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Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Zongpeng Li University of Calgary / INC, CUHK December 1 2011, at UNSW

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Joint work with: Xunrui Yin, Xin Wang, Jin Zhao, Xiangyang Xue School of Computer Science Fudan Univeristy, Shanghai, China To appear in IEEE INFOCOM 2012

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Benefits of Network Coding: Higher Multicast Rate

a a a a ⊕ b b b b a ⊕ b a ⊕ b s t1 t2 With Network Coding 2 bits / 1 sec [a, b] [a, b] [a, b] [a, c] [b, c] [b, c] [b, c] [a, c] [a, c] s t1 t2 Without Network Coding 3 bits / 2 secs

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Motivation

Theoretically, coding advantage can be arbitrarily large. In practice, observed coding advantage is marginal. We introduce two parameterized network models to characterize practical networks and bound the coding advantage accordingly

Bidirected Networks (with max link imbalance α) Hyper-Networks (with max link size β)

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

General Network Models

A network is represented as a (multi-)graph G(V , E) where

each link has unit capacity parallel links allowed

A multicast session (s,T):

s ∈ V : the multicast source T ⊂ V : the set of multicast receivers

A (symmetrical) multicast throughput R is achieved if each receiver receives information at rate R.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

• Max. Throughput with Network Coding

Theorem 1 (Ahlswede et al. IT2000) In a directed network, Rnc = min

t∈T{λG(s, t)}

Rnc : max multicast throughput with network coding λG(s, t) : edge connectivity from s to t.

i.e., the number of edge disjoint paths from s to t.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Example

Recall in the butterfly network, Rnc = 2:

s t1 t2 edge disjoint paths to t1 s t1 t2 edge disjoint paths to t2

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Max Throughput with Routing

Without network coding, symbols can still be replicated. The trace of each symbol forms a multicast tree. Packing multicast trees: deciding transmission rates for possible multicast trees, under link capacity constraints.

a a a a a

Proposition 1 Without network coding, the max multicast rate Rtree is achieved by an optimal packing of multicast trees.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Example

0.5 s t1 t2 +0.5 s t1 t2 +0.5 s t1 t2 = s t1 t2

In the butterfly network, Rtree = 0.5 + 0.5 + 0.5 = 1.5.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Coding Advantage

Definition Given topology G(V , E) and multicast session (s, T), Coding Advantage is defined as θ = Rnc/Rtree In the butterfly network, θ = 2/1.5 . = 1.33.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Coding Advantage

Question max

G,s,T θ =?

In terms of throughput improvement

How good can Network Coding be? In which scenario, Network Coding outperforms Routing the most?

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Previous Results

Scenario Coding Advantage Note In general[1] θ → ∞ Illustrated later. Unicast or Broadcast[3] θ = 1 |T| = 1 or T = V \{s} Most P2P Overlay Networks[4][5] θ = 1 sufficient down link capacity Undirected Networks[2] θ ≤ 2 f (u, v)+f (v, u) ≤ c({u, v})

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Previous Results

Scenario Coding Advantage Note In general[1] θ → ∞ Illustrated later. Unicast or Broadcast[3] θ = 1 |T| = 1 or T = V \{s} Most P2P Overlay Networks[4][5] θ = 1 sufficient down link capacity Undirected Networks[2] θ ≤ 2 f (u, v)+f (v, u) ≤ c({u, v})

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Previous Results

Scenario Coding Advantage Note In general[1] θ → ∞ Illustrated later. Unicast or Broadcast[3] θ = 1 |T| = 1 or T = V \{s} Most P2P Overlay Networks[4][5] θ = 1 sufficient down link capacity Undirected Networks[2] θ ≤ 2 f (u, v)+f (v, u) ≤ c({u, v})

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Previous Results

Scenario Coding Advantage Note In general[1] θ → ∞ Illustrated later. Unicast or Broadcast[3] θ = 1 |T| = 1 or T = V \{s} Most P2P Overlay Networks[4][5] θ = 1 sufficient down link capacity Undirected Networks[2] θ ≤ 2 f (u, v)+f (v, u) ≤ c({u, v})

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Example with Large Coding Advantage

n k k · · · · · · · · · · · · · · · s 1 2 n t1 t(

n k)

Rnc = k, Rtree ≤

n n−k+1.

θ ≥ k(n−k+1)

n

→ ∞, as n = 2k → ∞. Observation In practice, links are often bidirected.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Case Study

What happens when links are bidirected?

s t1 t2 θ = 4/3 s t1 t2 θ = 1

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Completely Balanced Networks

It is not a coincidence! Theorem 2 In completely link balanced networks, θ = 1. Let c(u, v) denote the actual link capacity from u to v, i.e., the number of parallel unit capacity links from u to v. A bidirected network is completely link balanced, if c(u, v) = c(v, u), ∀u, v ∈ V .

2 3 5 1 link balanced

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Proof of theorem 2

Proof sketch:

1 Convert the completely link balanced network B into a

broadcast network D such that

Rnc(B) = Rnc(D). Rtree(B) ≥ Rtree(D).

