Network coding an introduction. Playing The Butterfly Lovers melody

Transcription

1 Network coding an introduction Playing The Butterfly Lovers melody

2 Store-and-forward conventional mode of transport

3 Store-and-forward inherited by telecommunications Internet

4 Store-and-forward over the Butterfly Network Either x or y x = 0 or y = 0 or Every channels transmits one bit. 0 if x = y x y Store-and-forward = if x y Figure adapted from Scientific American, Chinese 7/2007 edition

5 Store-and-forward incurs traffic congestion. Either x or y x = 0 or y = 0 or Every channels transmits one bit. 0 if x = y x y Store-and-forward = if x y Figure adapted from Scientific American, Chinese 7/2007 edition

6 Store-and-forward fails. Traffic jam Figure adapted from Scientific American, Chinese 7/2007 edition

7 A new transport mode Decode y Decode x Figure adapted from Scientific American, Chinese 7/2007 edition

8 Network coding (NC) Figure adapted from Scientific American, Chinese 7/2007 edition

9 Intracellular communications Quantum information theory Information theory Channel coding Wireless networks Graph theory Optimization theory Network coding Computer networks Switching theory Game theory Matroid theory Cryptography Computer science 0

10 Intracellular communications Quantum information theory Information theory Channel coding Wireless networks Graph theory Optimization theory Discover an store-&-forward assumption in a network in any field. Liberate yourself from it and Bingo! Computer networks Switching theory Game theory Matroid theory Cryptography Computer science

11 The Butterfly Effect diff = 0 if x = y if x y diff diff diff Figure adapted from Scientific American, Chinese 7/2007 edition

12 Is NC always based on a field? Figure adapted from Scientific American, Chinese 7/2007 edition

13 Redundancy of Data Storage Disks A B Backup A Backup B

14 Redundant Array of Independent Disks (RAID) Disks A B A B A

15 Redundant Array of Independent Disks (RAID) Disks A B A B B

16 Equivalence to the Butterfly Network A Source B A B A A B B A B A

17 Equivalence to the Butterfly Network A Source B A B A A B B B A B

18 To sustain failure of 2 disks A B Backup A Backup B Still OK

19 To sustain failure of 2 disks A B Backup A Backup B Not OK

20 To sustain failure of 2 disks A B Backup A Backup B OK again Backup A Backup B

21 NC sustains all 2-disk failures. A B A+B A B OK again

22 Distributed storage (erasure coding) [W-D 09] 2 disks per site A, B A B C D C, D A+C, 2B+D Any 2 sites suffice for recovery (2 sites 4 linearly indep. vectors) 2A+C, B+D

23 Straightforward repair 2 disks per site A, B Total transmission = 4 disks A, B A B C, D C, D A+C, 2B+D C D A+C, 2B+D 2A+C, B+D

24 Repair by NC [W-D 09] 2 disks per site Total transmission = 3 disks A B A, B C+D A, B C, D A+2B+C+D C D A+C, 2B+D 2A+B+C+D 2A+C, B+D

25 Repair when two sites fail [K. Shum 20] 2 disks per site Total transmission = 8 disks A, B A, B A B C, D C, D A+C, 2B+D C D A+C, 2B+D C, D 2A+C, B+D A+C, 2B+D 2A+C, B+D

26 Repair 2 sites by NC 2 disks per site Total transmission = 6 disks A, B A, B A B C, D A, B C, D C D A+C, 2B+D 2A+C, B+D 2A+C, B+D 2A+C, B+D

27 To backup 8 disks via block coding Single backup Disks A B C D E F G H A B H D

28 To backup 8 disks via block coding Single backup Disks A B C D E F G H A B H Analogue: A parallel bus to transmit a byte Bits A B C D E F G H A B H = Parity check bit = An erasure code = A simple block code

29 To backup 8 disks via block coding Single backup Disks A B C D E F G H A B H Analogue: A parallel bus to transmit a byte Bits A B C D E F G H A B H = Parity check bit = Block coding = NC, too

30 NC = data mixing via linear algebra at intersection points of transmission paths

31 Transmission paths in the space-time domain Point A Space Point B Time

32 Data mixing at intersection points of transmission paths in the space-time domain Point A Space Point B Time

33 A tandem of noisy channel Allow m losses out of n Allow m losses 2 3 out of n The maximum throughput from node to node 3 is m/n. To achieve this throughput via conventional 2-stage erasure coding, the middle node incurs the delay of decoding plus re-coding.

