Contents Network coding NC Butterfly Network 11 Contents

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Contents 唐诗 李商隐 联想 Network coding NC = Butterfly Network 西昆诗派 11

Contents 唐诗 李商隐 联想 Network coding NC = Butterfly Network 西昆诗派 11

Contents 唐诗 李商隐 西昆诗派 联想 Network coding NC = Butterfly Network 简介NC 理论 及其

Contents 唐诗 李商隐 西昆诗派 联想 Network coding NC = Butterfly Network 简介NC 理论 及其 三大应用 范畴: (A) Wireless communications (B) Redundant storage (C) P 2 P content delivery 12

Part (A): Wireless communications 唐诗 李商隐 联想 (1) Network coding Butterfly Network Wireless Butterfly

Part (A): Wireless communications 唐诗 李商隐 联想 (1) Network coding Butterfly Network Wireless Butterfly Network 西昆诗派 Physical-layer NC 13

Part (B): Redundant storage 唐诗 李商隐 联想 Network coding Butterfly Network Wireless Butterfly Network

Part (B): Redundant storage 唐诗 李商隐 联想 Network coding Butterfly Network Wireless Butterfly Network 西昆诗派 Physical-layer NC (B) NC for redundant storage 杜牧 Coding redundancy 14

Part (C): P 2 P content delivery 唐诗 李商隐 联想 Network coding Butterfly Network

Part (C): P 2 P content delivery 唐诗 李商隐 联想 Network coding Butterfly Network Wireless Butterfly Network 西昆诗派 Physical-layer NC NC for redundant storage 杜牧 Coding redundancy (C) 简介NC 理论 禅师的偈子 NC for P 2 P content delivery 15

Part (A): Wireless communications 唐诗 联想 回想人生中的顿悟 李商隐 Network coding Butterfly Network Wireless Butterfly

Part (A): Wireless communications 唐诗 联想 回想人生中的顿悟 李商隐 Network coding Butterfly Network Wireless Butterfly Network 西昆诗派 Physical-layer NC (2) NC for redundant storage 杜牧 禅师的偈子 Coding redundancy (3) Introduction to NC theory NC for P 2 P content delivery

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In 科学人, 7/2007 络 Excerpted from Scientific American, 7/2007 Chinese edition. http: //sa. ylib.

In 科学人, 7/2007 络 Excerpted from Scientific American, 7/2007 Chinese edition. http: //sa. ylib. com/readshow. asp? FDoc. No=1041&Doc. No=1638 21

网络编码 与 蝴蝶 x = 0 or 1 y = 0 or 1 Every

网络编码 与 蝴蝶 x = 0 or 1 y = 0 or 1 Every channels transmits one bit. Store-and-forward Either x or y ? Figure adapted from Scientific American, Chinese 7/2007 edition 22

Store-and-forward Traffic jam Figure adapted from Scientific American, Chinese 7/2007 edition 23

Store-and-forward Traffic jam Figure adapted from Scientific American, Chinese 7/2007 edition 23

Network coding (NC) x y = Decode y 0 if x = y 1

Network coding (NC) x y = Decode y 0 if x = y 1 if x y Decode x Figure adapted from Scientific American, Chinese 7/2007 edition 24

In 科学人, 7/2007 25

In 科学人, 7/2007 25

1. Linear network coding (NC) 2. Convolutional NC 3. NC theory via commutative algebra

1. Linear network coding (NC) 2. Convolutional NC 3. NC theory via commutative algebra 4. Construction of NC over cyclic networks 5. Martingale of patterns 6. Computing by symmetry 7. Unified algebraic theory of sorting, routing, multicasting, & concentration networks 8. Cut-through coding 9. Algebraic transform of multistage interconnection networks 10. Scalable nonblocking switches and geometric intuition 数 学 与 程 的 对 话

Butterfly network 一举两得 Wireless transmission is multicasting in nature, perfect for applying NC. Multicasting

Butterfly network 一举两得 Wireless transmission is multicasting in nature, perfect for applying NC. Multicasting Figure adapted from Scientific American, Chinese 7/2007 edition 28

Wireless version of Butterfly Network T x T y R x+y Wireless transmission is

Wireless version of Butterfly Network T x T y R x+y Wireless transmission is multicasting in nature, perfect for applying NC. Physically, T x+y R x x+y R y 29

