Capacity Bounds for Diamond Networks
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1 Technische Universität München Capacity Bounds for Diamond Networks Gerhard Kramer (TUM) joint work with Shirin Saeedi Bidokhti (TUM & Stanford) DIMACS Workshop on Network Coding Rutgers University, NJ December 5, 205 Institute for Communications Engineering
2 Technische Universität München What is a Diamond Network? Cascade of a 2-receiver broadcast channel (BC) and a 2-transmitter multiaccess channel (MAC) Simplifications: () MAC is two bit-pipes; (2) BC is two bit-pipes Y R X W X Y Ŵ Src Enc BC MAC Dec Sink Y 2 R2 X 2
3 Technische Universität München What is a Diamond Network? Cascade of a 2-receiver broadcast channel (BC) and a 2-transmitter multiaccess channel (MAC) Simplifications: () MAC is two bit-pipes; (2) BC is two bit-pipes B bits n symbols Y R X W X Y Ŵ Src Enc BC MAC Dec Sink R = B/n Y 2 R2 X 2
4 Technische Universität München General Problem B. E. Schein, Distributed coordination in network information theory. PhD Dissertation, MIT, 200 MAC is 2 Bit Pipes A. Sanderovich, S. Shamai, Y. Steinberg, G. Kramer, Communication via decentralized processing, IEEE Trans. IT, 2008 BC is 2 Bit Pipes Background D. Traskov, G. Kramer, Reliable communication in networks with multiaccess interference, ITW 2007 W. Kang, N. Liu, and W. Chong, The Gaussian multiple access diamond channel, arxiv 20 (v) and 205 (v2)
5 Technische Universität München Here: BC is two bit pipes Capacity limitations C and C 2. Problem seems difficult! Gaussian MAC partially solved by Kang-Liu (20) using Ozarow s trick (980) Contribution: new capacity upper bound for discrete MACs Contribution: solved binary adder MAC capacity by extending Mrs. Gerber s Lemma Src W Enc V R X Y Ŵ MAC Dec Sink V 2 R2 X 2
6 Outline The Problem Setup ALowerBound An Upper-Bound Examples The Gaussian MAC The binary adder MAC 3/28
7 The Problem Setup Relay X n Source W Encoder MAC p(y x,x 2 ) Y n Decoder Ŵ Sink Relay 2 X n 2 I W message of rate R 4/28
8 The Problem Setup Relay X n Source W Encoder MAC p(y x,x 2 ) Y n Decoder Ŵ Sink Relay 2 X n 2 I W message of rate R I Bit-pipes of capacities C,C 2 4/28
9 The Problem Setup Relay X n Source W Encoder MAC p(y x,x 2 ) Y n Decoder Ŵ Sink Relay 2 X n 2 I W message of rate R I Bit-pipes of capacities C,C 2 I Goal: What is the highest rate R such that Pr(W 6= Ŵ )! 0? 4/28
10 ALowerBound Relay X n Source W Encoder MAC p(y x,x 2 ) Y n Decoder Ŵ Sink Relay 2 X n 2 I Rate splitting: W =(W 2, W, W 2 ) I Superposition Coding: W 2 encoded in V n. X n, Xn 2 superposed on V n. I Marton s Coding 5/28
11 Technische Universität München Rate-splitting bounds: Rate Bounds Now apply Fourier-Motzkin elimination
