The Multiple Unicast Network Coding Conjecture and a geometric - - PowerPoint PPT Presentation

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

Tang Xiahou, Chuan Wu, Jiaqing Huang, Zongpeng Li June 30 2012

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Multiple Unicast: Network Coding = Routing?

Undirected Network. Each link has unit capacity 1 in this example.

a a b b a+b a+b a+b

S1 S2 T 2 T 1 S1 S2 T 2 T 1

a1 b1 a1 b1 b2 a1 a2 b2 a2 b2 a2 b1

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Multiple Unicast: Network Coding = Routing?

Undirected Network. Each link has unit capacity 1 in this example.

a a b b c c a+b a+b a+b b+c b+c a+c a+b+c

a a b c c b

b+c

a b c a c b

a2 c1 a1 c1 b1 c1 b2 c1 c1 b2 c2 b2 a1 b1 a1 c2 a1 b2 c2 b1 c2 b1 c2 a2 a2 b1 a2 b2 a1 a2

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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.

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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

• pen”

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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.

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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 K3,2)

in general:

• coding advantage ≤ O(log k)

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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 K3,2)

in general:

• coding advantage ≤ O(log k) √

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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))

A B C D

x1 x2

1 1 2 2 3 3

Unit rate demand: A → B, A → C, B → D

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Space Information Flow: Multiple Unicast

A B C D

x1 x2

1 1 2 2 3 3

Cost =

• i

ridi Is optimal cost without network coding still optimal with network coding?

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MU-NC conjecture: Network vs. Space

• true in networks =

⇒ true in any geometric space with ‘distance’

• true in networks =

⇒ true in l2 (Euclidean distance)

• true in l2 =

⇒ not too far off in networks

• true in networks ⇐

⇒ true in l∞ (Chebyshev distance)

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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

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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 (r1, r2) is feasible with network coding if and only if it is feasible with routing.

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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

• f unicast terminals specified, and any desired throughput

vector r, the maximum coding advantage in (G, c) over all c ∈ QE

+, equals the maximum cost advantage in (G, w) over

all w ∈ QE

+.

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Example Application: Two Unicast Sessions

Step 2. Embedding: Isometric (distance-preserving) em- bedding of G into ln

∞.

||u, v||∞ = lim

p→∞

n

• i=1

|xui − xvi|p 1

p

= max

i

|xui − xvi| Embed each node ui in G to (xi1 = di1, xi2 = di2, . . . , xii = dii = 0, . . . , xi,n = di,n), where dij is the shortest path length between ui and uj in G No counter example for space information flow problem in ln

∞

= ⇒ no counter example for network information flow problem in G

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Example Application: Two Unicast Sessions

Step 3. Projection: (3.a.) from ln

∞ to l2 ∞, then (3.b.) from

l2

∞ to l1

(3.a.) Theorem: If network coding can outperform routing in ln

∞, then it can do so in lk ∞. k = 2 in this case.

— idea: keep k primary coordinates, truncate the other n − k coordinates (3.b.) idea: a unit vector in l2

∞, when projected to the two

diagonal lines, has constant total projected length

• x

y 1

• 1

1

• 1

M N C D E ̟/4

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Example Application: Two Unicast Sessions

Step 4. 1-D Proof: Integrating the cut inequality over the 1-D space

s1 s2 s3 t1 t2 t3 x0 x

∞

x=−∞

fxdx ≥ ∞

x=−∞

Demand((−∞, x) ↔ (x, ∞))dx LFH =

• e

(||e||1fe) RHS =

• i

||siti||1ri Therefore:

e(||e||1fe) ≥ i(||siti||1ri).

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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 coding, then routing can achieve at least

r O(log k).

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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

• f unicast terminals specified, and any desired throughput

vector r, the maximum coding advantage in (G, c) over all c ∈ QE

+, equals the maximum cost advantage in (G, w) over

all w ∈ QE

+.

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Example Application: The O(log k) Upper-bound

Step 2. Embedding: O(log k)-distortion embedding of G into l2 (Euclidean space). ||u, v||2 = n

• i=1

(xui − xvi)2 1

2

[Bourgain, 1985] The closure of an edge-weighted graph (G, w) with n nodes can be embedded into lp for any 1 ≤ p ≤ ∞, with distortion O(log n). No counter example for space information flow problem in ln

2

= ⇒ Throughput gap for network information flow problem in G upper-bounded by distortion O(log k)

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Example Application: The O(log k) Upper-bound

Step 3. Projection: from ln

2 to l1

Theorem: If network coding can outperform routing in ln

2,

then it can do so in l1 Find “good” 1-D space for projection onto: enumerate all possible

→

p, by integrating over Φ. Prove:

• ✞

✝ ☎ ✆

• Φ
• e

(fe|e ·

→

p|)dΦ <

• ✞

✝ ☎ ✆

• Φ
• i

(|

→

siti ·

→

p|ri)dΦ

p

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Example Application: The O(log k) Upper-bound

Step 4. 1-D Proof: Integrating the cut inequality over the 1-D space

s1 s2 s3 t1 t2 t3 x0 x

∞

x=−∞

fxdx ≥ ∞

x=−∞

Demand((−∞, x) ↔ (x, ∞))dx LFH =

• e

(||e||1fe) RHS =

• i

||siti||1ri Therefore:

e(||e||1fe) ≥ i(||siti||1ri).

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Example Application: Complete Networks

Network Coding is equivalent to routing in a complete network with uniform link weights. Isometrically embed G into ln

2, then project to l1:

for each vertex i, i = 1, 2, · · · , n, let all the coordinates of i be zero, except that the ith coordinate is

√ 2 2 :

• 0, . . . , 0, xi =

√ 2 2 , 0, . . . , 0

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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 lk

∞. done

Step 3. From lk

∞ to l1 (or l2 ∞): Projection. ???

(ln

2 to l1, done; l2 ∞ to l1, done)

Step 4. Proof in 1-D: Integrating Cut Inequality. done

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