Message exchange over Wireless Butterfly Network x 2021/2/27 y Store-and-forward, 4 steps 30

Message exchange over Wireless Butterfly Network x 2021/2/27 y Store-and-forward, 4 steps 30

Message exchange over Wireless Butterfly Network x 2021/2/27 y Store-and-forward, 4 steps to exchange

Message exchange over Wireless Butterfly Network x 2021/2/27 y Store-and-forward, 4 steps to exchange a message through the middle relay 31

Message exchange over Wireless Butterfly Network x y 2021/2/27 Store-and-forward, 4 steps to exchange

Message exchange over Wireless Butterfly Network x y 2021/2/27 Store-and-forward, 4 steps to exchange a message through the middle relay NC, 3 steps 32

Message exchange over Wireless Butterfly Network y x Store-and-forward, 4 steps to exchange a

Message exchange over Wireless Butterfly Network y x Store-and-forward, 4 steps to exchange a message through the middle relay NC, 3 steps x y 2021/2/27 33

B A 2021/2/27 A B B A Communications on Mars A+B 34

B A 2021/2/27 A B B A Communications on Mars A+B 34

Optical application of wireless Butterfly Network Did not happen 2000 years ago 35

Optical application of wireless Butterfly Network Did not happen 2000 years ago 35

如何可以略胜文采 ? x x x 2021/2/27 x y y Store-and-forward, 4 steps y NC,

如何可以略胜文采 ? x x x 2021/2/27 x y y Store-and-forward, 4 steps y NC, 3 steps y 2 steps (1 reception & 1 transmission, naturally) 37

如何可以略胜文采 ? x x x y x+y 2021/2/27 y Store-and-forward, 4 steps y NC,

如何可以略胜文采 ? x x x y x+y 2021/2/27 y Store-and-forward, 4 steps y NC, 3 steps 2 steps 38

Physical-layer NC (PNC) Step 1: Receive from multi-sources Superimposed electromagnetic waves with fading and

Physical-layer NC (PNC) Step 1: Receive from multi-sources Superimposed electromagnetic waves with fading and noise 2021/2/27 39

Digital PNC [ZLL 06] Interference outcome free higher throughput Step 1: Receive from multi-sources

Digital PNC [ZLL 06] Interference outcome free higher throughput Step 1: Receive from multi-sources 0/1 Cos( t) 2021/2/27 2 Cos( t) or 0 plus noise and fading (Assuming perfect sync) Cos( t) 0/1 40

Digital PNC [ZLL 06] Step 1 a: Translation into binary 0/1 Cos( t) 2

Digital PNC [ZLL 06] Step 1 a: Translation into binary 0/1 Cos( t) 2 Cos( t) or 0 (+ noise, fading) Cos( t) 0/1 (Not easy) Decode into: 0 1 (= XOR) 2021/2/27 41

Digital PNC [ZLL 06] Step 2: Broadcast XOR 2021/2/27 42

Digital PNC [ZLL 06] Step 2: Broadcast XOR 2021/2/27 42

Analog PNC Step 1: Receive from multi-sources Superimposed electromagnetic waves plus noise and fading

Analog PNC Step 1: Receive from multi-sources Superimposed electromagnetic waves plus noise and fading 2021/2/27 43

Analog PNC Step 2: Amplify and broadcast 2021/2/27 44

Analog PNC Step 2: Amplify and broadcast 2021/2/27 44

Analog PNC Step 2: Amplify and broadcast 2021/2/27 45

Analog PNC Step 2: Amplify and broadcast 2021/2/27 45

Analog PNC Step 2 a : Decoding at each end (not easy) 2021/2/27 46

Analog PNC Step 2 a : Decoding at each end (not easy) 2021/2/27 46

Digital or analog PNC x y Store-and-forward, 4 steps x y NC, 3 steps

Digital or analog PNC x y Store-and-forward, 4 steps x y NC, 3 steps x y PNC, 2 steps Decoding not easy 2021/2/27 47

Digital or analog PNC x y Store-and-forward, 4 steps x y NC, 3 steps

Digital or analog PNC x y Store-and-forward, 4 steps x y NC, 3 steps x y PNC, 2 steps Decoding not easy 宗法 PNC = A gold mine 现代版 西昆体 They all publish happily, with > 2 Chinese authors per paper. 2021/2/27 53