12 ALowerBound(Cont.) Theorem (Lower Bound) The rate R is achievable if it satisfies the following condition for some pmf p(v, x,x 2,y)=p(v, x,x 2 )p(y x,x 2 ): 8 9 C + C 2 I(X ; X 2 V ) >< C 2 + I(X ; Y X 2 V ) >= Rapplemin C + I(X 2 ; Y X V ) >: 2 (C +C 2 +I(X X 2 ; Y V ) I(X ; X 2 V )) >; I(X X 2 ; Y ) V 2V, V applemin{ X X 2 +2, Y +4} 6/28
13 The Cut-Set Bound Cut-Set bound: R is achievable only if it satisfies the following bounds for some p(x,x 2 ): R apple C + C 2 R apple C + I(X 2 ; Y X ) R apple C 2 + I(X ; Y X 2 ) source C 2 X 2 Y sink R apple I(X X 2 ; Y ). C X 7/28
14 Example I: binary adder MAC I X = X 2 = {0, }, Y = {0,, 2} I Y = X + X Rate R Cut-Set bound Lower bound Link Capacity C 8/28
15 Example I: binary adder MAC I X = X 2 = {0, }, Y = {0,, 2} I Y = X + X Rate R Cut-Set bound Lower bound and capacity Link Capacity C 8/28
16 Example II: Gaussian MAC I Y = X + X 2 + Z, Z N (0, ) I n P n i= E(X2,i ) apple P, n P n i= E(X2 2,i ) apple P 2, P = P 2 = Rate R Cut-Set bound Link Capacity C 9/28
17 Example II: Gaussian MAC I Y = X + X 2 + Z, Z N (0, ) I n P n i= E(X2,i ) apple P, n P n i= E(X2 2,i ) apple P 2, P = P 2 = Rate R Cut-Set bound Lower bound (no superposition coding) Link Capacity C 9/28
18 Example II: Gaussian MAC I Y = X + X 2 + Z, Z N (0, ) I n P n i= E(X2,i ) apple P, n P n i= E(X2 2,i ) apple P 2, P = P 2 = Rate R Cut-Set bound Lower bound (no superposition coding) Lower bound (Joint Gaussian dist.) Link Capacity C 9/28
19 Example II: Gaussian MAC I Y = X + X 2 + Z, Z N (0, ) I n P n i= E(X2,i ) apple P, n P n i= E(X2 2,i ) apple P 2, P = P 2 = Rate R Cut-Set bound Lower bound (no superposition coding) Lower bound (Joint Gaussian dist.) Lower bound (Mixture of two Gaussian dist.) Link Capacity C 9/28
20 Example II: Gaussian MAC I Y = X + X 2 + Z, Z N (0, ) I n P n i= E(X2,i ) apple P, n P n i= E(X2 2,i ) apple P 2, P = P 2 = Lower bound is tight Lower bound is tight! Rate R Cut-Set bound Lower bound (no superposition coding) Lower bound (Joint Gaussian dist.) Lower bound (Mixture of two Gaussian dist.) Link Capacity C 9/28
21 Is the Cut-Set Bound Tight? Cut-Set bound: R apple C + C 2 R apple C + I(X 2 ; Y X ) R apple C 2 + I(X ; Y X 2 ) R apple I(X X 2 ; Y ). source C 2 X 2 Y sink Maximize over p(x,x 2 ). C X 0 / 28
22 Is the Cut-Set Bound Tight? Cut-Set bound: R apple C + C 2 R apple C + I(X 2 ; Y X ) R apple C 2 + I(X ; Y X 2 ) R apple I(X X 2 ; Y ). source C 2 X 2 Y sink Maximize over p(x,x 2 ). C X It turns out that the cut-set bound is not tight. 0 / 28
23 Refining the Cut-Set Bound I Motivated by [Ozarow 80, KangLiu ] / 28
24 Refining the Cut-Set Bound I Motivated by [Ozarow 80, KangLiu ] nr apple nc + nc 2 I(X n ; X n 2 ) / 28
