2020 IEEE International Symposium on Information Theory On the Fraction of Capacity One Relay can Achieve in Gaussian Half-Duplex Diamond Networks Authors: Sarthak Jain (Presenter) Soheil Mohajer Martina Cardone University of Minnesota, T win Cities (The work of the authors was supported in part by the U.S. National Science Foundation under Grant CCF-1907785.) 1
System Model 1 r 1 ℓ 1 2 r 2 ℓ 2 Diamond Network S D r i ℓ i i r N ℓ N N 𝒪 = {( ℓ i , r i ) ∀ i ∈ [1 : N ]} Half-Duplex mode of operation 2
Capacity and Scheduling 1 C ( 𝒪 ) − G 1 ≤ ˜ C ≤ C ( 𝒪 ) + G 2 r 1 ℓ 1 r 2 ℓ 2 2 Approximate Capacity Shannon Capacity D S r 3 ℓ 3 3 A. S. Avestimehr, et al, IEEE Trans. Inf. Theory , 2011. A. Ozgur, et al., IEEE Trans. Inf. Theory , 2013. 3 S. Lim, et al. IEEE Trans. Inf. Theory , 2011.
Capacity and Scheduling 1 C ( 𝒪 ) − G 1 ≤ ˜ C ≤ C ( 𝒪 ) + G 2 r 1 ℓ 1 where, G 1 + G 2 = f ( N ) r 2 ℓ 2 2 D S r 3 ℓ 3 3 A. S. Avestimehr, et al, IEEE Trans. Inf. Theory , 2011. A. Ozgur, et al., IEEE Trans. Inf. Theory , 2013. 4 S. Lim, et al. IEEE Trans. Inf. Theory , 2011.
Capacity and Scheduling 1 C ( 𝒪 ) − G 1 ≤ ˜ C ≤ C ( 𝒪 ) + G 2 r 1 ℓ 1 where, G 1 + G 2 = f ( N ) r 2 ℓ 2 2 D S C ( 𝒪 ) = max t λ r 3 λ 𝒯 ( max ℓ 3 i ∈𝒯∩Ω r i ) 𝗍 . 𝗎 . t ≤ ∑ i ∈𝒯 c ∩Ω c ℓ i + max ∀Ω ⊆ [ N ] 𝒯⊆ [ N ] 3 ∑ λ 𝒯 = 1, λ 𝒯 ≥ 0, ∀𝒯 ⊆ [ N ] 𝒯⊆ [ N ] A. S. Avestimehr, et al, IEEE Trans. Inf. Theory , 2011. A. Ozgur, et al., IEEE Trans. Inf. Theory , 2013. 5 S. Lim, et al. IEEE Trans. Inf. Theory , 2011.
Capacity and Scheduling 1 C ( 𝒪 ) − G 1 ≤ ˜ C ≤ C ( 𝒪 ) + G 2 r 1 ℓ 1 where, G 1 + G 2 = f ( N ) r 2 ℓ 2 2 D S C ( 𝒪 ) = max t λ r 3 λ 𝒯 ( max ℓ 3 i ∈𝒯∩Ω r i ) 𝗍 . 𝗎 . t ≤ ∑ i ∈𝒯 c ∩Ω c ℓ i + max ∀Ω ⊆ [ N ] 𝒯⊆ [ N ] 3 ∑ λ 𝒯 = 1, λ 𝒯 ≥ 0, ∀𝒯 ⊆ [ N ] 𝒯⊆ [ N ] 𝒯 : ∅ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} 𝒯 ⊆ [1 : N ] λ 𝒯 : fraction of time in state 𝒯 λ ∅ , λ {1} , λ {2} , λ {3} , λ {1,2} , λ {1,3} , λ {2,3} , λ {1,2,3} i ∈ 𝒯 : Relay i Tx i ∈ 𝒯 c : Relay i Rx A. S. Avestimehr, et al, IEEE Trans. Inf. Theory , 2011. A. Ozgur, et al., IEEE Trans. Inf. Theory , 2013. 6 S. Lim, et al. IEEE Trans. Inf. Theory , 2011.