Part (B): Redundancy in Data Storage Disk A B Backup A B This corresponds

Part (B): Redundancy in Data Storage Disk A B Backup A B This corresponds to store-and-forward. 2021/2/27 54

Part (B): Redundancy in Data Storage Single backup RAIDs 2021/2/27 A B = NC

Part (B): Redundancy in Data Storage Single backup RAIDs 2021/2/27 A B = NC 55

Perform NC over an imaginary network Data Disks A B A’ 2021/2/27 56

Perform NC over an imaginary network Data Disks A B A’ 2021/2/27 56

Perform NC over an imaginary network Data Disks A B A’ B’ A B

Perform NC over an imaginary network Data Disks A B A’ B’ A B 57

Perform NC over an imaginary network Data Disks A B A’ B’ A B

Perform NC over an imaginary network Data Disks A B A’ B’ A B 58

Redundancy in backup storage Single backup RAIDs A B C D E F G

Redundancy in backup storage Single backup RAIDs A B C D E F G H A B … H 59

To backup n disks by NC Single backup RAIDs A B C D E

To backup n disks by NC Single backup RAIDs A B C D E F G H A B … H D’ 2021/2/27 60

To backup n disks by NC Single backup RAIDs 2021/2/27 A B C D

To backup n disks by NC Single backup RAIDs 2021/2/27 A B C D E D’ E’ F G H A B … H 61

Logical equivalent Single backup RAIDs A B C D E F G H A

Logical equivalent Single backup RAIDs A B C D E F G H A B … H A parallel bus to transmit, say, a byte Bits A B C D E F G H A B … H = Parity check bit = Block coding = NC, too 62

Network coding or block coding • Coding is an art in optimizing redundancy. •

Network coding or block coding • Coding is an art in optimizing redundancy. • The greatest artist on redundancy is 杜牧. 63

清明雨纷,路人断魂。4言 酒家何处?指杏花村。 x y x y 2021/2/27 Store-and-forward, NC, 4 steps 3 steps PNC,

清明雨纷,路人断魂。4言 酒家何处?指杏花村。 x y x y 2021/2/27 Store-and-forward, NC, 4 steps 3 steps PNC, 2 steps Decoding not easy 72

清明雨纷,路人断魂。4言 酒家何处?指杏花村。 x y x y 2021/2/27 Store-and-forward, NC, 4 steps 3 steps PNC,

清明雨纷,路人断魂。4言 酒家何处?指杏花村。 x y x y 2021/2/27 Store-and-forward, NC, 4 steps 3 steps PNC, 2 steps Decoding not easy 73

清明雨纷,路人断魂。 4言 x 酒家何处?指杏花村。 y Store-and-forward, 清明雨,人断魂。 3言 y x 酒何处?杏花村。 NC, x 2021/2/27

清明雨纷,路人断魂。 4言 x 酒家何处?指杏花村。 y Store-and-forward, 清明雨,人断魂。 3言 y x 酒何处?杏花村。 NC, x 2021/2/27 y 4 steps 3 steps PNC, 2 steps Decoding not easy 74

Part (C): Introduction to NC theory & P 2 P application hen channel d

Part (C): Introduction to NC theory & P 2 P application hen channel d ends at the node where channel e begins, the “adjacent pair” (d, e) corresponds by a red arrow inside the joining node. Channel d (d, e) Channel e s

Definition of a linear network code An F-linear network code assigns a coding coefficient

Definition of a linear network code An F-linear network code assigns a coding coefficient kd, e F, where the finite field F is the alphabet of data symbol. s kd, e

Example of a linear network code An F-linear network code assigns a coding coefficient

Example of a linear network code An F-linear network code assigns a coding coefficient F, where the finite field F is the alphabet of data symbol. Here, F = GRF(2) = {0, 1}. s 1 1

Transmitted symbol The source sends out symbols x and y through two channels. x

Transmitted symbol The source sends out symbols x and y through two channels. x Every intermediate node makes a linear combination of incoming symbols for each of its outgoing channels. Through top-down telescoping, the symbol transmitted over every channel is a linear combination of x and y. s y 1 1 x y x+y 1 1 x+y

