areduc(onapproachtothemul(ple2 …pages.cpsc.ucalgary.ca/~zongpeng/publications/slides... ·...
TRANSCRIPT
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A Reduc(on Approach to the Mul(ple-‐Unicast Conjecture in Network Coding
Zongpeng Li
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What is Network Coding?
• Encoding data during a mul(-‐hop transmission – mul(ple unicasts – mul(cast
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Coding Advantage
• Improve throughput for mul(cast
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Coding Advantage
• Improve throughput for mul(ple-‐unicast
t2 t1
s1 s2 a b
a+b a b
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Coding Advantage
• Save bandwidth – Network Coding: 9 bits – Rou(ng: 10 bits
t2 t1
s1 s2 a b
a+b a b
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Network Models
Directed Networks • Not necessarily bidirec(onal • A pair of reverse links each
has its own capacity
Undirected Networks • Bidirec(onal • Capacity can be freely
allocated to two direc(ons
2
3 6 4
4
5
6 4
4
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Coding Advantage in Undirected Networks
• Improve throughput for mul(cast – Up to a bounded factor
• Network Coding: 2 bps • Rou(ng: 1.875 bps
LeVer: 0.25bps; Number: 0.125bps
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Coding Advantage in Undirected Networks
• Reduce cost for mul(cast
Rou(ng: 4.64 Network Coding: 4.57
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Mul(ple-‐Unicast + Undirected Networks?
Coding advantage vanishes!
a
a
b
ba+b
a+ba+b
s1
t2
a1
b1
a1
b1
b2a1
a2
b2
a2
b2
a2 b1
s1
t2
2
t1
s2
t1
s
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Another Example
a
a b b c
ca+b
a+b
a+b
b+c
b+ca+ca+b+c
a
a
b c
c b
b+c
a b c
ac b
a2
c1
a1
c1
b1c1
b2
c1
c1
b2
c2
b2
a1 b1
a1
c2
a1b2
c2b1
c2b1
c2
a2 a2b1 a2
b2
a1a2
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The Conjecture
In terms of improving throughput or saving bandwidth, Network coding has no advantage over rou(ng for mul(ple unicast sessions in undirected networks. [Li and Li 2004]
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Comments
• Mitzenmacher : No.1 of seven open problems in network coding (2007)
• Chekuri : “bold conjecture”, the problem of fully understanding network coding for mul(ple unicast sessions is s(ll “wild open”.
• Adler : “arguably the most important open problem in the field of network coding” (2006)
• The conjecture implies an affirma(ve answer to a 28-‐year-‐old open problem.
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Verified Cases
• 2 unicast sessions • Terminal co-‐face planar networks • Complete networks with uniform link length • Grid networks with uniform link length and aligned source-‐receivers
• Each source is closer to its receiver than other receivers
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Verified Cases
• Okamura-‐Seymour Network (K3,2)
• Hu’s 3-‐commodity network
• Complete bipar(te networks with uniform link length
s1 t1
t3
s3
s2
t2
t4
s4
s1 t1 t3 s3
s2
t2
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Overview of our reduc(on approach
Undirected)Networks�
)�
Atom)Networks�
…�…�
Decompose�
Cut6set)Bound:)No� Theorem)1:)No�
) �
Require)Coding?�
Assemble� Theorem)3:)No)need)to)code)in)networks)that)can)be)decomposed)into)these)atoms)networks.�
Theorem)2:))when)&)how)to)decompose�
?�
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Highlights of our results
• Generalize proofs of verified cases • Prove the conjecture for up to 6 nodes & most 7-‐node networks
• Find an interes(ng example where new techniques may be necessary
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Cost Domain
• Link capacity is ignored • Each link is assigned with a non-‐nega(ve length le
