Lecture 6 Network Information Flow I-Hsiang Wang Department of - PowerPoint PPT Presentation

Single Unicast and Single Multicast Linear Network Coding and Algorithms Summary Lecture 6 Network Information Flow I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw December 2, 2014 1 / 63 I-Hsiang Wang NIT Lecture 6

Single Unicast and Single Multicast Observations : I-Hsiang Wang 2 / 63 and then transmit the processed outcome. ( Coding ). 3 Each terminal is able to collect the data it receives, process them, the succeeding link. 2 A receiving terminal of one link serves as a transmitting terminal of receiving terminal. 1 A link consists of two terminals: a transmitting terminal and a point-to-point link, but a collection of these links that form a network? Linear Network Coding and Algorithms Now, what if the physical medium carrying the information is no longer a coding and channel coding is optimal in terms of rate. 2 Source-channel separation theorem suggests that separate source communication link in the most efficient way. 1 Deliver an information source to a destination over a point-to-point So far we have put our focus on point-to-point communication systems: From Point-to-Point Channel to Multi-Terminal Network Summary NIT Lecture 6

Single Unicast and Single Multicast Linear Network Coding and Algorithms I-Hsiang Wang 3 / 63 NIT Lecture 6 Summary A Noisy Communication Channel Channel capacity is determined by the Channel Coding Theorem Memoryless channel: C = max p ( x ) I ( X ; Y ) Memoryless channel C = sup I ( X ; Y ) with input cost constraint: p ( x ): E [ b ( X )] ≤ B c X N Y N W W i j ENC Channel DEC

Single Unicast and Single Multicast Linear Network Coding and Algorithms I-Hsiang Wang 4 / 63 NIT Lecture 6 Summary Representing a Noisy Channel by an Edge Channel capacity is determined by the Channel Coding Theorem e := ( i, j ) , C ( e ) := C = max I ( X i ; Y j ) Edge e i j C ( e ) i j ENC Channel DEC Point-to-point communication link represented by edge e

Single Unicast and Single Multicast Linear Network Coding and Algorithms I-Hsiang Wang 5 / 63 NIT Lecture 6 Summary Abstraction: Layering Channel capacity is determined by the Channel Coding Theorem e := ( i, j ) , C ( e ) := C = max I ( X i ; Y j ) h 1 : 2 NC ( i,j ) i m ( i,j ) ∈ Edge e i j m ( i,j ) C ( e ) � b �� m ( i,j ) m ( i,j ) i j ENC Channel DEC Point-to-point communication link represented by edge e

Single Unicast and Single Multicast Linear Network Coding and Algorithms I-Hsiang Wang 6 / 63 NIT Lecture 6 A Network: Collection of Vertices and Edges Summary s t Source Destination m ( i,j ) − → i j C ( i, j ) m ( i,j ) m ( i,j ) i j ENC Channel DEC Point-to-point communication link represented by edge e

Single Unicast and Single Multicast Linear Network Coding and Algorithms I-Hsiang Wang 7 / 63 For simplicity, in this lecture we shall focus on graphs without cycles. deliver their own messages to their respective destinations. In the network, some terminals, as sources of information, would like to NIT Lecture 6 matter the links are incoming or outgoing. the links are modeled as edges of the graph. Summary Graphical Network (1) We shall focus on graphical networks, consisting of multiple terminals and links, where the terminals are modeled as vertices of a graph, and Moreover, links are assumed to be orthogonal at each terminal, no Orthogonal l i m ( i,k ) m ( k,l ) In ( k ) Out ( k ) k j m ( j,k ) m ( k,n ) n

Single Unicast and Single Multicast 3 Two kinds of special vertices (terminals) in the graph: I-Hsiang Wang 8 / 63 Linear Network Coding and Algorithms NIT Lecture 6 Summary Graphical Network (2) Hence, a network with message set { W 1 , . . . , W K } is specified by 1 A underlying directed acyclic graph ( V , E ) , where V is the collection of vertices (terminals), and E is the collection of edges (links). 2 Capacity function C : E → [0 , ∞ ) , Source vertices S = ∪ K i =1 S i , where S i ⊂ V for each i ∈ [1 : K ] . Each vertex of S i has message W i to send. Destination vertices T = ∪ K i =1 T i , where T i ⊂ V for each i ∈ [1 : K ] . Each vertex of T i would like to decode message W i . t 1 s 1 c W 1 W 1 W 2 G = ( V , E , C ( · )) t 3 S 1 = { s 1 } , S 2 = { s 1 , s 2 } W 1 c c W 2 T 1 = { t 1 , t 3 } , T 2 = { t 2 , t 3 } t 2 s 2 c W 2 W 2

Single Unicast and Single Multicast Linear Network Coding and Algorithms I-Hsiang Wang 9 / 63 5 Multi-message multi-source multi-destination multiple unicast: 4 Multi-message single-source multi-destination multiple unicast: 3 Multi-message multi-source single-group multiple multicast: NIT Lecture 6 In particular, some common traffic patterns of interest are the following: The traffic pattern of a K -message graphical network is determined by Traffic Patterns Summary the patterns of source vertices {S i } K i =1 and destination vertices {T i } K i =1 . 1 Single-message single-source unicast: K = 1 , |S| = 1 , |T | = 1 . 2 Single-message single-source multicast: K = 1 , |S| = 1 , |T | ≥ 2 . K ≥ 2 , |S i | = 1 , |S| = K , T i = T , ∀ i ∈ [1 : K ] . K ≥ 2 , |S i | = |S| = 1 , |T i | = 1 , |T | = K , ∀ i ∈ [1 : K ] . K ≥ 2 , |S i | = |T i | = 1 , |S| = |T | = K , ∀ i ∈ [1 : K ] .

