Upper Bounding the Performance ofArbitrary Finite LDPC Codes on
Binary Erasure ChannelsAn efficient exhaustion algorithm for
error-prone patternsC.-C. Wang, S.R. Kulkarni, and H.V. Poor
School of Electrical & Computer Engineering, Purdue Unversity
Department of Electrical Engineering, Princeton University
Wang, Kulkarni, & Poor – p. 1/24
ContentBrief introduction
Stopping sets in erasure channels. Hardness & Applications.
Existing approaches for upper / lower bounding thefixed
code performance:
A tree-basedupper bound forbit error rates.
Frame error rates and trapping sets.
Wang, Kulkarni, & Poor – p. 2/24
Stopping Sets (SSs)Definition: a set of variable nodes such that the induced graph
contains no check node of degree 1.
Example: The binary (7,4) Hamming Code
i =i
1i
2i
3i
4i
5i
6i
7
1 2 3j =
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(2, 5, 7) a validcodewordvs. (4, 6, 7) a purestopping set.
Hamming distance vs. Stopping distance
Wang, Kulkarni, & Poor – p. 3/24
Stopping Sets (SSs)Definition: a set of variable nodes such that the induced graph
contains no check node of degree 1.
Example: The binary (7,4) Hamming Code
i =i
1yi
2i
3i
4yi
5i
6yi
7
1 2 3j =
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HHHHH
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(2, 5, 7) a validcodewordvs. (4, 6, 7) a purestopping set.
Hamming distance vs. Stopping distance
Wang, Kulkarni, & Poor – p. 3/24
Stopping Sets (SSs)Definition: a set of variable nodes such that the induced graph
contains no check node of degree 1.
Example: The binary (7,4) Hamming Code
i =i
1i
2i
3yi
4i
5yi
6yi
7
1 2 3j =
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HHHHH
@@@
(2, 5, 7) a validcodewordvs. (4, 6, 7) a purestopping set.
Hamming distance vs. Stopping distance
Wang, Kulkarni, & Poor – p. 3/24
Upper Bounds vs. ExhaustingMinimum Stopping Sets
An n = 72 regular (3,6) LDPC code
10−3
10−2
10−1
100
10−20
10−15
10−10
10−5
100
erasure prob ε
ber
V60: Monte−CarloV60: UB
Wang, Kulkarni, & Poor – p. 4/24
Upper Bounds vs. ExhaustingMinimum Stopping Sets
An n = 72 regular (3,6) LDPC code
10−3
10−2
10−1
100
10−20
10−15
10−10
10−5
100
erasure prob ε
ber
V60: Monte−CarloV60: UB
An NP-Complete Problem !!by Krishnanet al.
Wang, Kulkarni, & Poor – p. 4/24
A Hard but Useful ProblemAs an upper bound: Guaranteed worst performance for extremely
low ber regimes.
As an SS exhaustion algorithm: Code annealing [Wang 06].
0.2 0.25 0.3 0.35 0.4 0.45 0.510
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Erasure Probability ε
Err
or P
roba
bilit
y
Typical CLCL+d2optCL+CA(DA+CA)+CL+CA(DA+CA)+CL+CA, asym.Random Constr.Direct CA
Wang, Kulkarni, & Poor – p. 5/24
Existing ApproachesEnsemble analysis:[Di et al. 02] avg. ber and FER curves,
[Amraoui et al. 04] the scaling law for water fall regions.
Fixed Code analysis:[Holzlöhneret al. 05] dual adaptive
importance sampling, [Stepanovet al. 06] instanton method
Enumerating bad patterns: [Richardson 03] error floors of
LDPC codes, [Yedidiaet al. 01] projection algebra, [Huet al.
04] the minimum distance of LDPC codes.
They are allinexhaustive enumeration.
