The Multiple Unicast Network Coding Conjecture. and a geometric framework for studying it

Transcription

1 The Multiple Unicast Network Coding Conjecture and a geometric framework for studying it Tang Xiahou, Chuan Wu, Jiaqing Huang, Zongpeng Li June

2 Multiple Unicast: Network Coding = Routing? Undirected Network. Each link has unit capacity 1 in this example. S 1 S 2 a b S1 S 2 a1 a1 a a+b b a2 b1 b1 b1 a1 b2 a+b a+b a2 a2 T 2 T 1 b2 b2 T 2 T 1 2

3 Multiple Unicast: Network Coding = Routing? Undirected Network. Each link has unit capacity 1 in this example. a a c a b b c a+b a+c a+b a+b b b+c b+c a+b+c b c c b+c a a2 c1 a b c c c1 a1 a1 b1 c2 b1 c2 b1 a2 c2 b1 c1 a1 b2 c1 c2 a2 b1 a2 b a1 b2 b2 c1 b2 c2 b2 a1 a2 a 3

4 The MU-NC Conjecture Network coding = routing, for multiple unicast sessions in an undirected network. Given k independent unicast sessions in an undirected linkcapacitated network, a throughput vector r is feasible with network coding if and only if it is feasible with routing. 4

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. 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 conjecture. A full understanding of the problem is wild open 5

6 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

7 known to be true: # of sessions = 1 or 2 The MU-NC Conjecture 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(logk) 7

8 known to be true: # of sessions = 1 or 2 The MU-NC Conjecture 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(logk) 8

9 Space Information Flow: Multiple Unicast Min-costnetworkinformationflow: cost= e (w(e)f(e)) Min-costspaceinformationflow: cost= e ( e f(e)) x D C 1 A B x 1 Unit rate demand: A B, A C, B D 9

10 Space Information Flow: Multiple Unicast x D C 1 A B x 1 Cost = i r i d i Is optimal cost without network coding still optimal with network coding? 10

11 MU-NC conjecture: Network vs. Space true in networks = true in any geometric space with distance trueinnetworks= trueinl 2 (Euclideandistance) true in l 2 = not too far off in networks true in networks true in l (Chebyshev distance) 11

12 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

13 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

14 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

15 Example Application: Two Unicast Sessions Step 2. Embedding: Isometric (distance-preserving) embedding of G into l. n ( n )1 p u,v = lim x ui x vi p = max x ui x vi p i i=1 Embed each node u i in G to (x i1 = d i1,x i2 = d i2,...,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

16 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 N 1 y C /4 M D -1 E o 1 x -16

17 Example Application: Two Unicast Sessions Step 4. 1-D Proof: Integrating the cut inequality over the 1-D space s1 t2 s3 t3 t1 s2 x0 x x= f x dx x= Demand((, x) (x, ))dx LFH = e ( e 1 f e ) RHS = i s i t i 1 r i Therefore: e ( e 1f e ) i ( s it i 1 r i ). 17

18 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(logk). 18

19 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

20 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 u,v 2 = (x ui x vi ) 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(logn). 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

21 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 )dφ < ( s i t i p r i )dφ Φ e Φ i p 21

22 Example Application: The O(log k) Upper-bound Step 4. 1-D Proof: Integrating the cut inequality over the 1-D space s1 t2 s3 t3 t1 s2 x0 x x= f x dx x= Demand((, x) (x, ))dx LFH = e ( e 1 f e ) RHS = i s i t i 1 r i Therefore: e ( e 1f e ) i ( s it i 1 r i ). 22

23 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 be zero, except that the ith coordinate is 2 2 : ( ) 2 0,...,0,x i = 2,0,...,0 23

24 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 Step 3. From l k to l 1 (or l 2 ): Projection.??? (l n 2 to l1, done; l 2 to l1, done) Step 4. Proof in 1-D: Integrating Cut Inequality. done 24