34 NC among successive packets instead of 2nd-stage coding [Lun 2006] Allow loss out of Allow loss out of 3 Packets A, B at source plus A+B A B A+B 3 scenarios of receiving 2 packets A, B or A, A+B or B, A+B

35 NC among successive packets instead of 2nd-stage coding [Lun 2006] Allow loss out of Allow loss out of 3 Packets A, B at source plus A+B A B A+B A, B or A, A+B or B, A+B Create a linear sum of the 2 received. A A+B 2A+B

36 NC among successive packets instead of 2nd-stage coding [Lun 2006] Allow loss out of Allow loss out of 3 Packets A, B at source plus A+B 3 scenarios of receiving 2 packets A A, B A A, A+B or or B A, A+B A+B A, 2A+B or or A+B B, A+B 2A+B A+B, 2A+B

37 Extension and variation (Linear combination) n = Linear combination Allow /3 loss 2 Allow /3 loss 3 Allow /3 loss 4 A network by relay or direct transmission [Lun 2006] x% loss 2 y% loss z% loss 3

38 NC for Security An eavesdropper may wiretap any one channel. x+y x+y x+y Data symbols are a and b. Let x and y be their linear combinations, say, x = a+8b, y = 2a+3b Two issues: Straightforward decoding involves multiplication and division in a finite field. Receiving nodes B and C must know the private coding coefficients, 8, 2, 3. Figure adapted from Scientific American, Chinese 7/2007 edition 43

39 Addition/shift in lieu of multiplication/division When nodes B and C have decoded x and y, there is the extra step to convert x and y into a and b. x+y x+y x+y (x y) = (a b) (x y) = (a b) = (9a 9b) // adjugate matrix Then, use a fast constant division routine to divide 9a and 9b by 9. [Li 85] IEEE Trans. on Computer.

40 Addition/shift in lieu of multiplication/division x+y z z*2 + z + 2 z z/8 + z z z/(2**8) + z z z/64 x+y x+y Then, use a fast constant division routine to divide 9a and 9b by 9. [Li 85] IEEE Trans. on Computer. (Application of Fermat s Little Thm)

41 Rid of code book vs. data rate a+x a+x a+x X a+2x a+2x a+2x Again, data symbols and coding coefficients are treated as integers modulo 2 n. Now consider publickey encryption. Coding coefficients are now public information. Something else has to be private. X X Keep a as data payload, and let X be a randomizer generated by the source node, which sends out two linear combinations of a and X, say, a+x and a+2x.

42 Variation: Secure NC with public middle channel a+x a+x a X a X The eavesdropper may wiretap a private channel and thereby receives a total of two pieces of information. a+x Public 2X a X X X

43 Variation: Secure NC with public middle channel a+x a+x M a X a X The eavesdropper may wiretap a private channel and thereby receives a total of two pieces of information. a+x X Y 2X+Y N X a X One solution is to install a new channel from node N to node M, through which N sends a new randomizer Y to M. Example by C. Chan. Node M then responds with 2X+Y by the public channel.

44 Interpreting butterfly network as 2-way relay channel T x y T R x+y T R x+y x+y R Physically, x y

45 Space T x y T R x+y T R x+y x+y R Time x y

46 Interpreting butterfly network as 2-way relay channel T x y T R x+y Wireless transmission is multicasting in nature. Perfect for applying NC! R x+y T Multicast x+y R Physically, x y

47 Half-duplex 2-way relay x y Store-and-forward, 4 steps to exchange a message through the middle relay 52

48 Half-duplex 2-way relay x y Store-and-forward, 4 steps to exchange a message through the middle relay 53

49 Half-duplex 2-way relay x y Store-and-forward, 4 steps to exchange a message through the middle relay x y NC, 3 steps 54

50 Half-duplex 2-way relay x y Store-and-forward, 4 steps to exchange a message through the middle relay x y NC, 3 steps M.I.T. prototype standard of wireless LAN (802. Wi-Fi) 3GPP2 selects NC as its potential technology in 4G wireless systems. 55

51 Communications on Mars A+B 56

52 Physical-layer NC (PNC) x y Store-and-forward, 4 steps x x y y NC, 3 steps x y 2 steps ( reception & transmission, naturally) 57