25 Refining the Cut-Set Bound I Motivated by [Ozarow 80, KangLiu ] nr apple nc + nc 2 I(X n ; X n 2 ) I For any U n : I(X n ; X n 2 )=I(X n X n 2 ; U n ) I(X n ; U n X n 2 ) I(X n 2 ; U n X n ) + I(X n ; X n 2 U n ) / 28
26 Refining the Cut-Set Bound I Motivated by [Ozarow 80, KangLiu ] nr apple nc + nc 2 I(X n ; X n 2 ) I For any U n : I(X n ; X n 2 ) I(X n X n 2 ; U n ) I(X n ; U n X n 2 ) I(X n 2 ; U n X n ) / 28
27 Refining the Cut-Set Bound (Cont.) nrapplenc + nc 2 I(X n X n 2 ; U n )+I(X n ; U n X n 2 )+I(X n 2 ; U n X n ) 2 / 28
28 Refining the Cut-Set Bound (Cont.) nrapplenc + nc 2 I(X n X n 2 ; U n )+I(X n ; U n X n 2 )+I(X n 2 ; U n X n ) choose U i as follows: Y i p U Y U i 2 / 28
29 Refining the Cut-Set Bound (Cont.) nrapplenc + nc 2 I(X n X n 2 ; U n )+I(X n ; U n X n 2 )+I(X n 2 ; U n X n ) choose U i as follows: Y i p U Y U i nr applei(x n X n 2 ; Y n ) nr applenc + nc 2 I(X n X n 2 ; U n )+I(X n ; U n X n 2 )+I(X n 2 ; U n X n ) 2 / 28
30 Refining the Cut-Set Bound (Cont.) nrapplenc + nc 2 I(X n X n 2 ; U n )+I(X n ; U n X n 2 )+I(X n 2 ; U n X n ) choose U i as follows: Y i p U Y U i nr applei(x n X n 2 ; Y n ) + nr applenc + nc 2 I(X n X n 2 ; U n )+I(X n ; U n X n 2 )+I(X n 2 ; U n X n ) 2 / 28
31 Refining the Cut-Set Bound (Cont.) nrapplenc + nc 2 I(X n X n 2 ; U n )+I(X n ; U n X n 2 )+I(X n 2 ; U n X n ) choose U i as follows: Y i p U Y U i nr applei(x n X n 2 ; Y n ) + nr applenc + nc 2 I(X n X n 2 ; U n )+I(X n ; U n X n 2 )+I(X n 2 ; U n X n ) 2nR applenc +nc 2 +I(X n X n 2 ; Y n U n )+I(X n ; U n X n 2)+I(X n 2 ; U n X n ) 2 / 28
32 Refining the Cut-Set Bound (Cont.) nrapplenc + nc 2 I(X n X n 2 ; U n )+I(X n ; U n X n 2 )+I(X n 2 ; U n X n ) choose U i as follows: Y i p U Y U i nr applei(x n X n 2 ; Y n ) + nr applenc + nc 2 I(X n X n 2 ; U n )+I(X n ; U n X n 2 )+I(X n 2 ; U n X n ) 2nR applenc +nc 2 +I(X n X n 2 ; Y n U n )+I(X n ; U n X n 2)+I(X n 2 ; U n X n )...applen (C + C 2 + I(X X 2 ; Y U)+I(X ; U X 2 )+I(X 2 ; U X )) 2 / 28
33 New Upper-Bounds () Theorem (Upper Bound I) The rate R is achievable only if there exists a joint distribution p(x,x 2 ) for which the following inequalities hold for every auxiliary channel p(u x,x 2,y)=p(u y) R apple C + C 2 R apple C 2 + I(X ; Y X 2 ) R apple C + I(X 2 ; Y X ) R apple I(X X 2 ; Y ) 2R apple C + C 2 + I(X X 2 ; Y U)+I(X ; U X 2 )+I(X 2 ; U X ) 3 / 28
34 New Upper-Bounds () Theorem (Upper Bound I) The rate R is achievable only if there exists a joint distribution p(x,x 2 ) for which the following inequalities hold for every auxiliary channel p(u x,x 2,y)=p(u y) R apple C + C 2 R apple C 2 + I(X ; Y X 2 ) R apple C + I(X 2 ; Y X ) R apple I(X X 2 ; Y ) 2R apple C + C 2 + I(X X 2 ; Y U)+I(X ; U X 2 )+I(X 2 ; U X ) 3 / 28
