The Multiple Unicast Network Coding Conjecture and a geometric framework for studying it Tang Xiahou, Chuan Wu, Jiaqing Huang, Zongpeng Li June 30 2012 1
Multiple Unicast: Network Coding = Routing? Undirected Network. Each link has unit capacity 1 in this example. S 1 S 2 S 1 S 2 a b a1 a1 b1 b1 a+b a b b1 a2 a1 b2 a+b a2 a+b a2 b2 b2 T 1 T 1 T 2 T 2 2
Multiple Unicast: Network Coding = Routing? Undirected Network. Each link has unit capacity 1 in this example. a b c a b c a1 b2 c1 a b b c b1 c1 c1 c1 b2 c a+b a b+c a1 b1 c2 a2 b2 a+b a1 c1 c2 a+b b1 b+c c2 c2 a1 a+c b+c a+b+c b1 b2 a2 a2 b1 a2 a1 a2 c2 b2 c c b a b a 3
The MU-NC Conjecture Network coding = routing, for multiple unicast sessions in an undirected network. Given k independent unicast sessions in an undirected link- capacitated network, a throughput vector r is feasible with network coding if and only if it is feasible with routing. 4
The MU-NC Conjecture For k pairs of independent unicast sessions in an undirected network, a throughput vector r is feasible with network coding if and only if it is feasible with routing. • Proposed in 2004, by Harvey et al. and by Li and Li • Sounds intuitive and simple • Studied extensively, not much progress so far • No counter example known yet • Mitzenmach, Ho, Sprintson, 2007: a list of 7 open problems in NC: MU-NC conjecture is problem #1 • Chekuri: Claiming an equivalence between network coding and routing for all undirected networks is a “bold con- jecture”. A full understanding of the problem is “wild open” 5
The MU-NC Conjecture For k pairs of independent unicast sessions in an undirected network, a throughput vector r is feasible with network coding if and only if it is feasible with routing. • Langberg, 2011: the MU-NC conjecture “has driven many crazy” • A growing agreement: probably need new tools, beyond a “simple blend” of graph theory and information theory • Network coding for general multiple sessions (multi-source, multi-destination) is hard, not much known • Multiple unicast is the most basic case of multiple session network coding. Good understanding desired. 6
The MU-NC Conjecture known to be true: • # of sessions = 1 or 2 • common sender/receiver location • planar network, all terminal nodes on same face – star – outer planar • Okamura-Seymour network (uniform-capacity K 3 , 2 ) in general: • coding advantage ≤ O (log k ) 7
The MU-NC Conjecture known to be true: • # of sessions = 1 or 2 √ • common sender/receiver location √ • planar network, all terminal nodes on same face – star √ – outer planar • Okamura-Seymour network (uniform-capacity K 3 , 2 ) in general: • coding advantage ≤ O (log k ) √ 8
Space Information Flow: Multiple Unicast Min-cost network information flow: cost = � e ( w ( e ) f ( e )) Min-cost space information flow: cost = � e ( || e || f ( e )) x 2 D 3 C 2 1 A B 1 0 3 2 x 1 Unit rate demand: A → B , A → C , B → D 9
Space Information Flow: Multiple Unicast x 2 D 3 C 2 1 A B 0 1 2 3 x 1 � Cost = r i d i i Is optimal cost without network coding still optimal with network coding? 10
MU-NC conjecture: Network vs. Space • true in networks = ⇒ true in any geometric space with ‘distance’ • true in networks = ⇒ true in l 2 (Euclidean distance) • true in l 2 = ⇒ not too far off in networks • true in networks ⇐ ⇒ true in l ∞ (Chebyshev distance) 11
The Geometric Framework Step 1. From Throughput to Cost: LP Duality Step 2. From Network to Space: Graph Embedding Step 3. From h -D to 1 -D: Projection Step 4. Proof in 1 -D: Integrating Cut Inequality 12
Example Application: Two Unicast Sessions For two unicast sessions in an undirected network ( G, c ) , network coding is equivalent to routing (MCF). i.e. , a throughput vector ( r 1 , r 2 ) is feasible with network coding if and only if it is feasible with routing. 13
Example Application: Two Unicast Sessions Step 1. Transformation: Apply the following result to all network configurations with k = 2 , to translate the statement from its throughput version to cost version. [Li and Li, 2004] Given an undirected network G with k pairs of unicast terminals specified, and any desired throughput vector r , the maximum coding advantage in ( G, c ) over all c ∈ Q E + , equals the maximum cost advantage in ( G, w ) over all w ∈ Q E + . 14
