VVL c
 ,22,2  kkk 22,2
sc
v
kkk VVL c
 ,22,12
  kkk 22,12
 


 
1
0
22,12212
k
i
v
iki
v
k
v
k
ccc
LHPP
 


 
1
0
22,2122
k
i
v
iki
Tv
k
v
k
ccc
LHPP
 


 
1
0
22,121212
k
i
v
iki
Tv
k
v
k
sss
LHPP
 


 
1
0
22,2122
k
i
v
iki
v
k
v
k
sss
LHPP
  IHPL Tv
k
v
k
cc
12)2,0( 
  IHPL ss v
k
v
k 12)2,0( 
   sc v
k
v
kk LTr
k
LTr
k
N )2,0()2,0(2
2
1
2
1

Number of cycles 2k length:

v c v c v c
1a 2a 3a ga 1ga 2ga
3ga
...
lollipop walk from to 2,g .13   gg gg1a
  sssssccs vv
g
vvv
g
v
g
v
g
v
g PILLPPLHILHL 2)2,2()2,0(22)2,1()2,1()2,( 0,1max   
v c v
c
c
1a 2a 3a
4a
2ga
lollipop walk from to g,2 .33 agg 
1a









3
6 2
),1(),1()2,(
s
ccs
v
v
g
v
g
v
g
P
ILHILHL
Substract non-trail ways
Substract non-trail ways
Lollipop-matrix equation

Lollipop-matrix equation
 
  vv
vcv
v
g
v
v
g
v
g
v
g
LL
HPHLL
)2,3()2,0(
1)2,2()2,1(
0,1max 


 
 
  vvvvv
ccv
vvv
g
v
g
v
v
g
v
g
v
g
IPPPLL
HLdiagHLL
222)2,2()2,0(
)2,1()2,1()2,(
0,1max 




cv
LTo get just replace toH T
H
vv
L To
cv
L
 
 
  
HPH
P
I
HPHL
HLP
LL
HPLLHLL
v
v
vv
cv
vv
vvvcv
v
v
v
g
v
g
vv
g
v
g
v
v
g
vv
g
v
g
v
g





























3
2
1)2,1(
)2,0(1
)2,1()2,0(
1)2,1(),0()2,()2,1(
2
2
2
0,2max2
0,1max
  








 
3
6
2
2
),1(),1(),2(
1),0(),1(
v
ccv
vcv
v
v
g
v
g
v
g
v
g
v
g
v
g
P
IHLdiagHLL
HPHLL 
 
HPH
P
I
HP
P
HI
H
P
I
P
I
LLHLL
v
v
v
v
vv
vvcv
v
v
v
v
vv
v
g
vv
g
v























































3
2
3
2
33
),1()2,0(),2()2,3(
2
4
2
6
2
4
3
6
0,1max
HPH
P
I
HP
P
HI
H
P
I
HPHLL
v
v
v
c
v
vcv
v
v
v
v
v
v
g
v
g
v
g















































 
3
2
3
2
3
1)2,0()2,1(
2
2
2
2
2
2
2

Time Complexity for girth+4 enumerate
376 g of matrix multiplication
Storage Complexity for g+4 enumerate
)64(
])(33[
211111
22
bitsdoubleof
locatorcyclesininvolveforsymbolschecks
symbolscheckscheckssymbols


Enumerate cycles in QC-LPDC 4x20 with circulant 500 (2000x10000):
40 minutes and around 20Gb of RAM
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
336 317 31 359 288 179 232 25 117 78 235 11 200 360 434 228 414 83 30 344
465 421 81 147 439 83 108 49 144 478 378 279 398 427 393 287 204 362 353 62
444 235 64 74 43 427 198 437 313 358 203 334 5 211 462 255 154 439 151 296
Number of cycles in the 4x20x500 HQ matrix BICC.txt
8 10 12
1 401 250 59 867 000 2.8729*10^9
).( 3
nO

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
336 317 31 359 288 179 232 25 117 78 235 11 200 360 434 228 414 83 30 344
465 421 81 147 439 83 108 49 144 478 378 279 398 427 393 287 204 362 353 62
444 235 64 74 43 427 198 437 313 358 203 334 5 211 462 255 154 439 151 296
134.25 133.75 138.5 139 138.5 131.5 141 137.75 139.25 142.75 141 141.75 144.75 137 147 136.25 151.25 142.25 142 143
694
714.5
690
703.5
Show “problem” part of parity-check matrix under BP decoder
Help to compare matrix, choice better symbols to puncture, use different decoder
under subgraph and etc…
Show number of cycle 8 in which involved every checks and symbols
This information help to:
1. shortening code: (ordered list of symbols to puncture)
2. code extension: use symbols in less number of cycles
3. redesign code to improve properties using different labeling