35 New Upper-Bounds () Theorem (Upper Bound I) The rate R is achievable only if there exists a joint distribution p(x,x 2 ) for which the following inequalities hold for every auxiliary channel p(u x,x 2,y)=p(u y) R apple C + C 2 R apple C 2 + I(X ; Y X 2 ) R apple C + I(X 2 ; Y X ) R apple I(X X 2 ; Y ) 2R apple C + C 2 + I(X X 2 ; Y U)+I(X ; U X 2 )+I(X 2 ; U X ) I max-min problem 3 / 28
36 New Upper-Bounds () Theorem (Upper Bound I) The rate R is achievable only if there exists a joint distribution p(x,x 2 ) for which the following inequalities hold for every auxiliary channel p(u x,x 2,y)=p(u y) R apple C + C 2 R apple C 2 + I(X ; Y X 2 ) R apple C + I(X 2 ; Y X ) R apple I(X X 2 ; Y ) 2R apple C + C 2 + I(X X 2 ; Y U)+I(X ; U X 2 )+I(X 2 ; U X ) I max-min problem I 2R apple C + C 2 + I(X X 2 ; Y ) I(X ; X 2 )+I(X ; X 2 U) 3 / 28
37 New Upper-Bounds (2) Theorem (Upper Bound II) The capacity is bounded from above by 8 9 C +C 2, >< C +I(X 2 ; Y X Q), >= max min max min C 2 +I(X ; Y X 2 Q), p(x,x 2 ) p(u x,x 2,y) p(q x,x 2,y,u) =p(u y) =p(q x,x 2 ) I(X >: X 2 ; Y Q), >; C +C 2 I(X ; X 2 Q)+I(X ; X 2 UQ) I Q apple X X / 28
38 New Upper-Bounds (2) Theorem (Upper Bound II) The capacity is bounded from above by 8 9 C +C 2, >< C +I(X 2 ; Y X Q), >= max min max min C 2 +I(X ; Y X 2 Q), p(x,x 2 ) p(u x,x 2,y) p(q x,x 2,y,u) =p(u y) =p(q x,x 2 ) I(X >: X 2 ; Y Q), >; C +C 2 I(X ; X 2 Q)+I(X ; X 2 UQ) I Q apple X X I last term is related to the Hekstra-Willems dependence balance bound and can be written as R apple C +C 2 I(X X 2 ; U Q)+I(X 2 ; U X Q)+I(X ; U X 2 Q) 4 / 28
39 New Upper-Bounds (2) Theorem (Upper Bound II) The capacity is bounded from above by 8 9 C +C 2, >< C +I(X 2 ; Y X Q), >= max min max min C 2 +I(X ; Y X 2 Q), p(x,x 2 ) p(u x,x 2,y) p(q x,x 2,y,u) =p(u y) =p(q x,x 2 ) I(X X 2 ; Y Q), >: C +C 2 I(X ; X 2 Q)+I(X ; X 2 UQ) >; I Q apple X X I last term is related to the Hekstra-Willems dependence balance bound and can be written as R apple C +C 2 I(X X 2 ; U Q) +I(X 2 ; U X Q)+I(X ; U X 2 Q) 4 / 28
40 The Gaussian MAC Y = X + X 2 + Z Z N (0, ), P n n i= E(X2,i ) apple P, P n n i= E(X2 2,i ) apple P 5 / 28
41 The Gaussian MAC Y = X + X 2 + Z Z N (0, ), P n n i= E(X2,i ) apple P, P n n i= E(X2 2,i ) apple P R apple 2C R apple C + I(X ; Y X 2 Q) R apple C + I(X 2 ; Y X Q) R apple I(X X 2 ; Y Q) Max-Min-Max problem R apple C + C 2 I(X X 2 ; U Q)+I(X ; U X 2 Q)+I(X 2 ; U X Q) 5 / 28
42 The Gaussian MAC Y = X + X 2 + Z Z N (0, ), P n n i= E(X2,i ) apple P, P n n i= E(X2 2,i ) apple P R apple 2C R apple C + I(X ; Y X 2 Q) R apple C + I(X 2 ; Y X Q) R apple I(X X 2 ; Y Q) Choose U = Y + Z N Z N N (0, N) N to be optimized. R apple C + C 2 I(X X 2 ; U Q)+I(X ; U X 2 Q)+I(X 2 ; U X Q) 5 / 28