Example Application: Two Unicast Sessions Step 2. Embedding: Isometric (distance-preserving) em- bedding of G into l n ∞ . � n � 1 p � | x ui − x vi | p || u, v || ∞ = lim = max | x ui − x vi | p →∞ i i =1 Embed each node u i in G to ( x i 1 = d i 1 , x i 2 = d i 2 , . . . , x ii = d ii = 0 , . . . , x i,n = d i,n ) , where d ij is the shortest path length between u i and u j in G No counter example for space information flow problem in l n ∞ = ⇒ no counter example for network information flow problem in G 15
Example Application: Two Unicast Sessions Step 3. Projection: (3.a.) from l n ∞ to l 2 ∞ , then (3.b.) from l 2 ∞ to l 1 (3.a.) Theorem: If network coding can outperform routing in l n ∞ , then it can do so in l k ∞ . k = 2 in this case. — idea: keep k primary coordinates , truncate the other n − k coordinates (3.b.) idea: a unit vector in l 2 ∞ , when projected to the two diagonal lines, has constant total projected length y 1 C M N ̟/4 D E 1 -1 o x 16 -1
Example Application: Two Unicast Sessions Step 4. 1 -D Proof: Integrating the cut inequality over the 1 -D space s 3 s 2 s 1 t 1 t 2 t 3 x 0 x � ∞ � ∞ f x dx ≥ Demand (( −∞ , x ) ↔ ( x, ∞ )) dx x = −∞ x = −∞ � LFH = ( || e || 1 f e ) e � RHS = || s i t i || 1 r i i Therefore: � e ( || e || 1 f e ) ≥ � i ( || s i t i || 1 r i ) . 17
Example Application: The O (log k ) Upper-bound For k unicast sessions in an undirected network ( G, c ) , if a throughput vector r can be achieved by network r coding, then routing can achieve at least O (log k ) . 18
Example Application: The O (log k ) Upper-bound Step 1. Transformation: Apply the following result to all network configurations with k unicast sessions, to translate the statement from its throughput version to cost version. [Li and Li, 2004] Given an undirected network G with k pairs of unicast terminals specified, and any desired throughput vector r , the maximum coding advantage in ( G, c ) over all c ∈ Q E + , equals the maximum cost advantage in ( G, w ) over all w ∈ Q E + . 19
Example Application: The O (log k ) Upper-bound Step 2. Embedding: O (log k ) -distortion embedding of G into l 2 (Euclidean space). � n � 1 2 � ( x ui − x vi ) 2 || u, v || 2 = i =1 [Bourgain, 1985] The closure of an edge-weighted graph ( G, w ) with n nodes can be embedded into l p for any 1 ≤ p ≤ ∞ , with distortion O (log n ) . No counter example for space information flow problem in l n 2 = ⇒ Throughput gap for network information flow problem in G upper-bounded by distortion O (log k ) 20
Example Application: The O (log k ) Upper-bound Step 3. Projection: from l n 2 to l 1 Theorem: If network coding can outperform routing in l n 2 , then it can do so in l 1 Find “good” 1-D space for projection onto: enumerate all → possible p , by integrating over Φ . Prove: � � � � → → → � � ✞ ☎ ✞ ☎ ( f e | e · p | ) d Φ < ( | s i t i · p | r i ) d Φ ✝ ✆ ✝ ✆ Φ Φ e i p 21
Example Application: The O (log k ) Upper-bound Step 4. 1 -D Proof: Integrating the cut inequality over the 1 -D space s 3 s 2 s 1 t 3 t 1 t 2 x 0 x � ∞ � ∞ f x dx ≥ Demand (( −∞ , x ) ↔ ( x, ∞ )) dx x = −∞ x = −∞ � LFH = ( || e || 1 f e ) e � RHS = || s i t i || 1 r i i Therefore: � e ( || e || 1 f e ) ≥ � i ( || s i t i || 1 r i ) . 22
Example Application: Complete Networks Network Coding is equivalent to routing in a complete network with uniform link weights. Isometrically embed G into l n 2 , then project to l 1 : for each vertex i , i = 1 , 2 , · · · , n , let all the coordinates of i √ 2 be zero, except that the ith coordinate is 2 : √ � � 2 0 , . . . , 0 , x i = 2 , 0 , . . . , 0 23
A Possible Proof to the MU-NC Conjecture? Step 1. From Throughput to Cost: translate to cost version. done Step 2. From Network to Space: Graph Embedding. Isometrically embed G into l k ∞ . done ∞ to l 1 (or l 2 Step 3. From l k ∞ ): Projection. ??? 2 to l 1 , done; l 2 ∞ to l 1 , done) ( l n Step 4. Proof in 1 -D: Integrating Cut Inequality. done 24
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