Solving the Single Rate 2-pair Network Coding Problem With the A-set Equation Kai Cai
[email protected] Institute of Computing Technology
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Chinese Academy of Sciences, Beijing, China
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Joint work with K. B. Letaief, Pingyi Fan and Rongquan Feng
Solving the Single Rate 2-pair Network Coding Problem With the A-set Equation – p. 1/23
Outline of Presentation
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Introduction
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Solving the Single Rate 2-pair Network Coding Problem With the A-set Equation – p. 2/23
Outline of Presentation Introduction
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Proof of the Main Result
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Solving the Single Rate 2-pair Network Coding Problem With the A-set Equation – p. 2/23
Outline of Presentation Introduction
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Proof of the Main Result
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Discussions
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Solving the Single Rate 2-pair Network Coding Problem With the A-set Equation – p. 2/23
Outline of Presentation Introduction
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Proof of the Main Result
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Discussions
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Solving the Single Rate 2-pair Network Coding Problem With the A-set Equation – p. 2/23
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1 Introduction
Solving the Single Rate 2-pair Network Coding Problem With the A-set Equation – p. 3/23
Definition A single rate 2-pair network coding (S2PNC) problem is specified as follows: 1. A communication network N = (V, E, {s1 , s2 }, {t1 , t2 }). 2. Two desired unit rate flows from si to ti for i = 1, 2. Remark: we assume each link e ∈ E has the unit capacity and si has no in-edge and a single out-edge and ti has no out-edge and a single in-edge for i = 1, 2.
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If the desired flows can be established by network coding, then the S2PNC problem is called solvable, otherwise, it is called unsolvable.
Solving the Single Rate 2-pair Network Coding Problem With the A-set Equation – p. 4/23
Known Result A S2PNC problem is solvable if and only if one of the following two (controlled edge overlap) conditions holds. •
[Condition 1] There exists a collection P of two paths Ps1 ,t1 and Ps2 ,t2 , such that maxe∈E esP (e) ≤ 1.
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[Condition 2] There exist a collection P of three paths {Ps1 ,t1 , Ps2 ,t2 , Ps2 ,t1 }, and a collection Q of three paths {Qs1 ,t1 , Qs2 ,t2 , Qs1 ,t2 }, such that maxe∈E esP (e) ≤ 2 and maxe∈E esQ (e) ≤ 2.
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[1] C.-C. Wang and N.B. Shroff, “Beyond the butterfly-a graph theoretic characterization
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of the feasibility of network coding with two simple unicast sessions,” ISIT 2007.
Solving the Single Rate 2-pair Network Coding Problem With the A-set Equation – p. 5/23
A-set Definition 1.1. Let N
= (V, E, s, t) be a point-to-point network with maximal flow f , then the A-set of N is defined as [ A= hVs , Vt i. hVs ,Vt iisM in−Cut
• A-set is composed of the most “important” links of the
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network.
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A-set Example: e6 e2 s
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• A = {e1 , e2 , e12 , e13 }
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The Method • Decomposed N = (V, E, {s1 , s2 }, {t1 , t2 }) into four
point-to-point networks Ni,j = (V, E, si , tj ), i, j = 1, 2. • The point-to-point network is easy to deal with. • The relations of Ai,j can completely determine the
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solvability of N , where Ai,j is the A-set of Ni,j .
Solving the Single Rate 2-pair Network Coding Problem With the A-set Equation – p. 8/23
Main Result • Let N = (V, E, {s1 , s2 }, {t1 , t2 }) be the underlying 2-pair
network. The S2PNC problem is solvable if and only if (A1,2 ∪ A2,1 ) ∩ (A1,1 ∩ A2,2 ) = ∅. • We call (A1,2 ∪ A2,1 ) ∩ (A1,1 ∩ A2,2 ) = ∅ as the A-set
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equation of N .