43 The Gaussian MAC Y = X + X 2 + Z Z N (0, ), P n n i= E(X2,i ) apple P, P n n i= E(X2 2,i ) apple P R apple 2C R apple C + log +P ( 2 ) /2 R apple C + I(X 2 ; Y X Q) R apple I(X X 2 ; Y Q) Choose U = Y + Z N Z N N (0, N) N to be optimized. R apple C + C 2 I(X X 2 ; U Q)+I(X ; U X 2 Q)+I(X 2 ; U X Q) 5 / 28
44 The Gaussian MAC Y = X + X 2 + Z Z N (0, ), P n n i= E(X2,i ) apple P, P n n i= E(X2 2,i ) apple P R apple 2C R apple C + log +P ( 2 ) /2 R apple C + log +P ( 2 ) /2 R apple I(X X 2 ; Y Q) Choose U = Y + Z N Z N N (0, N) N to be optimized. R apple C + C 2 I(X X 2 ; U Q)+I(X ; U X 2 Q)+I(X 2 ; U X Q) 5 / 28
45 The Gaussian MAC Y = X + X 2 + Z Z N (0, ), P n n i= E(X2,i ) apple P, P n n i= E(X2 2,i ) apple P R apple 2C R apple C + log +P ( 2 ) /2 R apple C + log +P ( 2 ) /2 R apple log ( + 2P ( + )) /2 Choose U = Y + Z N Z N N (0, N) N to be optimized. R apple C + C 2 I(X X 2 ; U Q)+I(X ; U X 2 Q)+I(X 2 ; U X Q) 5 / 28
46 The Gaussian MAC Y = X + X 2 + Z Z N (0, ), P n n i= E(X2,i ) apple P, P n n i= E(X2 2,i ) apple P R apple 2C R apple C + log +P ( 2 ) /2 R apple C + log +P ( 2 ) /2 R apple log ( + 2P ( + )) /2 R apple C + C 2 I(X X 2 ; U Q) + log Choose U = Y + Z N Z N N (0, N) N to be optimized. +N + P 2 +N! 5 / 28
47 The Gaussian MAC (Cont.) I U = Y + Z N, Z N N (0, N) I(X X 2 ; U Q) = h(u Q) h(u X X 2 ) EPI 2 log 2 en +2 2h(Y Q) log (2 e( + N)) 2 I(X X 2 ; Y Q) = h(y Q) log (2 e) R 2 6 / 28
48 The Gaussian MAC (Cont.) I U = Y + Z N, Z N N (0, N) I(X X 2 ; U Q) = h(u Q) h(u X X 2 ) EPI 2 log 2 en +2 2h(Y Q) log (2 e( + N)) 2 I(X X 2 ; Y Q) = h(y Q) log (2 e) R 2 R apple C + C 2 2 log N +22R log ( + N) 2 + log +N + P 2 6 / 28
49 The Gaussian MAC (Cont.) I U = Y + Z N, Z N N (0, N) I(X X 2 ; U Q) = h(u Q) h(u X X 2 ) EPI 2 log 2 en +2 2h(Y Q) log (2 e( + N)) 2 I(X X 2 ; Y Q) = h(y Q) log (2 e) R 2 R apple C + C 2 2 log N +22R log ( + N) 2 + log +N + P 2 I Strictly tighter than [KangLiu ] 6 / 28
50 The Gaussian MAC (Cont.) Rate R Cut-Set bound Lower bound (Mixture of two Gaussian dist.) Upper bound I Upper bound II Link Capacity C 7 / 28
51 The Gaussian MAC (Cont.) no cooperation Rate R Cut-Set bound Lower bound (Mixture of two Gaussian dist.) Upper bound I Upper bound II Link Capacity C 7 / 28
52 The Gaussian MAC (Cont.) no cooperation full cooperation Rate R Cut-Set bound Lower bound (Mixture of two Gaussian dist.) Upper bound I Upper bound II Link Capacity C 7 / 28
53 The Gaussian MAC (Cont.) no cooperation partial cooperation full cooperation Rate R Cut-Set bound Lower bound (Mixture of two Gaussian dist.) Upper bound I Upper bound II Link Capacity C 7 / 28
54 The Gaussian MAC (Cont.) no cooperation partial cooperation full cooperation Rate R Cut-Set bound Lower bound (Mixture of two Gaussian dist.) Upper bound I Upper bound II Link Capacity C 7 / 28
55 On The Capacity of The Gaussian MAC Theorem For a symmetric Gaussian diamond network, the upper bound meets the lower bound for all C such that C 2 log( + 4P ), or C apple 4 log +2P ( + (2) ) (2) 2 where (2) = r + 4P 2 2P 8 / 28
56 The Optimal Choice of N I U = Y + Z N (motivated by [Ozarow 80, KangLiu ]) I (X,X 2 ) an optimal jointly Gaussian input for the lower bound apple P? P?. P P? + I N = P?? 0: X U X 2 forms a Markov chain I P? new upper-bound I P?? apple 0: the cut-set bound 9 / 28
57 The Binary Adder MAC Y = X + X 2, X = X = {0, }, Y = {0,, 2} R apple C + C 2 R apple C 2 + I(X ; Y X 2 Q) R apple C + I(X 2 ; Y X Q) R apple I(X X 2 ; Y Q) R apple C + C 2 I(X X 2 ; U Q)+I(X ; U X 2 Q)+I(X 2 ; U X Q) 20 / 28
58 The Binary Adder MAC Y = X + X 2, X = X = {0, }, Y = {0,, 2} Y 0 q R apple C + C 2 R apple C 2 + I(X ; Y X 2 Q) R apple C + I(X 2 ; Y X Q) R apple I(X X 2 ; Y Q) Ỹ 0 q R apple C + C 2 I(X X 2 ; U Q)+I(X ; U X 2 Q)+I(X 2 ; U X Q) U 0 20 / 28
59 The Binary Adder MAC Y = X + X 2, X = X = {0, }, Y = {0,, 2} Y 0 q 2 R apple C + C 2 R apple C 2 + h 2 (q) R apple C + h 2 (q) 2 2 R apple +h 2 (q) q Ỹ 0 q R apple C + C 2 I(X X 2 ; U Q)+2h 2 ( q 2? ) 2( q)h 2( ) 2q U 0 20 / 28
60 The Interplay in the upper bound I(X X 2 ; U Q) = H(U Q) H(U X X 2 ) MGL h2?h2 H(Ỹ Q) ( q)h 2 ( ) q I(X X 2 ; Y Q) = H(Ỹ Q)+h 2(q) q R 2 / 28
61 The Binary Adder MAC (Cont.) Rate R Cut-Set bound Lower bound Link Capacity C 22 / 28
62 The Binary Adder MAC (Cont.) Cut-Set bound is tight! Rate R Cut-Set bound is tight Cut-Set bound Lower bound Link Capacity C 22 / 28
63 The Binary Adder MAC (Cont.) Cut-Set bound is tight! Rate R Cut-Set bound is tight Cut-Set bound Lower bound Upper bound I Link Capacity C 22 / 28
64 The Binary Adder MAC (Cont.) Cut-Set bound is tight!.54 Upper Bound I is tight! Rate R.52.5 Cut-Set bound is tight Cut-Set bound Lower bound Upper bound I Link Capacity C 22 / 28
65 The Binary Adder MAC (Cont.) Cut-Set bound is tight!.54 Upper Bound I is tight! Rate R.52.5 Cut-Set bound is tight Cut-Set bound Lower bound Upper bound I Upper bound II and Capacity Link Capacity C 22 / 28
66 The Binary Adder MAC (Cont.) Cut-Set bound is tight!.54 Upper Bound I is tight! Rate R.52 Cut-Set bound is tight Upper Bound II with Mrs. Gerber s lemma is tight! Cut-Set bound Lower bound Upper bound I Upper bound II and Capacity Link Capacity C 22 / 28
67 The interplay in the upper bounds R applei(x X 2 ; Y Q) R applec + C 2 I(X X 2 ; U Q)+I(X 2 ; U X Q)+I(X ; U X 2 Q) 23 / 28
68 The interplay in the upper bounds R applei(x X 2 ; Y Q) appleh(y Q) H(Y X X 2 ) R applec + C 2 I(X X 2 ; U Q)+I(X 2 ; U X Q)+I(X ; U X 2 Q) 23 / 28