Wang, Kulkarni, & Poor – p. 6/24
The Bit-Oriented Detection
i =i
1i
2i
3i
4i
5i
6
1 2 3 4j =
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Using
1: for erasure
0: for non-erasure
Wang, Kulkarni, & Poor – p. 7/24
The Bit-Oriented Detection
i =i
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3i
4i
5i
6
1 2 3 4j =
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Using
1: for erasure
0: for non-erasure
f2,0 = x2
Wang, Kulkarni, & Poor – p. 7/24
The Bit-Oriented Detection
i =i
1i
2i
3i
4i
5i
6
1 2 3 4j =
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Using
1: for erasure
0: for non-erasure
f2,0 = x2
f2,1 = x2(x1 + x3)(x4 + x6)
Wang, Kulkarni, & Poor – p. 7/24
The Bit-Oriented Detection
i =i
1i
2i
3i
4i
5i
6
1 2 3 4j =
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Using
1: for erasure
0: for non-erasure
f2,0 = x2
f2,1 = x2(x1 + x3)(x4 + x6)
f2,2 = x2 (x1( f5→4,1 + f6→4,1) + x3( f4→3,1 + f5→3,1))
· (x4( f3→3,1 + f5→3,1) + x6( f1→4,1 + f5→4,1))
= x2 (x1(x5 + x6) + x3(x4 + x5))
· (x4(x3 + x5) + x6(x1 + x5))
Wang, Kulkarni, & Poor – p. 7/24
The Bit-Oriented Detection
i =i
1i
2i
3i
4i
5i
6
1 2 3 4j =
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Using
1: for erasure
0: for non-erasure
f2,0 = x2
f2,1 = x2(x1 + x3)(x4 + x6)
f2,2 = x2 (x1( f5→4,1 + f6→4,1) + x3( f4→3,1 + f5→3,1))
· (x4( f3→3,1 + f5→3,1) + x6( f1→4,1 + f5→4,1))
= x2 (x1(x5 + x6) + x3(x4 + x5))
· (x4(x3 + x5) + x6(x1 + x5))
f2 := liml→∞
f2,l = f2,2
Wang, Kulkarni, & Poor – p. 7/24
The Bit-Oriented Detection
i =i
1i
2i
3i
4i
5i
6
1 2 3 4j =
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Using
1: for erasure
0: for non-erasure
f2,0 = x2
f2,1 = x2(x1 + x3)(x4 + x6)
f2,2 = x2 (x1( f5→4,1 + f6→4,1) + x3( f4→3,1 + f5→3,1))
· (x4( f3→3,1 + f5→3,1) + x6( f1→4,1 + f5→4,1))
= x2 (x1(x5 + x6) + x3(x4 + x5))
· (x4(x3 + x5) + x6(x1 + x5))
f2 := liml→∞
f2,l = f2,2
p2 = E{ f2}Wang, Kulkarni, & Poor – p. 7/24
The Bit-Oriented Detection
i =i
1i
2i
3i
4i
5i
6
1 2 3 4j =
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Using
1: for erasure
0: for non-erasure
Asymptotically, no repeated variables
f2,0 = x2
f2,1 = x2(x1 + x3)(x4 + x6)
f2,2 = x2 (x1( f5→4,1 + f6→4,1) + x3( f4→3,1 + f5→3,1))
· (x4( f3→3,1 + f5→3,1) + x6( f1→4,1 + f5→4,1))
= x2 (x1(x5 + x6) + x3(x4 + x5))
· (x4(x3 + x5) + x6(x1 + x5))
f2 := liml→∞
f2,l = f2,2
p2 = E{ f2}Wang, Kulkarni, & Poor – p. 7/24
Enumeration-Based LB & UBf2 = x2 (x1(x5 + x6) + x3(x4 + x5)) (x4(x3 + x5) + x6(x1 + x5))
= x1x2x6 + x2x3x4 + x1x2x4x5 + x2x3x5x6
Each product term corresponds to anirreducible stopping set.
Wang, Kulkarni, & Poor – p. 8/24
Enumeration-Based LB & UBf2 = x2 (x1(x5 + x6) + x3(x4 + x5)) (x4(x3 + x5) + x6(x1 + x5))
= x1x2x6 + x2x3x4 + x1x2x4x5 + x2x3x5x6
Each product term corresponds to anirreducible stopping set.