Results of RA QC-LDPC construction
Page 12
Broadcom 5x20, circulant 600
380 451 126 471 409 259 329 40 98 360 203 348 175 302 485 406 -1 -1 -1 -1
15 534 442 549 296 540 90 361 108 310 57 134 414 367 487 277 241 -1 -1 -1
137 412 473 117 233 250 72 599 356 13 438 154 591 7 233 -1 58 364 -1 -1
430 188 162 122 576 474 408 107 114 444 561 80 374 185 371 -1 -1 397 17 -1
280 486 452 485 54 332 178 102 48 198 38 340 536 585 -1 -1 -1 -1 105 277
483 554 -1 469 18 254 19 95 290 92 -1 142 556 118 510 308 -1 -1 -1 -1
273 -1 114 -1 75 541 329 581 67 -1 353 -1 99 251 153 147 76 -1 -1 -1
-1 137 592 219 -1 159 251 484 -1 143 102 110 -1 249 349 -1 371 153 -1 -1
86 -1 464 163 341 -1 571 -1 343 115 315 188 450 -1 154 -1 -1 469 405 -1
597 497 -1 124 529 432 -1 172 162 151 281 66 213 325 -1 -1 -1 -1 235 461
Our 5x20, circulant 600

Hamming (7,4)
1 1 1 0 1 0 0
0 1 1 1 0 1 0
1 1 0 1 0 0 1 ;0
,,
)0,0()0,0(
00


sc
sc
vv
vv
LL
IPIP
   













022
202
220
1
TTv
HHdiagHHP c
   



























0001011
0001110
0000111
1100121
0111021
1112202
1011120
1 HHdiagHHP TTvs

  













300
030
003
1max)2,0(
Tv
HHdiagL c
  



























0000000
0000000
0000000
0001000
0000100
0000020
0000001
1max)2,0( HHdiagL Tvs













0001021
0001120
0000121
)2,0()2,1(
cc vv
LHL



























300
030
003
330
033
333
303
)2,0()2,1(
cc vTv
LHL

 













0221421
2021124
2204121
)2,1(
ccc vv
g
v
g LHPP
 



























022
202
220
114
411
222
141
Tv
g
v
g
cs
PP











400
040
004
)4,0( IHPL Tv
g
v cc
























0000000
0000000
0000000
0002000
0000200
0000060
0000002
)4,0( IHPL Tv
g
v ss


  34/)4,0(4  cv
LtraceN
 























0
0
0
00000000.50000000
00000000.50000000
0000001.50000000
00000000.50000000
4/_4_ )4,0(
sv
Ldiagsymbolsgirt
 











1
1
1
4/_4_ )4,0(
cv
Ldiagcheckgirt
1 1 1 0 1 0 0
0 1 1 1 0 1 0
1 1 0 1 0 0 1

In similar way we can continue and enumerate cycle 6: 4.
1 1 1 0 1 0 0
0 1 1 1 0 1 0
1 1 0 1 0 0 1
1 1 1 0 1 0 0
0 1 1 1 0 1 0
1 1 0 1 0 0 1
1 1 1 0 1 0 0
0 1 1 1 0 1 0
1 1 0 1 0 0 1
1 1 1 0 1 0 0
0 1 1 1 0 1 0
1 1 0 1 0 0 1
  46/)6,0(6  cv
LtraceN























0
0
0
1
1
1
1
_6_ symbolsgirt











1.33
1.33
1.33
_6_ checkgirt

Thank you
www.huawei.com
Implementation of lollipop cycle count
https://yadi.sk/d/nMza892Y3PVR3U

Girth point of view
Labeling problem
Target girth >=6 length cycle
00
00
10
0***
0***
0***
II
II
II
?*is

Girth point of view
Target girth >=6 length cycle
Circulant 8
00
00
10
0***
0***
0***
II
II
II
}1,0{,
0***
0***
0**
00
00
10
A
II
II
IIIA
}1,0{,
0***
0**
0**
00
00
100
B
II
III
III
B
}6,7{
08mod)016(
,
0**
0**
0**
00
006
100


C
c
III
III
III
c
}0{,
0**
0**
0*
003
006
100
D
III
III
IIII D
}6,0{,
0**
0*
0*
003
006
1000
E
III
IIII
IIII
E
}7,5,1,0,3{,
0
0
0
00813
00376
10000
J
IIIII
IIIII
IIIII

and so on for
Girth>4

Symbolic methods to estimation of short cycles in graph

Enumerating cycles in bipartite graph using matrix approach

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