On Random Linear Network Coding for Butterfly Network

0.07 MB·English

by  Xuan Guang

1
On Random Linear Network Coding for Butterfly
Network
Xuan Guang, and Fang-Wei Fu
Abstract—Randomlinearnetworkcodingisafeasibleencoding
tool for network coding, specially for the non-coherent network,
d d
and its performance is important in theory and application. In 1 2
0 this letter, we study the performance of random linear network e s e
1 2
1 coding for the well-known butterfly network by analyzing the
0 failure probabilities. We determine the failure probabilities of s1 e e s2
2 random linear network coding for the well-known butterfly net- 4 5
work and the butterflynetwork with channel failureprobability
n p. i
a
J IndexTerms—Networkcoding,randomlinearnetworkcoding, e3 e7 e6
butterfly network, failure probabilities.
8
1 j

I. INTRODUCTION e e
] t 8 9 t
T IN the seminal paper [1], Ahlswede et al showed that with 1 2
I network coding, the source node can multicast informa-
s. tion to all sink nodes at the theoretically maximum rate as Fig. 1. Butterfly Network where d1,d2 represent two imaginary incoming
c channels ofsourcenodes.
[ the alphabet size approaches infinity, where the theoretically
maximum rate is the smallest minimum cut capacity between
2
the source node and any sink node. Li et al [2] showed that
v 1.Itisobviousthatbothvaluesofthemaximumflowsforthe
linear network coding with finite alphabet is sufficient for
7 sink nodes t and t are 2.
1 2
7 multicast. Moreover, the well-known butterfly network was Now we consider random linear network coding problem
0 proposedin papers [1] [2] to show the advantagesof network
with the information rate 2 for this butterfly network G. That
2 coding well compared with routing. Koetter and M´edard [3]
is to say, for source node s and internal nodes, all local
.
1 presentedanalgebraiccharacterizationofnetworkcoding.Ho
encoding coefficients are independently uniformly distributed
0 etal[4] proposedthe randomlinearnetworkcodingandgave
randomvariablestakingvaluesinthebasefieldF.Forsource
0 several upper bounds on the failure probabilities of random
1 node s, although it has no incoming channels, we use two
linearnetworkcoding.Balli, Yan,andZhang[5]improvedon
v: these bounds and discussed the limit behavior of the failure imaginary incoming channels d1,d2 and assume that they
provide the source messages for s. Let the global encoding
i
X probability as the field size goes to infinity. 1 0
r In this paper, we study the failure probabilities of random kernels of fd1 = (cid:18)0(cid:19) and fd2 = (cid:18)1(cid:19), respectively. The
a linear network coding for the well-known butterfly network local encoding kernel at the source node s is denoted as
when it is considered as a single source multicast network k d
K = d1,1 d1,2 .
described by Fig. 1, and discuss the limit behaviors of the s (cid:18)kd2,1 dd2,2(cid:19)
failure probabilities as the field size goes to infinity. Similarly, we denote by fi the global encoding kernel
of channel e (1 ≤ i ≤ 9), and k the local encoding
i i,j
II. FAILUREPROBABILITIES OFRANDOM LINEAR coefficient for the pair (ei,ej) with head(ei) = tail(ej) as
