Broadcasting With Side Information

Transcription

1 Department of Electrical and Computer Engineering Texas A&M Noga Alon, Avinatan Hasidim, Eyal Lubetzky, Uri Stav, Amit Weinstein, FOCS2008

2 Outline I shall avoid rigorous math and terminologies and be more intuitive, but last part might be more technical I shall be presenting : (FOCS 2008) Index Coding with Side Information (FOCS 2006) Coding on demand by an informed source (ISCOD) for efficient broadcast of different supplemental data to caching clients (IEEE/ACM Transactions on Networking 2006) Dissertations E. Lubetzky and Thesis by Amit Weinstein (Students of Noga Alon) Work by Chaudhry and Sprintson

3 Index Coding A Server with a set of packets A set of Clients Each Client wants some set of packets from server: Wants set Each Client has some set of packets available: Has (Cached) set, may be overheard from previous transmission Opportunistic Listening The server can broadcast the packets or their combinations (encoding)

4 Index Coding A Server with a set of packets A set of Clients Each Client wants some set of packets from server: Wants set Each Client has some set of packets available: Has (Cached) set, may be overheard from previous transmission Opportunistic Listening The server can broadcast the packets or their combinations (encoding)

5 Index Coding A Server with a set of packets A set of Clients Each Client wants some set of packets from server: Wants set Each Client has some set of packets available: Has (Cached) set, may be overheard from previous transmission Opportunistic Listening The server can broadcast the packets or their combinations (encoding)

6 Index Coding A Server with a set of packets A set of Clients Each Client wants some set of packets from server: Wants set Each Client has some set of packets available: Has (Cached) set, may be overheard from previous transmission Opportunistic Listening The server can broadcast the packets or their combinations (encoding)

7 Index Coding A Server with a set of packets A set of Clients Each Client wants some set of packets from server: Wants set Each Client has some set of packets available: Has (Cached) set, may be overheard from previous transmission Opportunistic Listening The server can broadcast the packets or their combinations (encoding)

8 Index Coding (cont.) Objective To satisfy the demands of all clients with the minimum possible number of transmissions from server by taking advantage of the Has sets NP-Hard to solve and approximate; measure of hardness?

9 Index Coding (cont.) Objective To satisfy the demands of all clients with the minimum possible number of transmissions from server by taking advantage of the Has sets NP-Hard to solve and approximate; measure of hardness?

10 Index Coding (cont.) Objective To satisfy the demands of all clients with the minimum possible number of transmissions from server by taking advantage of the Has sets NP-Hard to solve and approximate; measure of hardness?

11 Motivation: Content Distribution Network Number of Transmissions Without Index Coding 3 With Index Coding 1 Server needs to transmit only one packet A + B + C Advantages: Load balancing, efficient use of resources, lesser delay

12 Motivation: Content Distribution Network Number of Transmissions Without Index Coding 3 With Index Coding 1 Server needs to transmit only one packet A + B + C Advantages: Load balancing, efficient use of resources, lesser delay

13 Motivation: Content Distribution Network Number of Transmissions Without Index Coding 3 With Index Coding 1 Server needs to transmit only one packet A + B + C Advantages: Load balancing, efficient use of resources, lesser delay

14 Motivation: Content Distribution Network Number of Transmissions Without Index Coding 3 With Index Coding 1 Server needs to transmit only one packet A + B + C Advantages: Load balancing, efficient use of resources, lesser delay

15 Motivation: Content Distribution Network Number of Transmissions Without Index Coding 3 With Index Coding 1 Server needs to transmit only one packet A + B + C Advantages: Load balancing, efficient use of resources, lesser delay

16 Motivation: Content Distribution Network Number of Transmissions Without Index Coding 3 With Index Coding 1 Server needs to transmit only one packet A + B + C Advantages: Load balancing, efficient use of resources, lesser delay

17 Motivation: Content Distribution Network Number of Transmissions Without Index Coding 3 With Index Coding 1 Server needs to transmit only one packet A + B + C Advantages: Load balancing, efficient use of resources, lesser delay

18 Motivation: Content Distribution Network Number of Transmissions Without Index Coding 3 With Index Coding 1 Server needs to transmit only one packet A + B + C Advantages: Load balancing, efficient use of resources, lesser delay

19 Application: A Broadcast Network Number of Transmissions Without Index Coding = 4 With Index Coding = 2, i.e, Server needs to transmit only two packets p 1 + p 2 + p 3 and p 1 + p 4 Advantages: Power saving i.e, longer battery life, bandwidth saving, lesser latency at overall system level