Capacity and Scheduling 1 C ( 𝒪 ) − G 1 ≤ ˜ C ≤ C ( 𝒪 ) + G 2 r 1 ℓ 1 where, G 1 + G 2 = f ( N ) r 2 ℓ 2 2 D S C ( 𝒪 ) = max t λ r 3 λ 𝒯 ( max ℓ 3 i ∈𝒯∩Ω r i ) 𝗍 . 𝗎 . t ≤ ∑ i ∈𝒯 c ∩Ω c ℓ i + max ∀Ω ⊆ [ N ] 𝒯⊆ [ N ] 3 ∑ λ 𝒯 = 1, λ 𝒯 ≥ 0, ∀𝒯 ⊆ [ N ] 𝒯⊆ [ N ] 𝒯 : ∅ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} 𝒯 ⊆ [1 : N ] λ 𝒯 : fraction of time in state 𝒯 λ ∅ , λ {1} , λ {2} , λ {3} , λ {1,2} , λ {1,3} , λ {2,3} , λ {1,2,3} i ∈ 𝒯 : Relay i Tx i ∈ 𝒯 c : Relay i Rx A. S. Avestimehr, et al, IEEE Trans. Inf. Theory , 2011. A. Ozgur, et al., IEEE Trans. Inf. Theory , 2013. 7 S. Lim, et al. IEEE Trans. Inf. Theory , 2011.
Capacity and Scheduling 1 C ( 𝒪 ) − G 1 ≤ ˜ C ≤ C ( 𝒪 ) + G 2 r 1 ℓ 1 where, G 1 + G 2 = f ( N ) r 2 ℓ 2 2 D S C ( 𝒪 ) = max t λ r 3 λ 𝒯 ( max ℓ 3 i ∈𝒯∩Ω r i ) 𝗍 . 𝗎 . t ≤ ∑ i ∈𝒯 c ∩Ω c ℓ i + max ∀Ω ⊆ [ N ] 𝒯⊆ [ N ] 3 ∑ λ 𝒯 = 1, λ 𝒯 ≥ 0, ∀𝒯 ⊆ [ N ] 𝒯⊆ [ N ] 𝒯 : ∅ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} 𝒯 ⊆ [1 : N ] λ 𝒯 : fraction of time in state 𝒯 λ ∅ , λ {1} , λ {2} , λ {3} , λ {1,2} , λ {1,3} , λ {2,3} , λ {1,2,3} i ∈ 𝒯 : Relay i Tx i ∈ 𝒯 c : Relay i Rx A. S. Avestimehr, et al, IEEE Trans. Inf. Theory , 2011. A. Ozgur, et al., IEEE Trans. Inf. Theory , 2013. 8 S. Lim, et al. IEEE Trans. Inf. Theory , 2011.
Capacity and Scheduling 1 C ( 𝒪 ) − G 1 ≤ ˜ C ≤ C ( 𝒪 ) + G 2 r 1 ℓ 1 where, G 1 + G 2 = f ( N ) r 2 ℓ 2 2 D S C ( 𝒪 ) = max t λ r 3 λ 𝒯 ( max ℓ 3 i ∈𝒯∩Ω r i ) 𝗍 . 𝗎 . t ≤ ∑ i ∈𝒯 c ∩Ω c ℓ i + max ∀Ω ⊆ [ N ] 𝒯⊆ [ N ] 3 ∑ λ 𝒯 = 1, λ 𝒯 ≥ 0, ∀𝒯 ⊆ [ N ] 𝒯⊆ [ N ] 𝒯 : ∅ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} 𝒯 ⊆ [1 : N ] λ 𝒯 : fraction of time in state 𝒯 λ ∅ , λ {1} , λ {2} , λ {3} , λ {1,2} , λ {1,3} , λ {2,3} , λ {1,2,3} i ∈ 𝒯 : Relay i Tx Ω = ∅ , {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3} i ∈ 𝒯 c : Relay i Rx A. S. Avestimehr, et al, IEEE Trans. Inf. Theory , 2011. A. Ozgur, et al., IEEE Trans. Inf. Theory , 2013. 9 S. Lim, et al. IEEE Trans. Inf. Theory , 2011.