NETWORK CODING FORBUTTERFLY NETWORK described in [6, p.442], where tail(ei) represents the node
whose outgoing channels include e , head(e ) represents the
In this letter, we always denote G = (V,E) the butterfly i i
node whose incoming channels include e :
network shown by Fig. 1, where the single source node is i
s, the set of sink nodes is T = {t1,t2}, the set of internal f =k f +k f = kd1,1 ,
nodes is J = {s1,s2,i,j}, and the set of the channels is 1 d1,1 d1 d2,1 d2 (cid:18)kd2,1(cid:19)
E = {e ,e ,e ,e ,e ,e ,e ,e ,e }. Moreover, we assume
1 2 3 4 5 6 7 8 9 k
that each channel e∈E is error-free, and the capacity is unit f2 =kd1,2fd1 +kd2,2fd2 =(cid:18)kdd21,,22(cid:19),
This research is supported in part by the National Natural Science Foun- k k k k
datXio.nGoufanCghiinsawuinthdetrhteheChGerranntIsns6t0it8u7te20o2f5Manadthe1m09a9ti0c0s1,1N.ankai University, f3 =k1,3f1 =(cid:18)kdd21,,11k11,,33(cid:19),f4 =k1,4f1 =(cid:18)kdd21,,11k11,,44(cid:19),
Tianjin300071,P.R.China. Email:[email protected]
k k k k
F.-W. Fu is with the Chern Institute of Mathematics and LPMC, Nankai f =k f = d1,2 2,5 ,f =k f = d1,2 2,6 ,
University, Tianjin300071,P.R.China.Email:[email protected] 5 2,5 2 (cid:18)kd2,2k2,5(cid:19) 6 2,6 2 (cid:18)kd2,2k2,6(cid:19)
2
k k k +k k k
f7 =k4,7f4+k5,7f5 =(cid:18)kdd21,,11k11,,44k44,,77+kdd21,,22k22,,55k55,,77(cid:19), KThseBn2,.wMeokrenoovwerf,rformom(1L)eamnmda(21), wtheathFavt1e = KsB1, Ft2 =
k k k k +k k k k
f8 =k7,8f7 =(cid:18)kdd12,,11k11,,44k44,,77k77,,88+kdd12,,22k22,,55k55,,77k77,,88(cid:19), 1−Pe(t1)==PPrr(({Rdaentk(K(Ft)1)6==0}2)∩={dPert((Bdet)(F6=t10)}6=) 0)
s 1
f =k f = kd1,1k1,4k4,7k7,9+kd1,2k2,5k5,7k7,9 =Pr(det(Ks)6=0)·Pr(det(B1)6=0)
9 7,9 7 (cid:18)kd2,1k1,4k4,7k7,9+kd2,2k2,5k5,7k7,9(cid:19) (|F|+1)(|F|−1)2 |F|−1 4
and the decoding matrices of the sink nodes t1,t2 are = |F|3 (cid:18) |F| (cid:19)
k k k k k k +k k k k (|F|+1)(|F|−1)6
Ft1 =(cid:18)kdd12,,11k11,,33 kdd12,,11k11,,44k44,,77k77,,88+kdd12,,22k22,,55k55,,77k77,,88(cid:19), = |F|7 .
(1)
Similarly,wealsohave1−P (t )=(|F|+1)(|F|−1)6/|F|7.
k k k k k k +k k k k e 2
Ft2 =(cid:18)kdd12,,22k22,,66 kdd12,,11k11,,44k44,,77k77,,99+kdd12,,22k22,,55k55,,77k77,,99(cid:19). aboNveexta,nwaleyscios,nswideehratvhee failure probability Pe. Similar to the
(2)
1−P =Pr({Rank(F )=2}∩{Rank(F )=2})
e t1 t2
A. Butterfly Network without Fail Channels
=Pr({det(F )6=0}∩{det(F )6=0})
t1 t2
Next,webegintodiscussthefailureprobabilitiesofrandom
=Pr({det(K )6=0}∩{det(B )6=0}∩{det(B )6=0})
s 1 2
linear network coding for the butterfly network G. These
=Pr(det(K )6=0)·Pr(det(B )6=0)·Pr(det(B )6=0)
failure probabilities can illustrate the performance of the s 1 2
(|F|+1)(|F|−1)10
randomlinear networkcoding for the butterfly network.First,
= .
we give the definitions of failure probabilities. |F|11
Definition 1: For random linear network coding for the
Moreover, since P = Pe(t1)+Pe(t2), then we have
butterfly network G with information rate 2: av 2
• pPreo(btia)bil,ityoPfrs(iRnkannko(dFettii),th<ati2s)thiespcraolbleadbiltihtyethfaaitluthree Pav =1− (|F|+1|F)(||F7 |−1)6 .
messages cannot be decoded correctly at t (i=1,2);
i
• Pe ,Pr(∃ t∈T such thatRank(Ft)<2) is called the This completes the proof.