• Let fe denote the amount of informa(on transmiVed on link e
• Cost: Σe fe le
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Rela(ons Between Cost Domain and Throughput Domain
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The conjecture in Cost Domain
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Basic Techniques -‐-‐ inequali(es
• Cut-‐set: a set of edges dividing nodes into two parts
• Cut-‐set bound: F fe
e∈F∑ ≥ H (Xi )
i∈Sep(F )∑
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Example for the cut-‐set bound
• Unit link length • For each cut-‐set Fj:
• Sum up:
t2 t1
s1 s2
fee∈Fj
∑ ≥ H (X1)+H (X2 )
F1
F2 F3
fee∈E∑ ≥ 3H (X1)+3H (X2 )
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Overview of our reduc(on approach
Undirected)Networks�
)�
Atom)Networks�
…�…�
Decompose�
Cut6set)Bound:)No� Theorem)1:)No�
) �
Require)Coding?�
Assemble� Theorem)3:)No)need)to)code)in)networks)that)can)be)decomposed)into)these)atoms)networks.�
Theorem)2:))when)&)how)to)decompose�
?�
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An observa(on
• Under the condi(on Network coding is necessary in G1 iff it is necessary in G2
t2 t1
s1 s2
t2 t1
s1 s2
F2
fee∈F2
∑ ≥ H (X1)+H (X2 )
Contract edges in F2
G1 G2
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Generalize the idea
• Cut-‐set à Arbitrary edge set F • The problem is about the condi(on:
fee∈F2
∑ ≥ H (X1)+H (X2 )
fee∈F∑ ≥ ?
i∑ H (Xi )
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An Equivalent form of the conjecture
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Explana(on
• An edge set F decomposes G in to G/F and G/F.
t2 t1
s1 s2
t2 t1
s1 s2
F2
Decompose
t2 t1
s1 s2
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• As long as the decomposi(on preserves the distance between each pair of source-‐receiver:
• Network coding is unnecessary in G/F and G/F è it is unnecessary in G. – Cost of Network Coding: – Cost of Rou(ng:
fee∈E∑ = fe
e∈F∑ + fe
e∈F∑
dG (si, ti )H (Xi )i∑ = dG/F (si, ti )H (Xi )
i∑ + dG/F (si, ti )H (Xi )
i∑
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When there exists a decomposi(on
• An example – dG(s,t) = 2 – dG/F(s,t) =dG/F(s,t) = 0
• A path p in G à length |p F| in G/F length |p F| in G/F • There exist two shortest paths p1,p2 in G:
|p1 F|≠|p2 F|
s
t
F
∩
∩
∩ ∩
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When there exists a decomposi(on
• Another example • Observa(on: – Non-‐shortest paths have some redundancy
– Shortest paths intersect F the minimum (me
s
t F
1 > 0 + 0
2 = 2 + 0
scale up link length
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When there exists a decomposi(on
Theorem 2 If there is an edge set F that is compa(ble with all sessions, there exists a decomposi(on.
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Overview of our reduc(on approach
Undirected)Networks�
)�
Atom)Networks�
…�…�
Decompose�
Cut6set)Bound:)No� Theorem)1:)No�
) �
Require)Coding?�
Assemble� Theorem)3:)No)need)to)code)in)networks)that)can)be)decomposed)into)these)atoms)networks.�
Theorem)2:))when)&)how)to)decompose�
?�
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When the cut-‐set bound is insufficient
• Intui(vely, we need to combine several fe to show that their sum is no less than some H(Xi).
• Consider the following solu(on:
• LHS:
a d s1
b Xab
c
Xba
Xca
Xac
t1
s2
t2
fab + fac ≥ H (X1)
fab = H (Xab )+H (Xba )
fac = H (Xac )+H (Xca )
Xab = Xbd = X1 Xba = Xac = X2
fab + fac = H (X1)+ 2H (X2 )Loss!
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A Finer Technique -‐-‐ Informa(on Inequality
• Use instead of the combined version
• Submodularity
– Here A,B are sets of variables Xi , Xuv
fuv
H (Xuv ),H (Xvu )
H (A)+H (B) ≥ H (A∪B)+H (A∩B)
Flexible!
Might save some loss!
