Network Coding in Planar Networks

Transcription

1 Network Coding in Planar Networks Zongpeng Li Department of Computer Science, University of Calgary Institute of Network Coding, CUHK April 30, PolyU 1

2 joint work with: Tang Xiahou, University of Calgary Chuan Wu, The University of Hong Kong 2

3 Talk Outline 1. Network Coding, Field Size 2. Examples in Planar Networks 3. The Sufficiency of GF(3) in Planar Networks 4. The Sufficiency of Routing in Outerplanar Networks 5. The Case of Co-face Terminals 6. Conclusion and Open Problems 3

4 Multicast with Network Coding in Directed Networks [ACLY 2000] A multicast rate h is feasible in a directed network if and only if it is feasible as a unicast rate to each receiver separately. S S S a replication point a a T 1 T 2 T 1 T 2 T 1 T 2 S S a S b encoding point a b T 1 T 2 T 1 T 2 a a+b b T a+b a+b 1 T 2 4

5 Some Basic Questions on Network Coding When is network coding necessary? How much benefit can network coding bring, over routing? How and where to encode, in general networks? The overhead of network coding? How large a field is required for coding?... 5

6 Basic Questions on Network Coding When is network coding necessary? How much benefit can network coding bring, over routing? How and where to encode, in general networks? The overhead of network coding? How large a field is required for coding? The answer often closely depend on the network configuration. network topology link capacity vector source/receiver location 6

7 Basic Questions on Network Coding When is network coding necessary? How much benefit can network coding bring, over routing? How and where to encode, in general networks? The overhead of network coding? How large a field is required for coding? The answer often closely depend on the network configuration. network topology link capacity vector source/receiver location 7

8 Small vs. Large Fields A large field makes code assignment easy: each receiver obtains linearly independent info flows. A small field leads to efficient encoding and decoding operations. A very small field may also lead to efficient code assignment algorithms. 8

9 Field Size Requirement Let s focus on a single multicast session: one source, multiple receivers. We need GF(2) for C 3,2 x T 2 x x y T 1 x+y S x+y y y x+y T 3 9

10 We need GF(3) for C 4,2 Field Size Requirement S x y x+y 2x+y T1 T 2 T 3 T 4 T 5 T 6 10

11 We need GF(2 2 ) for C 5,2. Field Size Requirement For C n,2, we need GF(q) where q n 1. S v 1 v 2 v 3 v 4 v 5 x, y, x+y, 2x+y, 3x+y, over GF(2 2 ) x, y, x+y, 2x+y, x+2y, over GF(3)?? 11

12 Current Picture Arbitrary networks: no field of constant size is always sufficient. Best known result: a field GF(q) with q k is sufficient. (k: # of multicast receivers) (actually...) In practice: randomized network coding over GF(2 8 ) or GF(2 16 ). 12

13 Main Message of This Talk Compared to an arbitrary network, a planar network is often a much better reflection of a network from practice. Linear instead of quadratic number of links Planar mesh topology instead of totally random connections. Small finite fields suffice for network coding in planar networks, and most practical networks. New deterministic code assignment algorithms Requiring much smaller fields Low (linear) or Moderate (quadratic) time complexity 13

14 Planar Graphs A planar graph is a graph that can be drawn in a 2-D plane without crossing edges. Such a no-crossing drawing is called a planar embedding. Planar graphs have many nice properties, and allow very efficient algorithms to be designed, for classic problems such as max flow, shortest path. 14

15 The VTLWaveNet in Europe A real-world wide-area/backbone network, deployed along the surface of the globe, exhibits a natural planar embedding. UK Egham Lowestoft Crawley Brussels Antwerp Belgium Amiens Biaches Paris Nancy Netherlands Germany Interxion Strasbourg Routers VTLWaveNet Network Bordeaux Toulouse France Lyon Basel Italy Bern Swizerland Milan 15

16 Tier-1 Optical Fiber Network in China A canonical planar mesh network topology. 16

17 CERNET-2 TheIP-v6networkinChina,courtesyof: 17

18 Realword Networks Far From Planar A dense wireless sensor network. One of the most non-planar types of compute network. (example from [Alzoubi et al. 2003]) 18

19 Planar Backbone of Dense Networks Executing network protocols (broadcast, multicast etc) over a very dense network is extremely inefficient. A large series of work: extract a planar backbone, then run network protocols over the planar backbone. 19