53 Physical-layer NC (PNC) x y Store-and-forward, 4 steps x x y y NC, 3 steps x+y Physical-layer NC, 2 steps 58

54 Physical-layer NC (PNC) Interference is good free higher throughput Step : Receive from multi-sources Superimposed electromagnetic waves with fading and noise 59

55 Digital PNC [Zhang et al. 06] Step : Receive from multi-sources 0/ Cos( t) 2 Cos( t) or 0 Cos( t) 0/ (BPSK) plus noise and fading (BPSK) (assuming perfect sync) 60

56 Digital PNC [Zhang et al. 06] Not easy Step a: Translation into binary 0/ Cos( t) 2 Cos( t) or 0 (+ noise, fading) Cos( t) 0/ 0 (= XOR) 6

57 Digital PNC [Zhang et al. 06] Step 2: Broadcast XOR 62

58 Analog PNC Step : Receive from multi-sources Superimposed electromagnetic waves plus noise and fading 63

59 Analog PNC Step 2: Amplify and broadcast 64

60 Analog PNC Step 2: Amplify and broadcast 65

61 Analog PNC Not easy Step 2a : Decoding at each end 66

62 Full-duplex (B. Jiao of PKU) with NC x y y x x+y x"+y x +y y x x +y 67

63 More applications Protection from one link failure a a a b b b a a a+b b b a a b b a a+b b Conventional protection requires 0 channels. NC requires only 8. 68

64 NC in multicasting & conference Multicast both symbols a and b to 3 receivers. S S a b a b a b b a+b b a a a+b By store-and-forward, symbol b goes through 3 hops. With NC, just 2 hops. [Example by P. Chou] 69

65 All-to-all transmission: In each round of transmission, every node can multicast to or receive from the two neighbors. By store-and-forward, n 2 rounds of transmissions. With NC, just (n )/2 rounds of transmission. [Ex. by F-S]

66 All-to-all transmission: In each round of transmission, every node can multicast to or receive from the two neighbors. st round: a a 2 nd round: a+c a+b c b a+c a+b c b c+d b+d d d c+d b+d We illustrate green-to-all transmission in 4 rounds. Parallel red-to-all transmission is always in opposite directions.

67 NC in network tomography Network Tomography To characterize network behavior by taking measurements at the edge. Conventional active probing Many packets are multicast to several receivers via a multicast tree inside the network. Receivers experience the loss event which helps identifying shared links and estimating the loss rates. NC probing Coding coefficients contained in probe packets provide additional information about the original packets that were combined. 74

68 (2) Basic theory over acyclic networks + more applications Playing The Butterfly Lovers melody

69 Introduction to NC theory Represent the alphabet of data symbols in transmission by a finite field F. Here F = GF(2). The source S sends out symbols x and y, one through each outgoing channels. Then, every channel transmits a linear combination of x and y subject to (Rule ) x x x s x+y y y y x+y x+y

70 Transmitted symbol coding vector = (x y) Then, every channel transmits a linear combination of x and y subject to (Rule ) We can express each transmitted symbol as the dot product between (x y) and a coding vector: x x s y y x = (x y) 0 x x+y y y = (x y) 0 x+y x+y x+y = (x y)

71 Definition of a network code Definition. An F-linear network code on the network or, an F-code in short, assigns an -dim coding vector f e F to each channel e subject to (Rule ) For every node x on the network, let V x denote the vector space generated by incoming coding vectors to x. Then, the coding vector assigned to each outgoing vector of x belongs to V x s 0 0 0

72 Definition of a network code Definition. An F-linear network code on the network or, an F-code in short, assigns an -dim coding vector f e F to each channel e subject to (Rule ) For every node x on the network, let V x denote the vector space spanned by incoming coding vectors to x. Then, the coding vector assigned to each outgoing vector of x belongs to V x. Example s x V x =, = F 2 and F 2 0

73 Definition of a network code Definition. An F-linear network code on the network or, an F-code in short, assigns an -dim coding vector f e F to each channel e subject to (Rule ) For every node x on the network, let V x denote the vector space generated by incoming coding vectors to x. Then, the coding vector assigned to each outgoing vector of x belongs to V x s x (Rule 2) Coding vectors on outgoing channels from S form the natural basis of F 2.