Capacity and Scheduling 1 C ( 𝒪 ) − G 1 ≤ ˜ C ≤ C ( 𝒪 ) + G 2 Ω = {2,3} r 1 ℓ 1 where, G 1 + G 2 = f ( N ) r 2 ℓ 2 2 D S C ( 𝒪 ) = max t λ r 3 λ 𝒯 ( max ℓ 3 i ∈𝒯∩Ω r i ) 𝗍 . 𝗎 . t ≤ ∑ i ∈𝒯 c ∩Ω c ℓ i + max ∀Ω ⊆ [ N ] 𝒯⊆ [ N ] 3 ∑ λ 𝒯 = 1, λ 𝒯 ≥ 0, ∀𝒯 ⊆ [ N ] 𝒯⊆ [ N ] 𝒯 : ∅ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} 𝒯 ⊆ [1 : N ] λ 𝒯 : fraction of time in state 𝒯 λ ∅ , λ {1} , λ {2} , λ {3} , λ {1,2} , λ {1,3} , λ {2,3} , λ {1,2,3} i ∈ 𝒯 : Relay i Tx Ω = ∅ , {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3} i ∈ 𝒯 c : Relay i Rx A. S. Avestimehr, et al, IEEE Trans. Inf. Theory , 2011. A. Ozgur, et al., IEEE Trans. Inf. Theory , 2013. 10 S. Lim, et al. IEEE Trans. Inf. Theory , 2011.
Approximate Capacity of One Relay Network C ( 𝒪 ) = max t λ λ 𝒯 ( max i ∈𝒯∩Ω r i ) 𝗍 . 𝗎 . t ≤ ∑ i ∈𝒯 c ∩Ω c ℓ i + max ∀Ω ⊆ [ N ] r 1 ℓ 1 𝒯⊆ [ N ] 1 D S ∑ λ 𝒯 = 1, λ 𝒯 ≥ 0, ∀𝒯 ⊆ [ N ] 𝒯⊆ [ N ] 11
Approximate Capacity of One Relay Network C ( 𝒪 ) = max t λ λ 𝒯 ( max Ω = ∅ i ∈𝒯∩Ω r i ) 𝗍 . 𝗎 . t ≤ ∑ i ∈𝒯 c ∩Ω c ℓ i + max ∀Ω ⊆ [ N ] r 1 ℓ 1 𝒯⊆ [ N ] 1 D S ∑ λ 𝒯 = 1, λ 𝒯 ≥ 0, ∀𝒯 ⊆ [ N ] 𝒯⊆ [ N ] Ω = ∅ t ≤ λ ∅ . ℓ 1 + λ {1} .0 12
Approximate Capacity of One Relay Network C ( 𝒪 ) = max t λ λ 𝒯 ( max Ω = {1} Ω = ∅ i ∈𝒯∩Ω r i ) 𝗍 . 𝗎 . t ≤ ∑ i ∈𝒯 c ∩Ω c ℓ i + max ∀Ω ⊆ [ N ] r 1 ℓ 1 𝒯⊆ [ N ] 1 D S ∑ λ 𝒯 = 1, λ 𝒯 ≥ 0, ∀𝒯 ⊆ [ N ] 𝒯⊆ [ N ] Ω = ∅ t ≤ λ ∅ . ℓ 1 + λ {1} .0 Ω = {1} t ≤ λ ∅ .0 + λ {1} . r 1 13
Approximate Capacity of One Relay Network C ( 𝒪 ) = max t λ λ 𝒯 ( max Ω = {1} Ω = ∅ i ∈𝒯∩Ω r i ) 𝗍 . 𝗎 . t ≤ ∑ i ∈𝒯 c ∩Ω c ℓ i + max ∀Ω ⊆ [ N ] r 1 ℓ 1 𝒯⊆ [ N ] 1 D S ∑ λ 𝒯 = 1, λ 𝒯 ≥ 0, ∀𝒯 ⊆ [ N ] 𝒯⊆ [ N ] 𝖣 = max t λ Ω = ∅ t ≤ λ ∅ . ℓ 1 + λ {1} .0 Ω = {1} t ≤ λ ∅ .0 + λ {1} . r 1 λ ∅ + λ {1} = 1, λ ∅ , λ {1} ≥ 0 14