Solving the Single Rate 2-pair Network Coding Problem With the A-set Equation – p. 9/23
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2 Proof of the Main Result
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Basic Configuration We call (a), (b), (c), (d) as the basic configuration of solvable single rate 2-pair networks. s1
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Solving the Single Rate 2-pair Network Coding Problem With the A-set Equation – p. 11/23
Lemma If N contains (a), (b), (c), or (d), then N is solvable. X1
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What does “Contain” mean? We say N = (V, E, {s1 , s2 }, {t1 , t2 }) contains N0 = (V ′ , E ′ , {s′1 , s′2 }, {t′1 , t′2 }) if there exists a function f from the edges of N0 to the paths of N such that: If s′i −→ si , t′i −→ ti , for i = 1, 2;
(2)
If head(e′1 ) = tail(e′2 ), then head(f (e′1 )) = tail(f (e′2 )), for e′1 , e′2 ∈ E ′ ;
(3)
If e′1 6= e′2 , then f (e′1 ) and f (e′2 ) are edge-disjoint, for e′1 , e′2 ∈ E ′ .
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(1)
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Proof of “Necessity” Case 1:
A1,1 ∩ A2,2 = ∅
Theorem 2.1. Let N
= (V, E, {s1 , s2 }, {t1 , t2 }) be a 2-pair unicast network. If A1,1 ∩ A2,2 = ∅, then the network contains (a),(b), or (c). s1
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Proof of “Necessity” Case 2:
A1,1 ∩ A2,2 6= ∅
Theorem 2.2. If A1,1
∩ A2,2 6= ∅ and (A1,1 ∩ A2,2 ) ∩ (A1,2 ∪ A2,1 ) = ∅. Then N contains Fig. (d). s1
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Proof of “Sufficiency” ♦ Suppose N is solvable. Consider two cases: (1) A1,1 ∩ A2,2 = ∅; and (2) A1,1 ∩ A2,2 6= ∅. (1)
If A1,1 ∩ A2,2 = ∅, then (A1,2 ∪ A2,1 ) ∩ (A1,1 ∩ A2,2 ) = ∅.
(2)
If A1,1 ∩ A2,2 6= ∅, then one can find an s1 − t2 path and an s2 − t1 path disjoint with A1,1 ∩ A2,2 .
♦ (A1,2 ∪ A2,1 ) ∩ (A1,1 ∩ A2,2 ) = ∅ if and only if there exist an s1 -t2 path P1 and an s2 -t1 path P2 such that (P1 ∪ P2 ) ∩ (A1,1 ∩ A2,2 ) = ∅.
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A detailed proof was available at: http://arxiv.org/abs/1007.0465
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3 Discussions
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Capacity Factor Definition: Definition 3.1. Let N = (V, E, s, t) be a directed acyclic network with node set V , link set E , source node s, sink node t. F ⊆ E is called a capacity factor (CF) of N if and only if the following two conditions are satisfied.
2.
CN \F ′ (s, t) = CN (s, t) for any F ′
F, et
CN \F (s, t) < CN (s, t); .n
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[2] K. Cai and P. Fan, “An algebraic approach to link failures based on network coding,”
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IEEE Trans. Inf. Theory, vol. 53, no. 2, pp. 775-779, Feb. 2007.
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Capacity Factor Example: e6 e2 s
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{e3 , e4 }, {e3 , e9 }, {e5 , e6 }, {e5 , e11 }, {e6 , e10 }, {e8 , e9 }, {e10 , e11 }
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{e4 , e7 , e8 }
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{e1 }, {e2 }, {e12 }, {e13 }
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Capacity Rank Definition: • Capacity rank: the minimal cardinality of the CFs
containing e; denoted by CR(e). • The capacity rank characterize the degree of
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importance of a link, i.e., the smaller capacity rank a link has, the more important the link is.
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Capacity Rank Example: e6 e2 s
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• CR(e1 ) = CR(e2 ) = CR(e12 ) = CR(e13 ) = 1, • CR(e3 ) = CR(e4 ) = CR(e5 ) = CR(e6 ) = CR(e8 ) = et
CR(e9 ) = CR(e10 ) = CR(e11 ) = 2,
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• CR(e7 ) = 3, with the least importance.
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Questions • The links in the A-set is just the links with capacity rank
1. That is to say, the solvability of the S2PNC problem is determined by the most important links of N . • Can he solvability of a general multi-source multi-sink
network coding problem be determined by the links with capacity ranks ≤ k, a small integer? • Determine k or give bounds on k?
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• Is there a similar equation for the k-pair case?
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Thank You & Questions
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