20 Application: A Broadcast Network Number of Transmissions Without Index Coding = 4 With Index Coding = 2, i.e, Server needs to transmit only two packets p 1 + p 2 + p 3 and p 1 + p 4 Advantages: Power saving i.e, longer battery life, bandwidth saving, lesser latency at overall system level

21 Application: A Broadcast Network Number of Transmissions Without Index Coding = 4 With Index Coding = 2, i.e, Server needs to transmit only two packets p 1 + p 2 + p 3 and p 1 + p 4 Advantages: Power saving i.e, longer battery life, bandwidth saving, lesser latency at overall system level

22 Previous Work The simpler model of Index Coding was first suggested by [Birk and Kol (98)]. Given as a part of homework assignment ISCOD (Informed Source Coding) In this model each packet consists of only one bit Optimal linear schemes were characterized by [Bar-Yossef, Birk, Jayram and Kol (06)]. It was shown that linear encoding schemes are sometimes far from optimal [Lubetzky and Stav (07)]. Relationship to Network Coding What happens when packet size is not a bit? Some exact and heuristic approaches to solve Index Coding [Chaudhry and Spnitson]. Can we give some rates bounds?

23 Previous Work The simpler model of Index Coding was first suggested by [Birk and Kol (98)]. Given as a part of homework assignment ISCOD (Informed Source Coding) In this model each packet consists of only one bit Optimal linear schemes were characterized by [Bar-Yossef, Birk, Jayram and Kol (06)]. It was shown that linear encoding schemes are sometimes far from optimal [Lubetzky and Stav (07)]. Relationship to Network Coding What happens when packet size is not a bit? Some exact and heuristic approaches to solve Index Coding [Chaudhry and Spnitson]. Can we give some rates bounds?

24 Previous Work The simpler model of Index Coding was first suggested by [Birk and Kol (98)]. Given as a part of homework assignment ISCOD (Informed Source Coding) In this model each packet consists of only one bit Optimal linear schemes were characterized by [Bar-Yossef, Birk, Jayram and Kol (06)]. It was shown that linear encoding schemes are sometimes far from optimal [Lubetzky and Stav (07)]. Relationship to Network Coding What happens when packet size is not a bit? Some exact and heuristic approaches to solve Index Coding [Chaudhry and Spnitson]. Can we give some rates bounds?

25 Previous Work The simpler model of Index Coding was first suggested by [Birk and Kol (98)]. Given as a part of homework assignment ISCOD (Informed Source Coding) In this model each packet consists of only one bit Optimal linear schemes were characterized by [Bar-Yossef, Birk, Jayram and Kol (06)]. It was shown that linear encoding schemes are sometimes far from optimal [Lubetzky and Stav (07)]. Relationship to Network Coding What happens when packet size is not a bit? Some exact and heuristic approaches to solve Index Coding [Chaudhry and Spnitson]. Can we give some rates bounds?

26 Previous Work The simpler model of Index Coding was first suggested by [Birk and Kol (98)]. Given as a part of homework assignment ISCOD (Informed Source Coding) In this model each packet consists of only one bit Optimal linear schemes were characterized by [Bar-Yossef, Birk, Jayram and Kol (06)]. It was shown that linear encoding schemes are sometimes far from optimal [Lubetzky and Stav (07)]. Relationship to Network Coding What happens when packet size is not a bit? Some exact and heuristic approaches to solve Index Coding [Chaudhry and Spnitson]. Can we give some rates bounds?

27 Previous Work The simpler model of Index Coding was first suggested by [Birk and Kol (98)]. Given as a part of homework assignment ISCOD (Informed Source Coding) In this model each packet consists of only one bit Optimal linear schemes were characterized by [Bar-Yossef, Birk, Jayram and Kol (06)]. It was shown that linear encoding schemes are sometimes far from optimal [Lubetzky and Stav (07)]. Relationship to Network Coding What happens when packet size is not a bit? Some exact and heuristic approaches to solve Index Coding [Chaudhry and Spnitson]. Can we give some rates bounds?

28 Matrix Fits a Graph Let G be a directed graph on n vertices without self-loops. A 0-1 matrix A = (a ij ) fits G if for all i and j: a ii = 1, a ij = 0 whenever (i, j) is not an edge of G Means: A I is the adjacency matrix of an edge subgraph of G, where I denotes the identity matrix Figure: Ref:

29 Matrix Fits a Graph Let G be a directed graph on n vertices without self-loops. A 0-1 matrix A = (a ij ) fits G if for all i and j: a ii = 1, a ij = 0 whenever (i, j) is not an edge of G Means: A I is the adjacency matrix of an edge subgraph of G, where I denotes the identity matrix Figure: Ref:

30 minrk 2 (G) Let rk 2 ( ) denote the 2-rank of a 0-1 matrix(its rank over the field GF(2)), then: minrk 2 (G) = min{rk 2 (A) : A fits G} Figure: Ref:

31 minrk 2 (G) (contd.) Theorem Computing minrk 2 (G) is NP-Complete [Peeters] Can also be reduced to SAT [Chaudhry et al] Optimal Length of a linear Index Code (No. of packets to be transmitted)=minrk 2 (G) CliqueNumber(G) ShannonCapacity(G) minrk 2 (G) ChromaticNumber(G)

32 minrk 2 (G) (contd.) Theorem Computing minrk 2 (G) is NP-Complete [Peeters] Can also be reduced to SAT [Chaudhry et al] Optimal Length of a linear Index Code (No. of packets to be transmitted)=minrk 2 (G) CliqueNumber(G) ShannonCapacity(G) minrk 2 (G) ChromaticNumber(G)

33 minrk 2 (G) (contd.) Theorem Computing minrk 2 (G) is NP-Complete [Peeters] Can also be reduced to SAT [Chaudhry et al] Optimal Length of a linear Index Code (No. of packets to be transmitted)=minrk 2 (G) CliqueNumber(G) ShannonCapacity(G) minrk 2 (G) ChromaticNumber(G)

34 minrk 2 (G) Intuition (contd.) All clients that correspond to nodes in a clique can be satisfied by one transmission, which includes a linear combination of all packets in their wants sets Minimize the number of transmissions by finding the least number of cliques that cover whole graph, clique partition problem

35 minrk 2 (G) Intuition (contd.) All clients that correspond to nodes in a clique can be satisfied by one transmission, which includes a linear combination of all packets in their wants sets Minimize the number of transmissions by finding the least number of cliques that cover whole graph, clique partition problem

36 minrk 2 (G) Intuition (contd.) All clients that correspond to nodes in a clique can be satisfied by one transmission, which includes a linear combination of all packets in their wants sets Minimize the number of transmissions by finding the least number of cliques that cover whole graph, clique partition problem

37 Formal Model (Broadcasting with Side Information) A sender holds a word x = x 1, x 2 x n, each block xi consists of t bits There are m receivers Each receiver R i wants a specific block x f (i) and has prior side information consisting of some other blocks x j The goal: To construct an efficient encoding scheme such that each receiver can decode the block it wants

38 Figure: Ref: broadcasting slides.pdf?a At least 2 blocks (packets are needed). Why? x 1 x 2 and x 3 x 4 x 5

39 Figure: Ref: broadcasting slides.pdf?a At least 2 blocks (packets are needed). Why? x 1 x 2 and x 3 x 4 x 5

40 Presentation as Directed Hypergraph H = (V, E) Vertices: V = [n] = {1, 2,, n} Edges: For each receiver R j there is a directed edge (f (j), N(j)) E, where N(j) is the set of indices of all blocks known to R j Performance metric: Define β t (H) as the minimal number of bits required when the block length is t

41 Presentation as Directed Hypergraph H = (V, E) Vertices: V = [n] = {1, 2,, n} Edges: For each receiver R j there is a directed edge (f (j), N(j)) E, where N(j) is the set of indices of all blocks known to R j Performance metric: Define β t (H) as the minimal number of bits required when the block length is t

42 Presentation as Directed Hypergraph H = (V, E) Vertices: V = [n] = {1, 2,, n} Edges: For each receiver R j there is a directed edge (f (j), N(j)) E, where N(j) is the set of indices of all blocks known to R j Performance metric: Define β t (H) as the minimal number of bits required when the block length is t

43 Presentation as Directed Hypergraph H = (V, E) Vertices: V = [n] = {1, 2,, n} Edges: For each receiver R j there is a directed edge (f (j), N(j)) E, where N(j) is the set of indices of all blocks known to R j Performance metric: Define β t (H) as the minimal number of bits required when the block length is t

44 Example of Hypergrpah Figure: broadcasting slides.pdf?a β t (H) 2t β t (H) 2t by transmitting x 1 x 2 and x 3 x 4 x 5

45 Example of Hypergrpah Figure: broadcasting slides.pdf?a β t (H) 2t β t (H) 2t by transmitting x 1 x 2 and x 3 x 4 x 5

46 Example of Hypergrpah Figure: broadcasting slides.pdf?a β t (H) 2t β t (H) 2t by transmitting x 1 x 2 and x 3 x 4 x 5

47 Example of Hypergrpah Figure: broadcasting slides.pdf?a β t (H) 2t β t (H) 2t by transmitting x 1 x 2 and x 3 x 4 x 5

48 Example of Hypergrpah Figure: broadcasting slides.pdf?a β t (H) 2t β t (H) 2t by transmitting x 1 x 2 and x 3 x 4 x 5

49 Definition of β(h) Define β(h) = lim t β t t In words, β is the average asymptotic number of encoding bits needed per bit in each block In the example, β t (H) = 2t, hence β(h) = 2

50 Definition of β(h) Define β(h) = lim t β t t In words, β is the average asymptotic number of encoding bits needed per bit in each block In the example, β t (H) = 2t, hence β(h) = 2

51 Definition of β(h) Define β(h) = lim t β t t In words, β is the average asymptotic number of encoding bits needed per bit in each block In the example, β t (H) = 2t, hence β(h) = 2

52 Definition of β t Layers A related parameter β t = β 1 (t.h), where t.h is the disjoint union of t copies of H In words, β t is the minimum number of bits required if the network topology is repeated t times Similarly, define: β (H) = lim t β t t = inf t β t t

53 Definition of β t Layers A related parameter β t = β 1 (t.h), where t.h is the disjoint union of t copies of H In words, β t is the minimum number of bits required if the network topology is repeated t times Similarly, define: β (H) = lim t β t t = inf t β t t

54 Definition of β t Layers A related parameter β t = β 1 (t.h), where t.h is the disjoint union of t copies of H In words, β t is the minimum number of bits required if the network topology is repeated t times Similarly, define: β (H) = lim t β t t = inf t β t t

55 Results β(h) β (H) β 1 (H) β β 1 (t.h) (H) = lim t = n log 2 γ(h) t, where γ(h) is the maximal cardinality of the input strings which are non-confusable.

56 Proof Idea Definition: The confusion graph confusion(h) of a directed hypergraph H = ([n], E) describing a broadcast network is the undirected graph on {0, 1} n, where x, y are adjacent iff for some e = (i, j) E, x i y i and x j = y j j J Here each block is of length 1, the vertices are all possible input words, and two are adjacent iff they are confusable. Theorem 1 can be restated as follows: β (H) = lim t β 1 (t.h) t = n log 2 α(confusion(h)) The proof relies on the connection between the fractional chromatic number of confusion(h), and the chromatic number of confusion(h) s OR graph powers

57 Proof Idea Definition: The confusion graph confusion(h) of a directed hypergraph H = ([n], E) describing a broadcast network is the undirected graph on {0, 1} n, where x, y are adjacent iff for some e = (i, j) E, x i y i and x j = y j j J Here each block is of length 1, the vertices are all possible input words, and two are adjacent iff they are confusable. Theorem 1 can be restated as follows: β (H) = lim t β 1 (t.h) t = n log 2 α(confusion(h)) The proof relies on the connection between the fractional chromatic number of confusion(h), and the chromatic number of confusion(h) s OR graph powers

58 Proof Idea Definition: The confusion graph confusion(h) of a directed hypergraph H = ([n], E) describing a broadcast network is the undirected graph on {0, 1} n, where x, y are adjacent iff for some e = (i, j) E, x i y i and x j = y j j J Here each block is of length 1, the vertices are all possible input words, and two are adjacent iff they are confusable. Theorem 1 can be restated as follows: β (H) = lim t β 1 (t.h) t = n log 2 α(confusion(h)) The proof relies on the connection between the fractional chromatic number of confusion(h), and the chromatic number of confusion(h) s OR graph powers

59 Proof Idea Definition: The confusion graph confusion(h) of a directed hypergraph H = ([n], E) describing a broadcast network is the undirected graph on {0, 1} n, where x, y are adjacent iff for some e = (i, j) E, x i y i and x j = y j j J Here each block is of length 1, the vertices are all possible input words, and two are adjacent iff they are confusable. Theorem 1 can be restated as follows: β (H) = lim t β 1 (t.h) t = n log 2 α(confusion(h)) The proof relies on the connection between the fractional chromatic number of confusion(h), and the chromatic number of confusion(h) s OR graph powers

60 Proof Sketch Zero- Error Information Theory-vs- Information Theory (ECE 646) Confusion Graph Figure: Ref:

61 Proof Sketch Zero- Error Information Theory-vs- Information Theory (ECE 646) Confusion Graph Figure: Ref:

62 Proof Sketch (contd.) Shannon Capacity of a Graph Figure: minrk.pdf

63 Proof Sketch (contd.) Powers of the Graph Figure: minrk.pdf

64 Proof Sketch (contd.) Powers of the Graph Figure: minrk.pdf

65 Results β(h) β (H) β 1 (H) β β 1 (t.h) (H) = lim t = n log 2 γ(h) t, where γ(h) is the maximal cardinality of the input strings which non-confusable.

66 Proof Idea Definition: The confusion graph confusion(h) of a directed hypergraph H = ([n], E) describing a broadcast network is the undirected graph on {0, 1} n, where x,y are adjacent iff for some e = (i, j) E, x i y j j J Here each block is of length 1, the vertices are all possible input words, and two are adjacent iff they are confusable. Theorem 1 can be restated as follows: β (H) = lim t β 1 (t.h) t = n log 2 α(confusion(h)) The proof relies on the connection between the fractional chromatic number of confusion(h), and the chromatic number of confusion(h) s OR graph powers

67 Proof Idea Definition: The confusion graph confusion(h) of a directed hypergraph H = ([n], E) describing a broadcast network is the undirected graph on {0, 1} n, where x,y are adjacent iff for some e = (i, j) E, x i y j j J Here each block is of length 1, the vertices are all possible input words, and two are adjacent iff they are confusable. Theorem 1 can be restated as follows: β (H) = lim t β 1 (t.h) t = n log 2 α(confusion(h)) The proof relies on the connection between the fractional chromatic number of confusion(h), and the chromatic number of confusion(h) s OR graph powers

68 Proof Idea Definition: The confusion graph confusion(h) of a directed hypergraph H = ([n], E) describing a broadcast network is the undirected graph on {0, 1} n, where x,y are adjacent iff for some e = (i, j) E, x i y j j J Here each block is of length 1, the vertices are all possible input words, and two are adjacent iff they are confusable. Theorem 1 can be restated as follows: β (H) = lim t β 1 (t.h) t = n log 2 α(confusion(h)) The proof relies on the connection between the fractional chromatic number of confusion(h), and the chromatic number of confusion(h) s OR graph powers

69 Proof Idea Definition: The confusion graph confusion(h) of a directed hypergraph H = ([n], E) describing a broadcast network is the undirected graph on {0, 1} n, where x,y are adjacent iff for some e = (i, j) E, x i y j j J Here each block is of length 1, the vertices are all possible input words, and two are adjacent iff they are confusable. Theorem 1 can be restated as follows: β (H) = lim t β 1 (t.h) t = n log 2 α(confusion(h)) The proof relies on the connection between the fractional chromatic number of confusion(h), and the chromatic number of confusion(h) s OR graph powers

70 OR Graph Product: The OR Graph Product of G 1 an G 2, denoted by G 1 G 2, is the graph on the vertex set V (G 1 ) V (G 2 ), where (u, v) and (u, v ) are adjacent iff either uu E(G 1 ) or vv E(G 2 ) (or both). Let G k denote the k fold OR Product of a graph G.

71 Let H 1 and H 2 denote directed hypergraphs (as before) on the vertex-sets [m] and [n] respectively, and consider an encoding scheme for their disjoint union, H 1 + H 2. As there are no edges between H 1 and H 2, such a coding scheme cannot encode two input-words x, y 0, 1 m+n by the same codeword iff this forms an ambiguity either with respect to H 1 or with respect to H 2, (or both). Hence: For any pair H 1, H 2 of directed hypergraph, the graphs confusion(h 1 + H 2 ) and confusion(h 1 ) confusion(h 1 ) are isomorphic Then lim k (χ(g k )) 1/k = χ f (G) Then skipping many steps... Finally lim k (χ(g k )) 1/k = 2 n α(confusion(h)) = 2n γ

72 Let H 1 and H 2 denote directed hypergraphs (as before) on the vertex-sets [m] and [n] respectively, and consider an encoding scheme for their disjoint union, H 1 + H 2. As there are no edges between H 1 and H 2, such a coding scheme cannot encode two input-words x, y 0, 1 m+n by the same codeword iff this forms an ambiguity either with respect to H 1 or with respect to H 2, (or both). Hence: For any pair H 1, H 2 of directed hypergraph, the graphs confusion(h 1 + H 2 ) and confusion(h 1 ) confusion(h 1 ) are isomorphic Then lim k (χ(g k )) 1/k = χ f (G) Then skipping many steps... Finally lim k (χ(g k )) 1/k = 2 n α(confusion(h)) = 2n γ

73 Conclusion Index Coding and its applications Effective Rates

74 Questions