69 The interplay in the upper bounds R applei(x X 2 ; Y Q) appleh(y Q) H(Y X X 2 ) R applec + C 2 I(X X 2 ; U Q)+I(X 2 ; U X Q)+I(X ; U X 2 Q) applec + C 2 H(U Q) H(U X X 2 )+H(U X Q)+H(U X 2 Q) 23 / 28
70 The interplay in the upper bounds R applei(x X 2 ; Y Q) apple H(Y Q) H(Y X X 2 ) R applec + C 2 I(X X 2 ; U Q)+I(X 2 ; U X Q)+I(X ; U X 2 Q) applec + C 2 H(U Q) H(U X X 2 )+H(U X Q)+H(U X 2 Q) I Up to now: Entropy Power Inequality, Mrs. Gerber s Lemma. min {H(U) H(Y )=t} f(t) 2. f(t) is convex in t 23 / 28
71 The interplay in the upper bounds R applei(x X 2 ; Y Q) apple H(Y Q) H(Y X X 2 ) R applec + C 2 I(X X 2 ; U Q)+I(X 2 ; U X Q)+I(X ; U X 2 Q) applec + C 2 H(U Q) H(U X X 2 )+ H(U X Q)+H(U X 2 Q) I Up to now: Entropy Power Inequality, Mrs. Gerber s Lemma. min {H(U) H(Y )=t} f(t) 2. f(t) is convex in t 23 / 28
72 The interplay in the upper bounds R applei(x X 2 ; Y Q) apple H(Y Q) H(Y X X 2 ) R applec + C 2 I(X X 2 ; U Q)+I(X 2 ; U X Q)+I(X ; U X 2 Q) applec + C 2 H(U Q) H(U X X 2 )+ H(U X Q)+H(U X 2 Q) I Up to now: Entropy Power Inequality, Mrs. Gerber s Lemma. min {H(U) H(Y )=t} f(t) 2. f(t) is convex in t I What we want to do:. min {H(U) H(U X ) H(U X 2 ) H(Y )=t} f(t) 2. f(t) is convex in t 23 / 28
73 The Binary Adder MAC: Upper Bound R apple 2C R apple C + h 2 (q) R apple +h 2 (q) q q R apple 2C h 2? +( q)h2 min, (R h 2(q)) + 2 q ( q)h 2 ( ) q +2h 2? q 2 24 / 28
74 Capacity of The Binary Adder MAC Theorem The capacity of diamond networks with binary adder MACs is 8 C + C 2 +h 2 (p) >< C max min + h 2 (p) 0applepapple C 2 >: 2 + h 2 (p) h 2 (p)+ p. 25 / 28
75 The optimal Choice of I Let (X,X 2 ) be an optimizing doubly symmetric binary pmf with parameter p? for the lower bound I is such that p? ( ) = 2( p? ) and it makes the following Markov chain X U X / 28
76 Capacity of The Binary Adder MAC Cut-Set bound is tight!.54 Upper Bound I is tight! Rate R.52 Cut-Set bound is tight Upper Bound II with Mrs. Gerber s lemma is tight!.5.48 Upper Bound II with Generalized Mrs. Gerber s lemma is tight!.46 Cut-Set bound Lower bound Upper bound I Upper bound II and Capacity Link Capacity C 27 / 28
77 Summary and Work in Progress I Lower and Upper bounds on the capacity of a class of diamond networks I A new upper bound which is in the form of a max-min problem I Gaussian MACs: I improved previous lower and upper bounds I characterized the capacity for interesting ranges of bit-pipe capacities. I Binary adder MAC: fully characterized the capacity I Work in progress: the general class of 2-relay diamond networks, n-relay diamond networks with orthogonal BC components 28 / 28
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