Transmitted symbols = linear combinations of x & y s x x = (x

Transmitted symbols = linear combinations of x & y s x x = (x y) y 1 0 x y = (x y) y 0 1 x x+y = (x y) y x+y 1 1 x+y

Transmitted symbol = (x y) · Natural basis of F 1 0 The symbol

Transmitted symbol = (x y) · Natural basis of F 1 0 The symbol transmitted over a channel e is written as (x y) fe where fe is a column vector over F, called the coding vector. 0 1 1 0 1 1 1 0 0 1 1 1 1

Optimal network codes Fundamental Theorem of Linear NC [LY 1998, LYC 2003]. When |F|

Optimal network codes Fundamental Theorem of Linear NC [LY 1998, LYC 2003]. When |F| is large enough, there exists an optimal network code, that is: • For every node v, incoming coding vectors to v span the dimension equal to maxflow from source to v. 2021/2/27 87

What if |F| is not large enough? s Q. Can two symbols be transmitted

What if |F| is not large enough? s Q. Can two symbols be transmitted from s to all six receivers? // Here max flow from s to every receiver is 2. • No, when a symbol means a bit. • Yes, when a symbol means a byte. 2021/2/27 88

P 2 P content delivery by store-and-forward Constant problems in Bit Torrent (BT) operation:

P 2 P content delivery by store-and-forward Constant problems in Bit Torrent (BT) operation: n Which packet to send to neighboring peers? n Which packet to ask for from upstream? Heuristic algorithm of BT: n Randomness at the beginning n “Local rarest” scheme after a node has acquired a few packets. 2021/2/27 89

P 2 P content delivery by store-and-forward Constant problems in Bit Torrent (BT) operation:

P 2 P content delivery by store-and-forward Constant problems in Bit Torrent (BT) operation: n Which packet to send to neighboring peers? 神秀禅师: n Which packet to ask for from them? 身是菩提树, Heuristic algorithm of BT: n Randomness at the beginning n “Local rarest” scheme after a node has acquired a few packets. 2021/2/27 心如明镜台, 时时勤拂拭, 勿使惹尘埃。 90

P 2 P content delivery by store-and-forward Constant Vo problems in Bit Torrent (BT)

P 2 P content delivery by store-and-forward Constant Vo problems in Bit Torrent (BT) operation: ide n Which packet to send to neighboring peers? db n Which packet to ask for from them? y ran d om Heuristic algorithm of BT: n n l Randomness at the beginning inea r NC “Local rarest” scheme after a node has acquired a few packets. 2021/2/27 91

P 2 P content delivery by random linear NC • Use random coding coefficients

P 2 P content delivery by random linear NC • Use random coding coefficients a field of size 216 or 224. • Allow the linear NC the flexibility of being secret, random, dynamically changing, … • Every node includes the coding vector in the header of the transmitted packet. Eventually any set of received packets with linearly independent coding vectors suffice for decoding. 2021/2/27 5 6 7 8 … 1 2 3 4 … u v u+5 v 2 u+6 v 3 u+7 v 4 u+8 v … 92

P 2 P content delivery by random linear NC Constant Vo problems in Bit

P 2 P content delivery by random linear NC Constant Vo problems in Bit Torrent (BT) operation: ide n Which packet to send to neighboring peers? 六祖惠能: db n Which packet to ask for from them? y 菩提本无树, ran d om Heuristic algorithm of BT: n n l Randomness at the beginning inea r NC “Local rarest” scheme after a node has acquired a few packets. 明镜亦非台, 本来无一物, 何处惹尘埃。 一个文盲作出千古传诵的偈子 2021/2/27 93

NC 禅说 试问: “禅师, what is NC? ” 神秀答: “Read this & that papers

NC 禅说 试问: “禅师, what is NC? ” 神秀答: “Read this & that papers … If necessary, more papers渐渐就会懂. ” … 慢慢读、慢慢想 … 0 if x = y x y = 惠能答: “Look at Butterfly 1 if x y network. ” 当场就顿悟!

谢谢 11. Scalability of conditionally nonblocking switches 12. Coding by algebraic topology

谢谢 11. Scalability of conditionally nonblocking switches 12. Coding by algebraic topology