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

Transcription

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

2 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

3 Benefits of Network Coding: Higher Multicast Rate a s b [a, b] s [b, c] a b [a, b] [b, c] a a b b [a, b] [a, c] [b, c] a b a b [a, c] [a, c] t 1 t 2 With Network Coding 2 bits / 1 sec t 1 t 2 Without Network Coding 3 bits / 2 secs

4 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)

5 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.

6 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.

7 Example Recall in the butterfly network, R nc = 2: s s t 1 t 2 edge disjoint paths to t 1 t 1 t 2 edge disjoint paths to t 2

8 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. Proposition 1 a a a a a Without network coding, the max multicast rate R tree is achieved by an optimal packing of multicast trees.

9 Example s s s s = t 1 t 2 t 1 t 2 t 1 t 2 t 1 t 2 In the butterfly network, R tree = = 1.5.

10 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.

11 Coding Advantage Question max θ =? G,s,T In terms of throughput improvement How good can Network Coding be? In which scenario, Network Coding outperforms Routing the most?

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

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

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

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

16 Example with Large Coding Advantage s n 1 2 n k t 1 R nc = k, R tree n n k+1. θ k(n k+1) n, as n = 2k. k t ( n k) Observation In practice, links are often bidirected.

17 Case Study What happens when links are bidirected? s s t 1 θ = 4/3 t 2 t 1 θ = 1 t 2

18 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 link balanced

19 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

20 Proof of theorem 2 A relay node can be split off without affecting the edge connectivity among other nodes. R nc (B) = R nc (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. R tree (B) R tree (D)

21 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.

22 α-balanced Networks For a general bidirected network, define the link imbalance ratio α = c(v, u) max c(u,v)>0 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 α R nc.

23 α-balanced Networks Proposition 2 For α 1, there exists an α-balanced network, where θ α/4. s t 1 t 2 t 3 t 4

24 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.

25 Example

26 Example b b b a+b a+b a a a+b a

27 Example a b c d e f g h i

28 Example befgh bdefh defgh abcdi abcdi acegi afghi abcdi cefgh Total rate: R tree = = 1.8 Lower bound NP-hard

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

30 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.

31 Proof of theorem 5 Proof sketch: 1 Given Hyper-network H, construct a completely link balanced network B as follows: w w e v e1 x e 2 x 1 v e2 u y u y 2 According to theorem 2, R nc (B) = R tree (B), we only need to verify R nc (H) R nc (B) and R tree (H) 1 β R tree(b).

32 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.

33 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

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

35 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, 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, Keshavarz-Haddadt, A. and Riedi, R. Bounds on the Benefit of Network Coding: Throughput and Energy Saving in Wireless Networks INFOCOM, 2008.