LB: Graph-based search [Richardson 03], iteration-based
relaxation [Yedidiaet al. 01]
f2,LBa = x2x3x4
LBa = E{ f2,LB} = ǫ3
f2,LBb= x1x2x4x5
LBb = E{ f2,LBb} = ǫ4
Wang, Kulkarni, & Poor – p. 8/24
Enumeration-Based LB & UBf2 = x2 (x1(x5 + x6) + x3(x4 + x5)) (x4(x3 + x5) + x6(x1 + x5))
= x1x2x6 + x2x3x4 + x1x2x4x5 + x2x3x5x6
Each product term corresponds to anirreducible stopping set.
LB: Graph-based search [Richardson 03], iteration-based
relaxation [Yedidiaet al. 01]
f2,LBa = x2x3x4
LBa = E{ f2,LB} = ǫ3
f2,LBb= x1x2x4x5
LBb = E{ f2,LBb} = ǫ4
UB: Iteration-based relaxation [Yedidiaet al. 01]
f2,2 = x2x3x4 + x1x2x4x5 + x1x2x6 + x2x3x5x6
≤ x21x4 + 1x2x41 + 1x2x6 + x21 · 1x6 = x2x4 + x2x6
UB = 2ǫ2 − ǫ3Wang, Kulkarni, & Poor – p. 8/24
An Upper Bound
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Wang, Kulkarni, & Poor – p. 9/24
An Upper Bound
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EDE{ fa} < pe = E{ fa} = E{ fb} ≤ EDE{ fb}
Wang, Kulkarni, & Poor – p. 9/24
An Upper Bound
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Wang, Kulkarni, & Poor – p. 9/24
An Upper Bound
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Wang, Kulkarni, & Poor – p. 9/24
An Upper Bound (Cont’d)
Theorem 1 Suppose all messages entering any variable node are
independent, or equivalently, theyoungest common ancestorsof all
pairs of repeated bits arecheck nodes.
EDE{ f } ≥ E{ f }
Theorem 2 This upper bound is tight in order.
EDE{ f }(ǫ) = O(E{ f }(ǫ)), ∀ǫ
Wang, Kulkarni, & Poor – p. 10/24
A Pivoting Rule
f = x0 · f |x0=1 + f |x0=0
Wang, Kulkarni, & Poor – p. 11/24
A Pivoting Rule
f = x0 · f |x0=1 + f |x0=0
lf
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JJQQ
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x0
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lf |x0=0
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(a) Original (b) Decoupled
Wang, Kulkarni, & Poor – p. 11/24
The Two-Stage Algorithm
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Wang, Kulkarni, & Poor – p. 12/24
The Two-Stage Algorithm
f∞
Wang, Kulkarni, & Poor – p. 12/24
The Two-Stage Algorithm
f∞ ≤ fFINITE
Wang, Kulkarni, & Poor – p. 12/24
The Two-Stage Algorithm
f∞ ≤ fFINITE
Wang, Kulkarni, & Poor – p. 12/24
The Two-Stage Algorithm
f∞ ≤ fFINITE = f̃FINITE
Wang, Kulkarni, & Poor – p. 12/24
The Two-Stage Algorithm