Single Unicast and Single Multicast Single Unicast I-Hsiang Wang 10 / 63 Single Multicast Linear Network Coding and Algorithms NIT Lecture 6 Single Unicast and Single Multicast Summary G = ( V , E , C ( · )) t 1 s 1 c W 1 W 1 S 1 = { s 1 } , T 1 = { t 1 } t 1 c W 1 G = ( V , E , C ( · )) s 1 W 1 S 1 = { s 1 } , T 1 = { t 1 , t 2 } t 2 c W 1

Single Unicast and Single Multicast Multiple Access: Multi-Msg Multi-Src Single-Dest Multiple Unicast I-Hsiang Wang 11 / 63 Broadcast: Multi-Msg Single-Src Multi-Dest Multiple Unicast Linear Network Coding and Algorithms NIT Lecture 6 Summary Multiple-Access and Broadcast s 1 W 1 G = ( V , E , C ( · )) t 3 S 1 = { s 1 } , S 2 = { s 2 } W 1 c c W 2 T 1 = { t 3 } , T 2 = { t 3 } s 2 W 2 t 1 c W 1 G = ( V , E , C ( · )) s 1 S 1 = { s 1 } , S 2 = { s 1 } W 1 W 2 T 1 = { t 1 } , T 2 = { t 2 } t 2 c W 2

Single Unicast and Single Multicast Linear Network Coding and Algorithms I-Hsiang Wang 12 / 63 K -Unicast: Multi-Msg Multi-Src Multiple-Dest Multiple Unicast NIT Lecture 6 Summary K -Unicast t 1 s 1 c W 1 W 1 G = ( V , E , C ( · )) S 1 = { s 1 } , S 2 = { s 2 } T 1 = { t 1 } , T 2 = { t 2 } t 2 s 2 c W 2 W 2

Single Unicast and Single Multicast 3 Multiple Access and Broadcast: extensions of max-flow-min-cut I-Hsiang Wang 13 / 63 single source), which covers both single unicast and single multicast. Below, we formally set up the problem with single message only (hence Converse: cut-set bound Achievability: Ford-Fulkerson algorithm, network coding In particular, several achievability and converse ideas will be presented: 2 Single Multicast: Max-Flow (Network Coding) = Minimum Min-Cut Linear Network Coding and Algorithms 1 Single Unicast: Max-Flow (Routing) = Min-Cut complicated traffic patterns. In this lecture, we will visit the following: In general, we have no conclusive answers for the K -unicast and more delivered from the information source(s) to the respective destination(s)? Given a network, what is the highest data rate (tuples) that can be Key question to be answered in this lecture: Lecture Overview Summary NIT Lecture 6

Single Unicast and Single Multicast code over the network consists of: I-Hsiang Wang 14 / 63 to an outgoing Linear Network Coding and Algorithms s 1 a source encoding function (encoder) NIT Lecture 6 Summary Problem Setup: Single-Message Single-Source Multicast Given a single-message single-source multicast network G = ( V , E , C ( · )) with source node S = { s } and destination nodes T = { t 1 , . . . , t K } , a ( ) 2 NR , N [ 1 : 2 NR ] [ 1 : 2 NC ( s , k ) ] → × enc ( N ) : j ∈ Out ( s ) [ 1 : 2 NR ] that maps each source message w ∈ message m ( s , k ) for each outgoing edge ( s , k ) , k ∈ Out ( s ) . k 1 m ( s,k 1 ) m ( s,k 2 ) s k 2 w m ( s,k 3 ) k 3

Single Unicast and Single Multicast that maps incoming messages I-Hsiang Wang 15 / 63 f Linear Network Coding and Algorithms to an outgoing NIT Lecture 6 Problem Setup: Single-Message Single-Source Multicast j Summary 2 a relay encoding function (encoder) for each node j ∈ V \ { s } [ 1 : 2 NC ( i , j ) ] [ 1 : 2 NC ( j , k ) ] : × → × enc ( N ) i ∈ In ( j ) k ∈ Out ( j ) ( ) m ( i , j ) : i ∈ In ( j ) message m ( j , k ) for each outgoing edge ( j , k ) , k ∈ Out ( j ) . i 1 k 1 m ( i 1 ,j ) m ( j,k 1 ) m ( j,k 2 ) j k 2 m ( j,k 3 ) m ( i 2 ,j ) k 3 i 2 ( ) = m ( i 1 , j ) , m ( i 2 , j ) , l = 1 , 2 , 3 m ( j , k l )

Single Unicast and Single Multicast that maps maps incoming messages I-Hsiang Wang 16 / 63 e codes such that lim . A source data rate R is W e Linear Network Coding and Algorithms . to a NIT Lecture 6 t k Problem Setup: Single-Message Single-Source Multicast Summary 3 a decoding function (decoder) for each destination t k ∈ T [ 1 : 2 NC ( i , t k ) ] [ 1 : 2 NR ] : × dec ( N ) → i ∈ In ( t k ) ( ) m ( i , t k ) : i ∈ In ( t k ) [ 1 : 2 NR ] reconstructed message � w ∈ i 1 m ( i 1 ,t ) � � f t b w m ( i 1 ,t ) , m ( i 2 ,t ) = m ( i 2 ,t ) i 2 { } The error probability P ( N ) W ̸ = � := Pr ( ) N →∞ P ( N ) achievable , if ∃ a sequence of 2 NR , N = 0 .

Single Unicast and Single Multicast Linear Network Coding and Algorithms Summary 1 Single Unicast and Single Multicast 2 Linear Network Coding and Algorithms 3 Summary 17 / 63 I-Hsiang Wang NIT Lecture 6