74 Adjacent pair of channels When channel d ends at the node where channel e begins, an adjacent pair (d, e) is formed. It corresponds to a red arrow inside the joining node. Think of each channel as a highway and each adjacent pair an interconnecting ramp. Channel d (d, e) Channel e s

75 Alternative definition of a network code Definition. An F-code assigns a coding coefficient k d,e F to each adjacent pair (d, e). s Channel d (d, e) Channel e

76 Alternative definition of a network code Definition. An F-code assigns a coding coefficient k d,e F to each adjacent pair (d, e). Example. Every coding coefficient = GF(2) on the Butterfly Network. s

77 Transmitted symbol over a channel At an intermediate node, every outgoing symbol is a linear combination of incoming symbols so that each coding coefficient represents the linear gain. Through top-down telescoping on an acyclic network, the symbol transmitted over every channel e is a linear combination of x and y in correspondence to the coding vector f e. x x x+y s x y x+y y x+y y

78 Relationship between two definitions A set of coding coefficients (handy for encoding) top-down derivation A set of coding vectors (handy for decoding) The mapping is -to- only when all incoming coding vectors to each node are linearly indep s

79 Relationship between two definitions A set of coding coefficients (handy for encoding) 0 s 0 top-down derivation A set of coding vectors (handy for decoding) The mapping is -to- only when all incoming coding vectors to each node are linearly indep

80 Decoding at a node The two incoming channels to the node v are tv and uv. Juxtapose their associated coding vectors f tv and f uv into the matrix M v = 0 Given the message (a b) from the source, the two symbols received by node v form the row vector (a a+b) = (a b) M v The message is then decoded by: (a b) = (a a+b) M v t 0 s v u

81 Decoding at a node M v = 0 Given the message (a b) from the source, the two symbols received by node v form the row vector s (a a+b) = (a b) M v The message is then decoded by: t (a b) = (a a+b) M v Conditions for decodability at v: Incoming coding vectors span the full rank (= 2 here). The node v knows the incoming encoding vectors. v 0 u

82 Do incoming vectors span full rank? The dimension spanned by incoming coding vectors to node v = Information rate from s to node v maxflow(v) = max flow from s to v = min cut between s and v The cut beneath s = The message size s v

83 Intrinsic limitation on information rate Constraints on linear independence among coding vectors are: Information rate to v maxflow(v) the message size // A constraint by the network topology A subtle constraint by the choice of the symbol field F

84 Choice of the symbol field F matters. Q. Can 2 symbols be transmitted from s to all six receivers? // Here maxflow of every receiver = 2. No, when a symbol = a bit. Yes, when a symbol = a byte or two bits. s 9

85 Choice of the symbol field F matters. A symbol = a bit when F = GF(2). A symbol = a byte when F = GF(256). Here we need four 2-dim vectors that are pairwise linearly independent. There are at most F + such vectors. s 92

86 Fundamental theorem of NC Linear dependence among coding vectors are: Information rate to v maxflow(v) the message size // A constraint by the network topology A subtle constraint by the choice of the symbol field F Theory of NC will finesse this constraint by assuming large enough F so as to guarantee the existence of an optimal network code. 93

87 Existence/construction of optimal network code Theorem of Linear NC. Given a positive integer and an acyclic network, there exists an -dim NC F-valued generic NC (and hence an -dim NC that is optimal in every conceivable sense) when F is sufficiently large. Proof. Delayed till a later topic on Fundamental theorem of NC. 94

88 A hierarchy of NC optimality s Definition. An -dim F -NC qualifies as: a generic NC 95

89 A hierarchy of NC optimality s Definition. An -dim F -NC qualifies as: a generic NC a linear broadcast when info rate to v = maxflow(v) for all v 97

90 A hierarchy of NC optimality s Definition. An -dim F -NC qualifies as: a generic NC a linear broadcast when info rate to v = maxflow(v) for all v a linear multicast when info rate to v = for all eligible v // A node v is eligible when maxflow(v) =. Decodability 99

91 A hierarchy of NC optimality s Definition. An -dim F -NC qualifies as: a generic NC a linear broadcast when info rate to v = maxflow(v) for all v a linear multicast when info rate to v = for all eligible v // A node v is eligible when info rate to v is. optimal w.r.t. some designated eligible receivers // Good enough for most applications 00

92 How are coding vectors known? Method. Predetermined All coding coefficients throughout the network are fixed, hence so are all coding vectors. 0