f∞ ≤ fFINITE = f̃FINITE
≤w. the same order
≤pe = E{ f∞} E{ fFINITE} EDE{ f̃FINITE}
Wang, Kulkarni, & Poor – p. 12/24
The Two-Stage Algorithm
f∞ ≤ fFINITE = f̃FINITE
≤w. the same order
≤pe = E{ f∞} E{ fFINITE} EDE{ f̃FINITE}
Wang, Kulkarni, & Poor – p. 12/24
The Two-Stage Algorithm
f∞ ≤ fFINITE = f̃FINITE
≤w. the same order
≤pe = E{ f∞} E{ fFINITE} EDE{ f̃FINITE}
Wang, Kulkarni, & Poor – p. 12/24
The Two-Stage Algorithm
f∞ ≤ fFINITE = f̃FINITE
≤w. the same order
≤pe = E{ f∞} E{ fFINITE} EDE{ f̃FINITE}
Wang, Kulkarni, & Poor – p. 12/24
The Two-Stage Algorithm
f∞ ≤ fFINITE = f̃FINITE
≤w. the same order
≤pe = E{ f∞} E{ fFINITE} EDE{ f̃FINITE}
Wang, Kulkarni, & Poor – p. 12/24
The Two-Stage Algorithm
f∞ ≤ fFINITE = f̃FINITE
≤w. the same order
≤pe = E{ f∞} E{ fFINITE} EDE{ f̃FINITE}
Wang, Kulkarni, & Poor – p. 12/24
The Two-Stage Algorithm
f∞ ≤ fFINITE = f̃FINITE
≤w. the same order
≤pe = E{ f∞} E{ fFINITE} EDE{ f̃FINITE}
Wang, Kulkarni, & Poor – p. 12/24
The Two-Stage Algorithm
f∞ ≤ fFINITE = f̃FINITE
≤w. the same order
≤pe = E{ f∞} E{ fFINITE} EDE{ f̃FINITE}
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Wang, Kulkarni, & Poor – p. 12/24
A Narrowing SearchTheorem 3 (Asymptotically Tight) With an“optimal" growing
module, we have
UB(ǫ) = O(pe(ǫ)).
Wang, Kulkarni, & Poor – p. 13/24
A Narrowing SearchTheorem 3 (Asymptotically Tight) With an“optimal" growing
module, we have
UB(ǫ) = O(pe(ǫ)).
Theorem 4 (Exhaustive Enumeration)
d := min{size(x)|∀x such thatf̃FINITE(x) = 1}. Then the
stopping distance≥ d.
If there exists such a minimumx being a SS, then ALL minimum
SSs are in the set of all minimumx.
The exhaustive list of minimum SSs leads to a lower boundtight
in bothorderandmultiplicity.
Wang, Kulkarni, & Poor – p. 13/24
The Two-Stage Algorithm
f∞ ≤ fFINITE = f̃FINITE
≤ ≤pe = E{ f∞} E{ fFINITE} EDE{ f̃FINITE}
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Wang, Kulkarni, & Poor – p. 14/24
Numerical ExperimentsThe (7,4,3) Hamming code:
H =
0 0 1 1 1 0 1
0 1 0 1 0 1 1
1 0 0 0 1 1 1
Wang, Kulkarni, & Poor – p. 15/24
Numerical ExperimentsThe (7,4,3) Hamming code:
H =
0 0 1 1 1 0 1
0 1 0 1 0 1 1
1 0 0 0 1 1 1
∀i ∈ [1, 7], ∀ǫ ∈ (0, 1], UBi(ǫ)pi(ǫ)
≤ 1.3.
Wang, Kulkarni, & Poor – p. 15/24
Numerical ExperimentsThe (7,4,3) Hamming code:
H =
0 0 1 1 1 0 1
0 1 0 1 0 1 1
1 0 0 0 1 1 1
∀i ∈ [1, 7], ∀ǫ ∈ (0, 1], UBi(ǫ)pi(ǫ)
≤ 1.3.