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• If messages B are determined by messages A – H(A) ≥ H(B)
• Input-‐output Inequality – The messages leaving node set U are determined by the messages entering U
• Crypto Inequality – A source message is determined by the messages transmiVed through a cut-‐set separa(ng the source and the receiver.
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Example using informa(on inequali(es
s1 t1
t3
s3
s2
t2
t4
s4
d a b
c
e
H (Xac )+H (Xbc )+H (X2 )≥ H (Xac,Xbc,X2 )≥ H (Xac,Xbc,X2,X4,Xca,Xcb )
brought in by the Input-‐output Inequality
Similarly, combine messages enters d and e, respec(vely. We obtain
H (Xad,Xbd,X3,X2,Xda,Xdb )H (Xae,Xbe,X4,X3,Xea,Xeb )
Borrowed
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Example using informa(on inequali(es
s1 t1
t3
s3
s2
t2
t4
s4
d a b
c
e
H (Xac,Xbc,X2,X4,Xca,Xcb )+H (Xad,Xbd,X3,X2,Xda,Xdb )≥ H (...,X2,X3,X4 )+H (X2 )
Then combine the 3 resul(ng entropies:
H (...,X2,X3,X4 )+H (Xae,Xbe,X4,X3,Xea,Xeb )≥ H (XE,X2,X3,X4 )+H (X3,X4 )
returned
set of messages on every link
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Example using informa(on inequali(es
s1 t1
t3
s3
s2
t2
t4
s4
d a b
c
e
To sum up,
H (XE,X2,X3,X4 )≥ H (XE,X2,X3,X4,X1)≥ H (X1)+H (X2 )+H (X3)+H (X4 )
crypto inequality
source independent
H (Xuv )u=a,b; v=c,d,e∑ ≥ H (Xi )
i=1,2,3,4∑
Similarly, we can derive H (Xvu )
u=a,b; v=c,d,e∑ ≥ H (Xi )
i=1,2,3,4∑
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Lessons Learned from the example
• Splixng fe into H(Xuv) and H(Xvu) is helpful. • Entropy terms H(A) can be combined in a cascade way. – we first combine the entropies of messages entering each node, then combine the resul(ng entropies.
• Borrowing source messages to trigger the input-‐output inequality is OK. – what actually maVers is the number of source messages brought into the deriva(on by the input-‐output/crypto inequality.
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• We study the edge sets that are a liVle bit more complicate than cut-‐sets – the union of two cut-‐sets
• For such an edge set F, we find a way to combine the entropy terms to derive that
where zi equals dG/F(si,ti) for one session and min{2, dG/F(si,ti)} for the other sessions.
H (Xuv )+H (Xvu )e=uv∈F∑ ≥ ziH (Xi )
i∑
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s1 t1
t3
s3
s2
t2
t4
s4 F1 F2
s1 t1 t3 s3
s2
t2 F1 F2
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Proof of Theorem 1
• Each component is labeled according to its distance to s in G/F.
s t
F1
F2
Connected Component
U0
U1
U’1
U3
U2 U4
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Proof of Theorem 1 (cont.)
• Step 1: combine the entropies of messages entering each component Ui;
• Step 2: combine the resul(ng entropies of U1 and U’1
• Step 3: similarly, combine U1 ,U’1,U3; combine U0, U2, U4.
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Combine results together
A cut-‐set F is orthogonal to session i, if each shortest si-‐ti path crosses F at most once.
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Remarks
• Condi(ons P1 and P2 only relate to cut-‐sets and shortest paths.
• Can be verified in (me O(2^|V|), in contrast to O(2^|E|) for the state-‐of-‐art LP outer-‐bound.
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The Next Atom Network
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Examine the Next Atom Network
s1
t1
t3
s3
s2
t2
s1
t1
t3
s3
s2
t2
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Conclusion
• A Reduc(on Approach – brings the abstract conjecture to concrete small networks
• Prove the conjecture for up to 6 nodes • An interes(ng example for future research
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Q&A
• Thanks for your (me!