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 s s [ a, b ] [ b, c ] a b [ a, b ] [ b, c ] a b a [ a, b ] [ a, c ] [ b, c ] a ⊕ b b [ a, c ] [ a, c ] a ⊕ b a ⊕ b t 1 t 2 t 1 t 2 With Network Coding Without Network Coding 2 bits / 1 sec 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, R nc = min t ∈ T { λ G ( s , t ) } R nc : max multicast throughput with network coding λ G ( 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, R nc = 2: s s t 1 t 2 t 1 t 2 edge disjoint paths to t 1 edge disjoint paths to t 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 Max Throughput with Routing Without network coding, symbols can still a be replicated. The trace of each symbol forms a a a multicast tree. Packing multicast trees: deciding a a transmission rates for possible multicast trees, under link capacity constraints. Proposition 1 Without network coding, the max multicast rate R tree 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 s s s s = 0 . 5 +0 . 5 +0 . 5 t 1 t 2 t 1 t 2 t 1 t 2 t 1 t 2 In the butterfly network, R tree = 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 = R nc / R tree 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 G , s , T θ =? max 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 | T | = 1 or θ = 1 Broadcast [3] T = V \{ s } Most P2P sufficient down Overlay θ = 1 link capacity Networks [4][5] Undirected f ( u , v )+ f ( v , u ) ≤ θ ≤ 2 Networks [2] 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 | T | = 1 or θ = 1 Broadcast [3] T = V \{ s } Most P2P sufficient down Overlay θ = 1 link capacity Networks [4][5] Undirected f ( u , v )+ f ( v , u ) ≤ θ ≤ 2 Networks [2] 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 | T | = 1 or θ = 1 Broadcast [3] T = V \{ s } Most P2P sufficient down Overlay θ = 1 link capacity Networks [4][5] Undirected f ( u , v )+ f ( v , u ) ≤ θ ≤ 2 Networks [2] 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 | T | = 1 or θ = 1 Broadcast [3] T = V \{ s } Most P2P sufficient down Overlay θ = 1 link capacity Networks [4][5] Undirected f ( u , v )+ f ( v , u ) ≤ θ ≤ 2 Networks [2] 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 s n · · · 1 2 n · · · · · · · · · k k · · · t 1 t ( k ) n n R nc = k , R tree ≤ n − k +1 . θ ≥ k ( n − k +1) → ∞ , as n = 2 k → ∞ . n 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 s t 1 t 2 t 1 t 2 θ = 4 / 3 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 2 from u to v , i.e. , the number of parallel unit capacity links from u to v . 1 3 A bidirected network is completely link 5 balanced, if c ( u , v ) = c ( v , u ), ∀ u , v ∈ V . 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 R nc ( B ) = R nc ( D ). R tree ( B ) ≥ R tree ( D ). 2 Apply the fact network coding can not increase multicast rate when T = V \{ s } , we have R tree ( D ) = R nc ( D ) and thereby, R tree ( B ) = R nc ( B ). For the conversion, we employ edge splitting to isolate each relay node. z z u v u v Zongpeng Li University of Calgary / INC, CUHK On Benefits of Network Coding in Bidirected Networks and Hyper-networks
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