The (23,12,7) Golay code:
H = [H′I] in which H
′ =
1 0 0 1 1 1 0 0 0 1 1 1
1 0 1 0 1 1 0 1 1 0 0 1
1 0 1 1 0 1 1 0 1 0 1 0
1 0 1 1 1 0 1 1 0 1 0 0
1 1 0 0 1 1 1 0 1 1 0 0
1 1 0 1 0 1 1 1 0 0 0 1
1 1 0 1 1 0 0 1 1 0 1 0
1 1 1 0 0 1 0 1 0 1 1 0
1 1 1 0 1 0 1 0 0 0 1 1
1 1 1 1 0 0 0 0 1 1 0 1
0 1 1 1 1 1 1 1 1 1 1 1
Wang, Kulkarni, & Poor – p. 15/24
Numerical Experiments (cont’d)The (23,12,7) Golay code:
bit 0, (4*,75*)
10−3
10−2
10−1
100
10−10
10−8
10−6
10−4
10−2
100
erasure prob ε
ber
V0: MC−SV0: UBV0: C−UBV0: LB
Wang, Kulkarni, & Poor – p. 16/24
Numerical Experiments (cont’d)A fixed, finite LDPC code with var deg. 3, chk deg. 6,n = 50:
bit 0, (4*,1*)
10−3
10−2
10−1
100
10−15
10−10
10−5
100
erasure prob ε
ber
V0: MC−SV0: UBV0: C−UBV0: LB
Wang, Kulkarni, & Poor – p. 17/24
Numerical Experiments (cont’d)A fixed, finite LDPC code with var deg. 3, chk deg. 6,n = 50:
bit 26, (6*,2*)
10−3
10−2
10−1
100
10−15
10−10
10−5
100
erasure prob ε
ber
V26: MC−SV26: UBV26: C−UBV26: LB
Wang, Kulkarni, & Poor – p. 17/24
Numerical Experiments (cont’d)A fixed, finite LDPC code with var deg. 3, chk deg. 6,n = 50:
bit 19, (7*,10 → 5*)
10−3
10−2
10−1
100
10−15
10−10
10−5
100
erasure prob ε
ber
V19: MC−SV19: UBV19: C−UBV19: LB
Wang, Kulkarni, & Poor – p. 17/24
Frame Error Rates and IrregularCodes
Frame error rate (FER):fFER = ∑ni=1 fi.
Wang, Kulkarni, & Poor – p. 18/24
Frame Error Rates and IrregularCodes
Frame error rate (FER):fFER = ∑ni=1 fi.
Efficiency:
More cycles=⇒ less efficiency. (The Golay code is the
worst.)
The largern, the easier for our algorithm.
Wang, Kulkarni, & Poor – p. 18/24
Frame Error Rates and IrregularCodes
Frame error rate (FER):fFER = ∑ni=1 fi.
Efficiency:
More cycles=⇒ less efficiency. (The Golay code is the
worst.)
The largern, the easier for our algorithm.
Experimental results
Consider codes ofn = 500–1000.
For regular codes,dSS ≤ 12 can be exhausted.
((51212 ) = 5.9 × 1023 trials)
TheFERof irregular codessuits the algorithm most.
dSS ≤ 13 can be exhausted. ((51213 ) = 2.5 × 1025 trials)
Wang, Kulkarni, & Poor – p. 18/24
The FER of A Rate 1/2 IrregularCode
λ(x) = 0.416667x + 0.166667x2 + 0.416667x5
ρ(x) = x5. 61% variable nodes of degree 2
A rate 1/2n = 572 irregular LDPC code, (order, multi)=(13*,104*)
0.2 0.25 0.3 0.35 0.4 0.45 0.510
−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Erasure Probability ε
Err
or P
roba
bilit
y
Typical CLCL+d2optCL+CA(DA+CA)+CL+CA(DA+CA)+CL+CA, asym.Random Constr.Direct CA
Wang, Kulkarni, & Poor – p. 19/24
SummaryUpper bounding and exhausting the bad patterns
An NP-completeproblem butfeasiblefor at least short practical
LDPC codes.
Wang, Kulkarni, & Poor – p. 20/24
SummaryUpper bounding and exhausting the bad patterns
An NP-completeproblem butfeasiblefor at least short practical
LDPC codes.
Taking advantages of thetree-likestructure.
Wang, Kulkarni, & Poor – p. 20/24
SummaryUpper bounding and exhausting the bad patterns
An NP-completeproblem butfeasiblefor at least short practical
LDPC codes.
Taking advantages of thetree-likestructure.
Suitable for bothFERandber, punctured codes, unequal
sub-channel analysis, non-sparse codes, etc.
Wang, Kulkarni, & Poor – p. 20/24
SummaryUpper bounding and exhausting the bad patterns
An NP-completeproblem butfeasiblefor at least short practical
LDPC codes.
Taking advantages of thetree-likestructure.
Suitable for bothFERandber, punctured codes, unequal
sub-channel analysis, non-sparse codes, etc.
Our algorithm can be easily modified fortrapping sets.
Wang, Kulkarni, & Poor – p. 20/24
SummaryUpper bounding and exhausting the bad patterns
An NP-completeproblem butfeasiblefor at least short practical
LDPC codes.
Taking advantages of thetree-likestructure.
Suitable for bothFERandber, punctured codes, unequal
sub-channel analysis, non-sparse codes, etc.
Our algorithm can be easily modified fortrapping sets.
A useful tool forfinite code optimization.
Wang, Kulkarni, & Poor – p. 20/24
SummaryUpper bounding and exhausting the bad patterns
An NP-completeproblem butfeasiblefor at least short practical
LDPC codes.
Taking advantages of thetree-likestructure.
Suitable for bothFERandber, punctured codes, unequal
sub-channel analysis, non-sparse codes, etc.
Our algorithm can be easily modified fortrapping sets.
A useful tool forfinite code optimization.
A starting point for more efficient algorithms.
Wang, Kulkarni, & Poor – p. 20/24
Thank you for the attention.
Wang, Kulkarni, & Poor – p. 21/24
Searching for Trapping SetsWe define thek-out trapping set, namely, the induced graph hask
check node of degree 1.
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Theorem 5 Consider a fixedk. Determining thek-out trapping
distance is NP-complete.
Our algorithm can be modified for trapping set exhaustion.
Wang, Kulkarni, & Poor – p. 22/24
Some NotesComplexity of evaluatingUB: O(|T | log(|T |)), even for
arbitrarily smallǫ.
Wang, Kulkarni, & Poor – p. 23/24
Some NotesComplexity of evaluatingUB: O(|T | log(|T |)), even for
arbitrarily smallǫ.
A pleasant byproduct: anexhaustivelist of minimum SS leading
to a lower bound tight in bothorderandmultiplicity.
Wang, Kulkarni, & Poor – p. 23/24
Some NotesComplexity of evaluatingUB: O(|T | log(|T |)), even for
arbitrarily smallǫ.
A pleasant byproduct: anexhaustivelist of minimum SS leading
to a lower bound tight in bothorderandmultiplicity.
Complexity and performance tradeoff.
Wang, Kulkarni, & Poor – p. 23/24
Some NotesComplexity of evaluatingUB: O(|T | log(|T |)), even for
arbitrarily smallǫ.
A pleasant byproduct: anexhaustivelist of minimum SS leading
to a lower bound tight in bothorderandmultiplicity.
Complexity and performance tradeoff.
Puncturedvs. ShortenedCodes
Wang, Kulkarni, & Poor – p. 23/24
Some NotesComplexity of evaluatingUB: O(|T | log(|T |)), even for
arbitrarily smallǫ.
A pleasant byproduct: anexhaustivelist of minimum SS leading
to a lower bound tight in bothorderandmultiplicity.
Complexity and performance tradeoff.
Puncturedvs. ShortenedCodes
A partitioned approach— a hybrid search
E{ f } = ∑j
E{Aj}E{ f |Aj}.
Ex: A1 = {x3 = 0}, A2 = {x3 = 1, x7 = 0}, and
A3 = {x3 = 1, x7 = 1}.
Wang, Kulkarni, & Poor – p. 23/24
The Tree Converting Algorithm1: repeat2: Find the next leaf variable node, sayxj.
3: if there exists another non-leafxj in T then4: if theyoungest common ancestorof the leafxj and existing
non-leafxj, denoted asyca(xj), is a check nodethen5: Include the newxj in T .
6: else if yca(xj) is avariable nodethen7: Do thepivotingconstruction.
8: end if9: end if
10: Construct the immediate children ofxj.
11: until the size ofT exceeds the preset limit.
Wang, Kulkarni, & Poor – p. 24/24