2 Apply the fact network coding can not increase multicast rate

when T = V \{s}, we have Rtree(D) = Rnc(D) and thereby, Rtree(B) = Rnc(B). For the conversion, we employ edge splitting to isolate each relay node.

u v z u v z

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Proof of theorem 2

A relay node can be split off without affecting the edge connectivity among other nodes. ⇒ Rnc(B) = Rnc(D) Lemma [Frank and Jackson 1995] Let D = (V + z, E) be a node balanced directed graph. For each link

→

uz ∈ E, there exists a link

→

zv ∈ E, such that after splitting off

→

uz,

→

zv, the edge connectivity between every pair of nodes in V is unchanged. Each multicast tree in the resulting network D can be converted into a multicast tree in the original network B. ⇒ Rtree(B) ≥ Rtree(D)

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Remarks

link balanced node balanced

‘link balanced’ can be relaxed to ‘node balanced’. The core of Internet is close to a link balanced network. From the proof: neither coding (network coding) or replication (IP multicast) is necessary at interior routers.

Each splitting operation corresponds to a forwarding at the interior routers.

A polynomial time algorithm for tree-packing can be extracted from the proof.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

α-Balanced Networks

For a general bidirected network, define the link imbalance ratio α = max

c(u,v)>0

c(v, u) c(u, v) Theorem 3 In bidirected networks, θ ≤ α. Proof: Ignoring the excess capacity, we can perform an optimal tree packing in the resulting link balanced network, achieving a multicast throughput no less than 1

αRnc.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

α-Balanced Networks

Proposition 2 For α ≥ 1, there exists an α-balanced network, where θ ≥ √α/4.

12 13 14 23 24 34 1 2 3 4 t1 t2 t3 t4 s

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Undirected Networks

Bidirected network: for each pair of adjacent nodes u, v, c(u, v) and c(v, u) are fixed and independent of each other. Undirected network: the two directions share a total link capacity c({u, v})

Let f (u, v) denote the information flow rate from u to v Link capacity constraints: f (u, v) + f (v, u) ≤ c({u, v}).

Theorem 4 (Li and Li, CISS2004) In an undirected multicast network, θ ≤ 2.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Example

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Example

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

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Example

a b c d e f g h i

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Example

abcdi abcdi cefgh defgh acegi befgh bdefh abcdi afghi

Total rate: Rtree = 0.2 × 9 = 1.8 Lower bound NP-hard

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Hyper-Network as an Extension

Hyper-network A (hyper-)link connect 2 or more nodes. When one node transmits through a hyper-link, all other nodes can simultaneously receive. For example: wireless link, Ethernet bus.

u v w c({u, v, w})

Let f (u → vw) denote the transmission rate from u to v, w. Link capacity constraint: f (u → vw) + f (v → uw) + f (w → uv) ≤ c({u, v, w}).

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Hyper-networks

The size/cardinality of a hyper-link is defined as the number of nodes it covers. Theorem 5 In a hyper-network with max edge size β, θ ≤ β. An undirected network is a special case with β = 2.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Proof of theorem 5

Proof sketch:

1 Given Hyper-network H, construct a completely link balanced

network B as follows:

u y w x ve1 ve2 e1 e2 u y w x

2 According to theorem 2, Rnc(B) = Rtree(B), we only need to

verify Rnc(H) ≤ Rnc(B) and Rtree(H) ≥ 1

βRtree(B).

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Lower bound

Proposition 3 There exists a hyper-network (H, s, T) with max edge size β, such that θ ≥ 1

2 log β.

Proof sketch: Consider the relay nodes of the combination network as hyper-links connecting the source and the receivers. Coding advantage in this hyper-network is the same as in the directed combination network, while the size of each hyper-link is n−1

k−1

• + 1.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

Future Work

Tightness of the bound α for bidirected networks

Efficient Steiner tree packing algorithm for bidirected networks

Tightness of the bound 2 for undirected networks Multiple unicast Cost advantage (almost done) Other Constraints & Models

Number of receiver nodes Dynamic Capacity

Network Coding in Planar Networks

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

References I

Jaggi, S. and Sanders, P. and Chou, P.A. and Effros, M. and Egner, S. and Jain, K. and Tolhuizen, L.M.G.M. Polynomial time algorithms for multicast network code construction Information Theory, IEEE Trans. on, 2005. Zongpeng Li and Baochun Li and Lap Chi Lau. A Constant Bound on Throughput Improvement of Multicast Network Coding in Undirected Networks. Information Theory, IEEE Trans. on , 2009.

• J. Edmonds.

Edge-disjoint branchings. Combinatorial Algorithms, ed. R. Rustin, 1973.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks

Motivation Problem Formulation Bidirected Networks Hyper-Networks Future Work References

References II

Dah Ming Chiu and Yeung, R.W. and Jiaqing Huang and Bin Fan. Can Network Coding Help in P2P Networks? Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, 2006. Ziyu Shao and Li, S.-Y.R. To Code or Not to Code: Rate Optimality of Network Coding versus Routing in Peer-to-Peer Networks Communications, IEEE Transactions on, 2011. Keshavarz-Haddadt, A. and Riedi, R. Bounds on the Benefit of Network Coding: Throughput and Energy Saving in Wireless Networks INFOCOM, 2008.

Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks