The Role of Directed Information in Network Capacity
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1 The Role of Directed Information in Network Capacity Sudeep Kamath 1 and Young-Han Kim 2 1 Department of Electrical Engineering Princeton University 2 Department of Electrical and Computer Engineering University of California, San Diego IEEE Information Theory Workshop Jerusalem, Israel April 27, 2015
2 Communication over a general network M 1 N p(y 1,...,y N x 1,...,x N ) ˆM Nodes: (X 1,Y 1 ),...,(X N,Y N ) Network: p(y N x N ) captures noise, interference, broadcast, multiple access,... Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 2 / 17
3 Communication over a general network M 1 N p(y 1,...,y N x 1,...,x N ) ˆM Nodes: (X 1,Y 1 ),...,(X N,Y N ) Network: p(y N x N ) captures noise, interference, broadcast, multiple access,... Network information flow questions Network capacity Optimal coding schemes Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 2 / 17
4 Cutset bound S Sc M 1 N p(y 1,...,y N x 1,...,x N ) w2 Cutset bound (El Gamal 1981) I(X(S);Y(S c ) X(S c )) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 3 / 17
5 Cutset bound S Sc M 1 N p(y 1,...,y N x 1,...,x N ) w2 Cutset bound (El Gamal 1981) min S I(X(S);Y(S c ) X(S c )) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 3 / 17
6 Cutset bound S Sc M 1 N p(y 1,...,y N x 1,...,x N ) w2 Cutset bound (El Gamal 1981) C max min I(X(S);Y(S c ) X(S c )) p(x N ) S Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 3 / 17
7 Cutset bound S Sc M 1 N p(y 1,...,y N x 1,...,x N ) w2 Cutset bound (El Gamal 1981) C max min I(X(S);Y(S c ) X(S c )) p(x N ) S amount of information nodes in S can transmit to nodes in S c Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 3 / 17
8 Networks with multiple flows (Cover Thomas 2006) 2 M j 2 j ˆM 2 j M 1 N,M 1 k 1 p(y 1,...,y N x 1,...,x N ) N ˆM 1 N M 3 k 3 k ˆM 1 k, ˆM 3 k Multiple information flows: M 1 k, M 1 N, M 2 j, M 3 k,... Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 4 / 17
9 Networks with multiple flows (Cover Thomas 2006) p(y 1,...,y N x 1,...,x N ) 2 j 1 N 3 k Multiple information flows: M 1 k, M 1 N, M 2 j, M 3 k,... Circles: nodes Edges: unicast flows (can be generalized to subset-to-subset multicast) Background: physical network p(y 1,...,y N x 1,...,x N ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 4 / 17
10 Cutset bound for multiple flows p(y 1,...,y N x 1,...,x N ) 1 2 S Sc j N 3 k Cutset bound R j k I(X(S);Y(S c ) X(S c )) j S,k S c for all S for some p(x N ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 5 / 17
11 Cutset bound for multiple flows p(y 1,...,y N x 1,...,x N ) 2 S S c j 1 N 3 k Cutset bound R j k I(X(S);Y(S c ) X(S c )) j S,k S c for all S for some p(x N ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 5 / 17
12 Cutset bound for multiple flows p(y 1,...,y N x 1,...,x N ) 2 S S c j 1 N 3 k Cutset bound j S,k S c R j k I(X(S);Y(S c ) X(S c )) Simple, intuitive, general for all S for some p(x N ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 5 / 17
13 Cutset bound for multiple flows p(y 1,...,y N x 1,...,x N ) 2 j 1 N 3 k Cutset bound j S,k S c R j k I(X(S);Y(S c ) X(S c )) Simple, intuitive, general Good for single destination for all S for some p(x N ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 5 / 17
14 Cutset bound for multiple flows p(y 1,...,y N x 1,...,x N ) 2 j 1 N 3 k Cutset bound R j k I(X(S);Y(S c ) X(S c )) j S,k S c for all S for some p(x N ) Simple, intuitive, general Good for single destination Fair for single source Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 5 / 17
15 Cutset bound for multiple flows p(y 1,...,y N x 1,...,x N ) 2 j 1 N 3 k Cutset bound R j k I(X(S);Y(S c ) X(S c )) j S,k S c for all S for some p(x N ) Simple, intuitive, general Good for single destination Fair for single source Not good for multiple unicast Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 5 / 17
16 How to improve the cutset bound Auxiliary random variables (Gallager 1974, Nair El Gamal 2007, Telatar Tse 2007) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 6 / 17
17 How to improve the cutset bound Auxiliary random variables (Gallager 1974, Nair El Gamal 2007, Telatar Tse 2007) Restricted input distributions (Cover Thomas 2006, Hekstra Willems 1989) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 6 / 17
18 How to improve the cutset bound Auxiliary random variables (Gallager 1974, Nair El Gamal 2007, Telatar Tse 2007) Restricted input distributions (Cover Thomas 2006, Hekstra Willems 1989) Specialized inequalities (Bergmans 1973, Weingarten Steinberg Shamai 2006, Aleksic Razaghi Yu 2009, Geng Nair 2014, Polyanskiy Wu 2015) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 6 / 17
19 How to improve the cutset bound Auxiliary random variables (Gallager 1974, Nair El Gamal 2007, Telatar Tse 2007) Restricted input distributions (Cover Thomas 2006, Hekstra Willems 1989) Specialized inequalities (Bergmans 1973, Weingarten Steinberg Shamai 2006, Aleksic Razaghi Yu 2009, Geng Nair 2014, Polyanskiy Wu 2015) Graphical/deterministic networks (Yeung 2008, Kramer Savari 2006, Harvey Kleinberg Lehman 2006, Kamath Tse Anantharam 2011, Shomorony Avestimehr 2014) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 6 / 17
20 How to improve the cutset bound Auxiliary random variables (Gallager 1974, Nair El Gamal 2007, Telatar Tse 2007) Restricted input distributions (Cover Thomas 2006, Hekstra Willems 1989) Specialized inequalities (Bergmans 1973, Weingarten Steinberg Shamai 2006, Aleksic Razaghi Yu 2009, Geng Nair 2014, Polyanskiy Wu 2015) Graphical/deterministic networks (Yeung 2008, Kramer Savari 2006, Harvey Kleinberg Lehman 2006, Kamath Tse Anantharam 2011, Shomorony Avestimehr 2014) What we are missing is a new bound Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 6 / 17
21 How to improve the cutset bound Auxiliary random variables (Gallager 1974, Nair El Gamal 2007, Telatar Tse 2007) Restricted input distributions (Cover Thomas 2006, Hekstra Willems 1989) Specialized inequalities (Bergmans 1973, Weingarten Steinberg Shamai 2006, Aleksic Razaghi Yu 2009, Geng Nair 2014, Polyanskiy Wu 2015) Graphical/deterministic networks (Yeung 2008, Kramer Savari 2006, Harvey Kleinberg Lehman 2006, Kamath Tse Anantharam 2011, Shomorony Avestimehr 2014) What we are missing is a new bound Simple, intuitive, general (just like cutset) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 6 / 17
22 How to improve the cutset bound Auxiliary random variables (Gallager 1974, Nair El Gamal 2007, Telatar Tse 2007) Restricted input distributions (Cover Thomas 2006, Hekstra Willems 1989) Specialized inequalities (Bergmans 1973, Weingarten Steinberg Shamai 2006, Aleksic Razaghi Yu 2009, Geng Nair 2014, Polyanskiy Wu 2015) Graphical/deterministic networks (Yeung 2008, Kramer Savari 2006, Harvey Kleinberg Lehman 2006, Kamath Tse Anantharam 2011, Shomorony Avestimehr 2014) What we are missing is a new bound Simple, intuitive, general (just like cutset) Yet better than cutset Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 6 / 17
23 Not-relay channel p(y 2,y 3 x 1,x 2 ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 7 / 17
24 Not-relay channel p(y 2,y 3 x 1,x 2 ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 7 / 17
25 Not-relay channel p(y 2,y 3 x 1,x 2 ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 7 / 17
26 Not-relay channel p(y 2,y 3 x 1,x 2 ) S Sc Cutset bound R 1 2 I(X 1 ;Y 2 X 2 ) for some p(x 1,x 2 ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 7 / 17
27 Not-relay channel p(y 2,y 3 x 1,x 2 ) S c S Cutset bound R 1 2 I(X 1 ;Y 2 X 2 ) R 2 3 I(X 2 ;Y 3 X 1 ) for some p(x 1,x 2 ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 7 / 17
28 Not-relay channel p(y 2,y 3 x 1,x 2 ) Improved bound (Kamath Kim 2014) R 1 2 I(X 1 ;Y 2 X 2 ) R 2 3 I(X 2 ;Y 3 X 1 ) R 1 2 +R 2 3 I(X 1 ;Y 2,Y 3 X 2 )+I(X 2 ;Y 3 ) for some p(x 1,x 2 ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 7 / 17
29 The third inequality R 1 2 I(X 1 ;Y 2 X 2 ) =: I 1 R 2 3 I(X 2 ;Y 3 X 1 ) =: I 2 R 1 2 +R 2 3 I(X 1 ;Y 2,Y 3 X 2 )+I(X 2 ;Y 3 ) =: I 3 Not too tight: I 1 I 3, I 2 I 3 Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 8 / 17
30 The third inequality R 1 2 I(X 1 ;Y 2 X 2 ) =: I 1 R 2 3 I(X 2 ;Y 3 X 1 ) =: I 2 R 1 2 +R 2 3 I(X 1 ;Y 2,Y 3 X 2 )+I(X 2 ;Y 3 ) =: I 3 Not too tight: I 1 I 3, I 2 I 3 Not always active: I 3 I 1 +I 2 Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 8 / 17
31 The third inequality R 1 2 I(X 1 ;Y 2 X 2 ) =: I 1 R 2 3 I(X 2 ;Y 3 X 1 ) =: I 2 R 1 2 +R 2 3 I(X 1 ;Y 2,Y 3 X 2 )+I(X 2 ;Y 3 ) =: I 3 Not too tight: I 1 I 3, I 2 I 3 Not always active: I 3 I 1 +I 2 Not too loose: I 3 I(X 1,X 2 ;Y 2,Y 3 ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 8 / 17
32 The third inequality R 1 2 I(X 1 ;Y 2 X 2 ) =: I 1 R 2 3 I(X 2 ;Y 3 X 1 ) =: I 2 R 1 2 +R 2 3 I(X 1 ;Y 2,Y 3 X 2 )+I(X 2 ;Y 3 ) =: I 3 Not too tight: I 1 I 3, I 2 I 3 Not always active: I 3 I 1 +I 2 Not too loose: I 3 I(X 1,X 2 ;Y 2,Y 3 ) The proof follows mechanically from Fano s inequality: n(r 1 2 +R 2 3 ) I(M 1 2 ;Y n 2,Y n 3 M 2 3)+I(M 2 3 ;Y n 3 )+nǫ n Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 8 / 17
33 The third inequality R 1 2 I(X 1 ;Y 2 X 2 ) =: I 1 R 2 3 I(X 2 ;Y 3 X 1 ) =: I 2 R 1 2 +R 2 3 I(X 1 ;Y 2,Y 3 X 2 )+I(X 2 ;Y 3 ) =: I 3 Not too tight: I 1 I 3, I 2 I 3 Not always active: I 3 I 1 +I 2 Not too loose: I 3 I(X 1,X 2 ;Y 2,Y 3 ) The proof follows mechanically from Fano s inequality: n(r 1 2 +R 2 3 ) I(M 1 2 ;Y n 2,Y n 3 M 2 3)+I(M 2 3 ;Y n 3 )+nǫ n More questions than answers Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 8 / 17
34 The third inequality R 1 2 I(X 1 ;Y 2 X 2 ) =: I 1 R 2 3 I(X 2 ;Y 3 X 1 ) =: I 2 R 1 2 +R 2 3 I(X 1 ;Y 2,Y 3 X 2 )+I(X 2 ;Y 3 ) =: I 3 Not too tight: I 1 I 3, I 2 I 3 Not always active: I 3 I 1 +I 2 Not too loose: I 3 I(X 1,X 2 ;Y 2,Y 3 ) The proof follows mechanically from Fano s inequality: n(r 1 2 +R 2 3 ) I(M 1 2 ;Y n 2,Y n 3 M 2 3)+I(M 2 3 ;Y n 3 )+nǫ n More questions than answers Is it really good? Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 8 / 17
35 The third inequality R 1 2 I(X 1 ;Y 2 X 2 ) =: I 1 R 2 3 I(X 2 ;Y 3 X 1 ) =: I 2 R 1 2 +R 2 3 I(X 1 ;Y 2,Y 3 X 2 )+I(X 2 ;Y 3 ) =: I 3 Not too tight: I 1 I 3, I 2 I 3 Not always active: I 3 I 1 +I 2 Not too loose: I 3 I(X 1,X 2 ;Y 2,Y 3 ) The proof follows mechanically from Fano s inequality: n(r 1 2 +R 2 3 ) I(M 1 2 ;Y n 2,Y n 3 M 2 3)+I(M 2 3 ;Y n 3 )+nǫ n More questions than answers Is it really good? Can we generalize and simplify? Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 8 / 17
36 The third inequality R 1 2 I(X 1 ;Y 2 X 2 ) =: I 1 R 2 3 I(X 2 ;Y 3 X 1 ) =: I 2 R 1 2 +R 2 3 I(X 1 ;Y 2,Y 3 X 2 )+I(X 2 ;Y 3 ) =: I 3 Not too tight: I 1 I 3, I 2 I 3 Not always active: I 3 I 1 +I 2 Not too loose: I 3 I(X 1,X 2 ;Y 2,Y 3 ) The proof follows mechanically from Fano s inequality: n(r 1 2 +R 2 3 ) I(M 1 2 ;Y n 2,Y n 3 M 2 3)+I(M 2 3 ;Y n 3 )+nǫ n More questions than answers Is it really good? Can we generalize and simplify? How should we interpret? Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 8 / 17
37 Directed information Mutual information I(A 1,...,A N ;B 1,...,B N ) N = I(A N ;B j B j 1 ) j=1 A 1 A 2 A 3 A 4 A 5 A 6 B 1 B 2 B 3 B 4 B 5 B 6 Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 9 / 17
38 Directed information Mutual information I(A 1,...,A N ;B 1,...,B N ) N = I(A N ;B j B j 1 ) j=1 A 1 A 2 A 3 A 4 A 5 A 6 B 1 B 2 B 3 B 4 B 5 B 6 Directed information (Marko 1973, Massey 1990) I(A 1,...,A N B 1,...,B N ) N = I(A j ;B j B j 1 ) j=1 A 1 A 2 A 3 A 4 A 5 A 6 B 1 B 2 B 3 B 4 B 5 B 6 Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 9 / 17
39 Directed information Mutual information I(A 1,...,A N ;B 1,...,B N ) N = I(A N ;B j B j 1 ) j=1 A 1 A 2 A 3 A 4 A 5 A 6 B 1 B 2 B 3 B 4 B 5 B 6 Directed information (Marko 1973, Massey 1990) I(A 1,...,A N B 1,...,B N ) N = I(A j ;B j B j 1 ) j=1 A 1 A 2 A 3 A 4 A 5 A 6 B 1 B 2 B 3 B 4 B 5 B 6 Amount of information A N causally provides about B N (Permuter et al. 2011) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 9 / 17
40 Directed cutset bound for the not-relay channel p(y 2,y 3 x 1,x 2 ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 10 / 17
41 Directed cutset bound for the not-relay channel p(y 2,y 3 x 1,x 2 ) Directed cutset bound (Kamath Kim 2014, 2015) R 1 2 I(X 1 ;Y 2 X 2 ) R 2 3 I(X 2 ;Y 3 X 1 ) R 1 2 +R 2 3 I(Y 3,Y 2 X 2,X 1 ) for some p(x 1,x 2 ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 10 / 17
42 Directed cutset bound for the not-relay channel p(y 2,y 3 x 1,x 2 ) Directed cutset bound (Kamath Kim 2014, 2015) R 1 2 I(Y 2 X 1 X 2 ) R 2 3 I(Y 3 X 2 X 1 ) R 1 2 +R 2 3 I(Y 3,Y 2 X 2,X 1 ) for some p(x 1,x 2 ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 10 / 17
43 Directed cutset bound for the not-relay channel p(y 2,y 3 x 1,x 2 ) Directed cutset bound (Kamath Kim 2014, 2015) R 1 2 I(Y 2 X 1 X 2 ) R 2 3 I(Y 3 X 2 X 1 ) R 1 2 +R 2 3 I(Y 3,Y 2 X 2,X 1 ) for some p(x 1,x 2 ) Amount of information Y 3,Y 2 spatiocausally TM provide about X 2,X 1 Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 10 / 17
44 Directed cutset bound for the not-relay channel p(y 2,y 3 x 1,x 2 ) Directed cutset bound (Kamath Kim 2014, 2015) R 1 2 I(Y 2 X 1 X 2 ) R 2 3 I(Y 3 X 2 X 1 ) R 1 2 +R 2 3 I(Y 3,Y 2 X 2,X 1 ) for some p(x 1,x 2 ) Amount of information Y 3,Y 2 spatiocausally TM provide about X 2,X 1 Fano s inequality: n(r 1 2 +R 2 3 ) I(Y n 3,Y n 2 M 2 3,M 1 2 )+nǫ n Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 10 / 17
45 Directed cutset bound for the general network p(y 1,...,y 5 x 1,...,x 5 ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 11 / 17
46 Directed cutset bound for the general network p(y 1,...,y 5 x 1,...,x 5 ) Partition the network into V V 0 V 1 V L V V 0 5 Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 11 / 17
47 Directed cutset bound for the general network p(y 1,...,y 5 x 1,...,x 5 ) Partition the network into V 2 V V 0 V 1 V L Consider the flows cut by the partition information flows from top to bottom V 0 5 Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 11 / 17
48 Directed cutset bound for the general network p(y 1,...,y 5 x 1,...,x 5 ) Partition the network into V V 0 V 1 V L V Consider the flows cut by the partition information flows from top to bottom Bound the sum rate by V 0 5 I(Y(V 0 ),...,Y(V L 1 ) X(V 1 ),...,X(V L ) X(V 0 )) amount of information nodes in V 0,...,V L 1 can receive spatiocausally from nodes in V 1,...,V L Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 11 / 17
49 Directed cutset bound for the general network p(y 1,...,y 5 x 1,...,x 5 ) Partition the network into V V 0 V 1 V L V Consider the flows cut by the partition information flows from top to bottom Bound the sum rate by V 0 5 I(Y(V 0 ),...,Y(V L 1 ) X(V 1 ),...,X(V L ) X(V 0 )) amount of information nodes in V 0,...,V L 1 can receive spatiocausally from nodes in V 1,...,V L f cut by V L 0 R f I(Y(V 0 ),...,Y(V L 1 ) X(V 1 ),...,X(V L ) X(V 0 )) for every (L+1) partition V L 0 and every L = 1,...,N for some p(x 1,...,x N ) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 11 / 17
50 Feature 1: High SNR performance Y 2 = X 1 +Z 2 Y 3 = X 1 +X 2 +Z Cutset: R 1 2 I(X 1 ;Y 2 X 2 ) = C(P) R 2 3 I(X 2 ;Y 3 X 1 ) = C(P) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 12 / 17
51 Feature 1: High SNR performance Y 2 = X 1 +Z 2 Y 3 = X 1 +X 2 +Z Cutset: R 1 2 I(X 1 ;Y 2 X 2 ) = C(P) R 2 3 I(X 2 ;Y 3 X 1 ) = C(P) Time division with power control: R 1 2 +R 2 3 C(2P) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 12 / 17
52 Feature 1: High SNR performance Y 2 = X 1 +Z 2 Y 3 = X 1 +X 2 +Z Cutset: R 1 2 I(X 1 ;Y 2 X 2 ) = C(P) R 2 3 I(X 2 ;Y 3 X 1 ) = C(P) Time division with power control: R 1 2 +R 2 3 C(2P) Directed cutset: R 1 2 +R 2 3 I(Y 3,Y 2 X 2,X 1 ) C(8P) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 12 / 17
53 Feature 2: Scaling law Y 2 = X 1 +Z 2 Y N = X 1 + +X N 1 +Z N N Cutset: R sum = O(N) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 13 / 17
54 Feature 2: Scaling law Y 2 = X 1 +Z 2 Y N = X 1 + +X N 1 +Z N N Cutset: R sum = O(N) Time division with power control: R sum = Ω(logN) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 13 / 17
55 Feature 2: Scaling law Y 2 = X 1 +Z 2 Y N = X 1 + +X N 1 +Z N N Cutset: R sum = O(N) Time division with power control: R sum = Ω(logN) Directed cutset: R sum = O(logN) Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 13 / 17
56 Feature 3: Graphical networks X k = (X kl : (k,l) E), Y k = (X jk : (j,k) E), X jk [1 : 2 C jk] 2 j C 12 1 C 14 4 N C 13 3 k Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 14 / 17
57 Feature 3: Graphical networks X k = (X kl : (k,l) E), Y k = (X jk : (j,k) E), X jk [1 : 2 C jk] 2 j 1 4 N 3 k Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 14 / 17
58 Feature 3: Graphical networks X k = (X kl : (k,l) E), Y k = (X jk : (j,k) E), X jk [1 : 2 C jk] S S c 2 j 1 4 N 3 k Cutset (node-cut, not edge-cut): R f H(Y(S c ) X(S c )) C(S,S c ) f Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 14 / 17
59 Feature 3: Graphical networks X k = (X kl : (k,l) E), Y k = (X jk : (j,k) E), X jk [1 : 2 C jk] V V j V N 3 k Cutset (node-cut, not edge-cut): R f H(Y(S c ) X(S c )) C(S,S c ) f Directed cutset (still node-cut): R f H(Y(V l ) X(V0),Y(V l 0 l 1 )) C(V(l +1),V(l)) f l l Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 14 / 17
60 Feature 3: Graphical networks Cutset (node-cut, not edge-cut): R f H(Y(S c ) X(S c )) C(S,S c ) f Directed cutset (still node-cut): R f H(Y(V l ) X(V0),Y(V l 0 l 1 )) C(V(l +1),V(l)) f l l Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 14 / 17
61 References Aleksic, M., Razaghi, P., and Yu, W. (2009). Capacity of a class of modulo-sum relay channels. IEEE Trans. Inf. Theory, 55(3), Bergmans, P. P. (1973). Random coding theorem for broadcast channels with degraded components. IEEE Trans. Inf. Theory, 19(2), Cover, T. M. and Thomas, J. A. (2006). Elements of Information Theory. 2nd ed. Wiley, New York. El Gamal, A. (1981). On information flow in relay networks. In Proc. IEEE National Telecomm. Conf., vol. 2, pp. D4.1.1 D New Orleans, LA. Gallager, R. G. (1974). Capacity and coding for degraded broadcast channels. Probl. Inf. Transm., 10(3), Geng, Y. and Nair, C. (2014). The capacity region of the two-receiver Gaussian vector broadcast channel with private and common messages. IEEE Trans. Inf. Theory, 60(4), Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 15 / 17
62 References (cont.) Harvey, N. J. A., Kleinberg, R., and Lehman, A. R. (2006). On the capacity of information networks. IEEE Trans. Inf. Theory, 52(6), Hekstra, A. P. and Willems, F. M. J. (1989). Dependence balance bounds for single-output two-way channels. IEEE Trans. Inf. Theory, 35(1), Kamath, S. and Kim, Y.-H. (2014). Chop and roll: Improving the cutset bound. In Proc. 52nd Ann. Allerton Conf. Comm. Control Comput., Monticello, IL, pp Kamath, S. and Kim, Y.-H. (2015). The directed cutset bound on network capacity. Kamath, S., Tse, D. N. C., and Anantharam, V. (2011). Generalized network sharing outer bound and the two-unicast problem. In Proc. Int. Symp. Netw. Coding, Beijing, China. Kramer, G. and Savari, S. A. (2006). Edge-cut bounds on network coding rates. J. Network Syst. Manage., 14(1), Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 16 / 17
63 References (cont.) Nair, C. and El Gamal, A. (2007). An outer bound to the capacity region of the broadcast channel. IEEE Trans. Inf. Theory, 53(1), Polyanskiy, Y. and Wu, Y. (2015). Wasserstein continuity of entropy and outer bounds for interference channels. Shomorony, I. and Avestimehr, S. (2014). A generalized cut-set bound for deterministic multi-flow networks and its applications. In Proc. IEEE Int. Symp. Inf. Theory, Honolulu, Hawaii. Telatar, İ. E. and Tse, D. N. C. (2007). Bounds on the capacity region of a class of interference channels. In Proc. IEEE Int. Symp. Inf. Theory, Nice, France, pp Weingarten, H., Steinberg, Y., and Shamai, S. (2006). The capacity region of the Gaussian multiple-input multiple-output broadcast channel. IEEE Trans. Inf. Theory, 52(9), Yeung, R. W. (2008). Information Theory and Network Coding. Springer, New York. Kamath/Kim (Princeton/UCSD) Index Coding ITW 2015 Jerusalem 17 / 17
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