failure probability of the butterfly network G, that is the It is known that if |F|≥|T|, there exists a linear network
probabilitythatthemessagescannotbedecodedcorrectly code C such that all sink nodes can decode successfully w=
at at least one sink node in T; mint∈T Ct symbolsgeneratedbythesourcenodes,whereCt
• Pav , Pt∈|TTP|e(t) iscalledtheaveragefailureprobability is the minimum cut capacity between s and the sink node t.
of all sink nodes for the butterfly network G. For butterfly network, |F|≥2 is enough. However, when we
consider the performance of random linear network coding,
Todeterminethefailureprobabilitiesofrandomlinearnetwork
the result is different.
coding for the butterfly network G, we need the following
For|F|=2,wehave1−P = 3 ≈0.001.Inotherwords,
lemma. e 211
this successful probability is too much low for application.
Lemma 1: Let M be a uniformly distributed random 2×2
In the same way, for |F| = 3 and |F| = 4, the successful
matrix over a finite field F, then the probability that M is
invertible is (|F|+1)(|F|−1)2/|F|3. probabilitiesare approximately0.023 and 0.070, respectively.
Thesesuccessfulprobabilitiesarestilltoolowforapplication.
Theorem 1: For random linear network coding for the but-
terfly network G with information rate 2: If 1−Pe ≥ 0.9 is thought of enough for application, we
have to choose a very large base field F with cardinality not
• the failure probability of the sink node ti (i=1,2) is
less than 87. This is because
(|F|+1)(|F|−1)6
P (t )=1− ; (|F|+1)(|F|−1)10
e i |F|7 ≥0.9⇐⇒|F|≥87 .
|F|11
• the failure probability of the butterfly network G is
(|F|+1)(|F|−1)10 B. Butterfly Network with Fail Channels
P =1− ;
e |F|11
In this subsection, as in [4], we will consider the failure
• the average failure probability is probabilitiesofrandomlinearnetworkcodingforthebutterfly
(|F|+1)(|F|−1)6 network with channel failure probability p (i.e., each channel
Pav =1− |F|7 . is possibly deleted from the network with probability p).
Generallyspeaking,channelfailureis alowprobabilityevent,
Proof: Recall that Ks = (f1 f2) is the local encoding i.e.,0≤p≪ 1.Infact,ifp=0,itisjustthebutterflynetwork
2
kernel at the source node s. Denote without fail channels discussed in the above subsection.
k k k k 0 k k k Theorem 2: For the random linear network coding of the
B , 1,3 1,4 4,7 7,8 ,B , 1,4 4,7 7,9 .
1 (cid:18) 0 k2,5k5,7k7,8(cid:19) 2 (cid:18)k2,6 k2,5k5,7k7,9(cid:19) butterfly network G with channel failure probability p,
3
=Pr({Rank((f f ))=2}∩{δ ,δ ,δ ,δ ,δ ,δ ,δ ,δ ,δ }
• the failure probability of the sink node ti (i=1,2) is 1 2 1 2 3 4 5 6 7 8 9
∩{k ,k ,k ,k ,k ,k ,k ,k 6=0})
1,3 2,5 5,7 7,8 2,6 1,4 4,7 7,9
(|F|+1)(|F|−1)6
P˜e(ti)=1− |F|7 (1−p)6 ; =Pr(Rank((f1 f2))=2)·Pr(δ1,δ2,δ3,δ4,δ5,δ6,δ7,δ8,δ9)
·Pr({k ,k ,k ,k ,k ,k ,k ,k 6=0})
1,3 2,5 5,7 7,8 2,6 1,4 4,7 7,9
• the failure probability of the butterfly network G is (|F|+1)(|F|−1)10
= (1−p)9.
(|F|+1)(|F|−1)10 |F|11
P˜ =1− (1−p)9 ;
e |F|11 At last, the average probability P˜ is given by
av
• the average failure probability is P˜ (t )+P˜ (t ) (|F|+1)(|F|−1)6
P˜ =1− (|F|+1)(|F|−1)6(1−p)6 . P˜av = e 1 2 e 2 =1− |F|7 (1−p)6 .
av |F|7
The proof is completed.
Proof:Atfirst,weconsiderthefailureprobabilityP˜ (t ).
e 1
Foreachchannele ,we call“channele issuccessful”if e is III. THELIMIT BEHAVIORS OF THEFAILURE
i i i
not deleted from the network, and define δ as the event that PROBABILITIES
i
“ei is successful”. We define f˜i as the active globalencoding In this section, we will discuss the limit behaviors of the
kernel of ei, where failure probabilities. From Theorem 2, we have
f˜ = fi, ei is successful, lim P˜e(ti)= lim P˜av =1−(1−p)6 ,(i=1,2) ,
i (cid:26) 0 , otherwise. |F|→∞ |F|→∞
lim P˜ =1−(1−p)9 .
e
andf = k f˜.Thenthedecodingmatrix |F|→∞
i j:head(ej)=tail(ei) j,i j
of the sinPk node t is F = (f˜ f˜). Hence, P˜ (t ) = Specially, if we consider the random linear network coding
1 t1 3 8 e 1
Pr(Rank((f˜ f˜))6=2). for the butterfly network without fail channels, i.e. p=0, we
3 8
Note that the event “Rank((f˜ f˜)) = 2” is equivalent to have (also from Theorem 1)
3 8
the event “Rank((f f )) = 2, δ , and δ ”. Moreover, since
3 8 3 8 lim P (t )= lim P = lim P =0 , (i=1,2).
(f3 f8) = (k1,3f˜1 k7,8f˜7), we have Rank((f3 f8)) = 2 if |F|→∞ e i |F|→∞ e |F|→∞ av
and only if Rank((f˜1 f˜7)) = 2 and k1,3 6= 0,k7,8 6= 0. That is, for the sufficient large base field F, the failure
Similarly, the event “Rank((f˜1 f˜7)) = 2” is equivalent probabilities can become arbitrary small. In fact, this result
to the event “Rank((f1 f7)) = 2, δ1, and δ7”. Note that is correct for all single source multicast network coding [4].
f˜4 = 0 or f4, and f4 = k1,4f˜1, so f˜4 = 0 or k1,4f1. Since It is easy to see from Theorem 1 that the rates of Pe(ti)
iff7R=ankk4,(7(ff˜41+f˜5k)5),7=f˜5,2waendhakv5e,7R6=an0k.(G(fo1infg7)o)n=w2ithiftahnedsoanmlye oapfpProaacphpinrogac0hianngd0Piasv a9p.proaching 0 are |F5|, and the rate
analysisasabove,wehavethattheevent“Rank((f f˜))=2” e |F|
1 5
is equivalent to the event “Rank((f f )) = 2, and δ ”, and
1 5 5 IV. CONCLUSION
the event “Rank((f f )) = 2” is equivalent to the event
“Rank((f f )) = 21, k5 6= 0, and δ ” since f = k f˜. The performance of random linear network coding is illus-
Therefore,1th2e event “R2a,5nk((f˜ f˜)) =2 2” is eq5uivale2n,5t t2o trated by the failure probabilities of random linear network
3 8 coding. In general, it is very difficult to calculate the failure
the event “Rank((f f )) = 2, δ , δ , δ , δ , δ , δ , and
1 2 1 2 3 5 7 8 probabilities of random linear network coding for a general
k ,k ,k ,k 6=0”. Hence, we have
1,3 2,5 5,7 7,8 communicationnetwork.Inthisletter,wedeterminethefailure
1−P˜ (t )=Pr(Rank((f˜ f˜))=2) probabilities of random linear network coding for the well-
e 1 3 8
knownbutterfly network and the butterfly network with chan-
=Pr({Rank(f f )=2}∩{δ ,δ ,δ ,δ ,δ ,δ }
1 2 1 2 3 5 7 8
nel failure probability p.
∩{k ,k ,k ,k 6=0})
1,3 2,5 5,7 7,8
=Pr({Rank(f1 f2)=2})·Pr({δ1,δ2,δ3,δ5,δ7,δ8}) REFERENCES
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In the same way, we can get
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3 8 6 9