93 Method 2. Include f e in the packet header for transmission over the channel e. An intermediate node chooses the coding coefficients, which may be secret, random, dynamically changing, for flexible applications. In this case, the coding coefficients are 0 and Any linearly independent packets suffice for decoding

94 Method 2. Include f e in the packet header for transmission over the channel e. Example. 00 packets to be disseminated A packet = 000 data symbols A data symbol = 2 bytes = an element in a field of the size 2 6 NC overhead = 00 extra symbols per transmission of a 000-symbol payload = 0%

95 Random NC Coding coefficients x and y are randomly distributed over a sizable base field F. x x+5y 2x+6y 3x+7y 4x+8y y 04

96 Problems of P2P by Bit Torrent Constant problems in Bit Torrent (BT) operation: Which packet to send to neighboring peers? Which packet to ask for from upstream? Heuristic algorithm of BT: Randomness at the beginning Local rarest scheme after a node has acquired a few packets. 05

97 P2P content delivery by random NC [Ho et al.] Constant problems in Bit Torrent (BT) operation: Which packet to send to neighboring peers? Which packet to ask for from them? Heuristic algorithm of BT: Randomness at the beginning Local rarest scheme after a node has acquired a few packets. x x+5y 2x+6y 3x+7y 4x+8y y 06

98 NC for P2P file-sharing, video streaming 08

99 Benefit of random NC over BT Benefits: Performance insensitive to topology of the overlay P2P network Robustness w.r.t. to dynamical change of topology (high churn rate). Speed, i.e., increased bandwidth Favorable analysis of Avalanche [G-R] Opposite analysis. In practice, there is little increase in bandwidth, because the critical constraint in P2P is node capacity [Chen et al.]. 09

100 蝴蝶网的蝴蝶效应 Figure adapted from Scientific American, Chinese 7/2007 edition

101 蝴蝶网的蝴蝶效应 Figure adapted from Scientific American, Chinese 7/2007 edition

102 蝴蝶网的蝴蝶效应 Figure adapted from Scientific American, Chinese 7/2007 edition

103 Initial development of NC theory Li discovered the example of the Butterfly Network and generalized it into a linear-algebraic theory of network coding (997 manuscript). Li-Yeung, Proc. ISORA Conference, 8/998. Li-Yeung, Proc. IT Workshop, 7/999. Li et al., Linear network coding, IEEE Trans. IT, 2/2003. Kotter-Medard, IEEE/ACM Trans. on Networking, 0/2003. The Butterfly Network later became the icon of network coding. About 0,000 papers followed up. Initial publications of the Butterfly Network Li-Sun, IEEE Trans. IT, /20, (about convolutional network coding theory on non-acyclic networks)

104 Before NC, there were work on joint coding at intersection points of transmission paths Earliest papers with a flavor of NC: Celebiler-Stette, Proceedings of IEEE, /978 (about increasing down-link capacity of regenerative satellite repeater.) Patterson et al., Proceedings of VLDB, 4/988 (about redundant array of independent disks in magnetic data storage.) Li, Proceedings of IT Workshop, 7/988, (about optimizing transmission capacity by mix coding among parallel channels).

105 Network coding Conference paper in 8/998. Li et al., Linear network coding, IEEE Trans. IT, 2/2003. Kotter-Medard, An algebraic approach to network coding, IEEE/ACM Tr. Netw., 0/ ,000 papers followed up. Theories with NC flavor Celebiler-Stette, 978 Patterson et al., 988 Li, 988 Ahlswede et al., conference paper in 8/998. Ahlswede et al., Network Information Flow, IEEE Trans. IT, 7/2000 (Information theory in the form of limiting probability when something ). An incompatible theory in terms of angle of viewpoint, formulation, mathematical approach, elegance, result, transparency, applicability, and implementability.

106 What is NC theory is made murky. Conference paper in 8/998. Li et al., Linear network coding, IEEE Trans. IT, 2/2003. Kotter-Medard, An algebraic approach to network coding, IEEE/ACM Tr. Netw., 0/ ,000 papers followed up. Theories with NC flavor Celebiler-Stette, 978 Patterson et al., 988 Li, 988 Ahlswede et al., conference paper in 8/998. Ahlswede et al., Network Information Flow, IEEE Trans. IT, 7/2000 (Information theory in the form of limiting probability when something ). An incompatible theory in terms of angle of viewpoint, formulation, mathematical approach, elegance, result, transparency, applicability, and implementability.