20 Example Planar Networks Requires coding at many nodes Yet coding over GF(2) suffices. S S v 1 1 v 2 1 v 3 1 v 1 1 v 2 1 v 3 1 v 4 1 u 2 1 u 3 1 u 2 1 u 3 1 u 4 1 v 2 2 v 3 2 v 2 2 v 3 2 v 4 2 u 2 3 v 3 3 u 2 3 v 3 3 u 2 4 v 3 4 u 4 3 T 1 T 2 T 1 T 2 v 4 4 (example from [LSB 2006]) 20

21 Another Example Planar Network A minimal multicast network that requires network coding for multicasting two flows. Yet no particular node must perform encoding. Coding over GF(2) suffices. T1 A T2 T3 S B T4 T5 21

22 The Sufficiency of GF(3) in Planar Networks Theorem. For multicasting h = 2 flows in a planar network, coding over GF(3) is sufficient. Inspired by [FSS 2004] and [EGS 2006]. Conjecture: holds for any h 2. 22

23 Sufficiency of GF(3) subtree decomposition Decompose a multicast flow into non-overlapping subtrees Each subtree has 1 root, 1 leaves Each root has in-degree 2 x s y x+2y x y x+2y x+y x+y (A planar bipartite network that mimics C 4,2, and hence requires GF(3).) 23

24 Sufficiency of GF(3) node expansion Decompose the plane into faces, each containing one subtree If a node has two opposite faces feeding into it, perform expansion Prepare for four-coloring a planar network 24

25 Sufficiency of GF(3) 4-coloring a planar graph Every planar graph is 5-colorable ([Kempe 1879]), and such coloring can be done in O(n) time ([CNS 1981]). Every planar graph is 4-colorable ([Appel & Haken 1976]), and such coloring can be done in O(n 2 ) time ([RSST 1996]). 25

26 Sufficiency of GF(3) four-coloring a planar graph Code assignment over GF(3). The four colors: x, y, x+y, x+2y. (Another planar multicast network requiring GF(3).) 26

27 GF(3) vs. GF(2 2 ) GF(2 2 ) may be preferred over GF(3) in practice, for: + between two packets/flows is simply bit-wise xor Code assignment complexity is O(n) instead of O(n 2 ) 2-bit symbol representation without wasted symbols 27

28 Randomized Network Coding in Planar Networks Randomized network coding has the same O(n) complexity as 5-coloring. Appears incapable of exploiting planarity. Success probability of randomized code assignment for multicast in random planar networks: field size success rate

29 Necessity of GF(3) - Example # 3 x y S x+y x x y x+y T1 T2 T3 T4 T5 x+2y x+2y x+2y x x+2y T6 x+y 29

30 Outerplanar Multicast Networks Outerplanar: all nodes adjacent to a common face Contracting the bottleneck link in the butterfly network leads to an outerplanar network Network coding not necessary anymore S S T 1 T 2 T 1 T 2 30

31 Outerplanar Multicast Networks Theorem. Network coding is equivalent to routing in an outerplanar network, for h = 2. Conjecture: holds for any h 2. 31

32 Outerplanar networks face merging Subtree decomposition, as usual. Face merging. v F α F β F F β α 32

33 Outerplanar networks two types of regions Region 1: boundary faces. Region 2: the inner region. region 2 33

34 Outerplanar networks coloring region 1 Coloring faces in region 1, using two colors only. F L F R 34

35 Outerplanar networks coloring region 2 Coloring chords in region 2, one at a time, without using a third color. Expanded Node 35

36 The Case of Co-face Terminals The case between planar and outerplanar: all multicast terminals lie on a common face. S v 1 1 S v 2 1 v 3 1 v 4 1 u 2 1 u 3 1 u 4 1 v 2 2 v 3 2 v 4 2 u 2 3 v 3 3 u 2 4 v 3 4 u 4 3 T 1 T 2 T 1 T 2 v 4 4 Conjecture: Coding over GF(2) is sufficient in this case. 36

37 Conclusion We proved that, for multicasting h = 2 flows: GF(3) is sufficient for general planar networks. Routing is sufficient for outerplanar networks. 37

38 Future Work We conjecture that, for multicasting any h 2 flows: GF(3) is sufficient for general planar networks. GF(2) is sufficient for terminal co-face networks. Routing is sufficient for outerplanar networks. Field size requirement for multiple unicast sessions in a planar network. 38

39 A Graph Minor Perspective to Network Coding We conjecture that: If a directed multicast network G requires network coding for achieving maximum throughput, then G contains a K 4 minor. If an udirected multicast network G requires network coding for achieving maximum throughput, then G contains a C 3,2 minor. If a multicast network G requires GF(2 2 ) for achieving maximum throughput, then G contains a K 5 minor. There exists a function f(q), such that if a multicast network G requires GF(q), then G contains a K f(q) minor, and f(2) = f(3) = 4, f(4) = 5. 39