Approximate Capacity of One Relay Network C ( 𝒪 ) = max t λ λ 𝒯 ( max Ω = {1} Ω = ∅ i ∈𝒯∩Ω r i ) 𝗍 . 𝗎 . t ≤ ∑ i ∈𝒯 c ∩Ω c ℓ i + max ∀Ω ⊆ [ N ] r 1 ℓ 1 𝒯⊆ [ N ] 1 D S ∑ λ 𝒯 = 1, λ 𝒯 ≥ 0, ∀𝒯 ⊆ [ N ] 𝒯⊆ [ N ] 𝖣 = max t λ ℓ 1 r 1 Ω = ∅ t ≤ λ ∅ . ℓ 1 + λ {1} .0 C = ℓ 1 + r 1 Ω = {1} t ≤ λ ∅ .0 + λ {1} . r 1 λ ∅ + λ {1} = 1, λ ∅ , λ {1} ≥ 0 15
Goal: Find the fraction of approximate capacity guaranteed by a single relay in an N-relay half-duplex diamond network 1 r 1 ℓ 1 r 2 2 ℓ 2 S D r i ℓ i i r N ℓ N N 𝒪 = {( ℓ j , r j ), j ∈ [1 : N ]} Approximate Capacity = C ( 𝒪 ) 16
Goal: Find the fraction of approximate capacity guaranteed by a single relay in an N-relay half-duplex diamond network 1 r 1 ℓ 1 r 2 2 ℓ 2 S D S D r i r i ℓ i ℓ i i i r N ℓ N 𝒪 i = {( ℓ i , r i )} N ℓ i r i C ( 𝒪 i ) = 𝒪 = {( ℓ j , r j ), j ∈ [1 : N ]} ℓ i + r i Approximate Capacity = C ( 𝒪 ) 17
Goal: Find the fraction of approximate capacity guaranteed by a single relay in an N-relay half-duplex diamond network 1 r 1 ℓ 1 C ( 𝒪 i ) C ( 𝒪 ) r 2 2 ℓ 2 S D S D r i r i ℓ i ℓ i i i r N ℓ N 𝒪 i = {( ℓ i , r i )} N ℓ i r i C ( 𝒪 i ) = 𝒪 = {( ℓ j , r j ), j ∈ [1 : N ]} ℓ i + r i Approximate Capacity = C ( 𝒪 ) 18
Goal: Find the fraction of approximate capacity guaranteed by a single relay in an N-relay half-duplex diamond network 1 r 1 ℓ 1 C ( 𝒪 i ) C ( 𝒪 ) r 2 2 ℓ 2 S D S D r i r i ℓ i ℓ i max i C ( 𝒪 i ) i i r N ℓ N C ( 𝒪 ) 𝒪 i = {( ℓ i , r i )} N ℓ i r i C ( 𝒪 i ) = 𝒪 = {( ℓ j , r j ), j ∈ [1 : N ]} ℓ i + r i Approximate Capacity = C ( 𝒪 ) 19
Goal: Find the fraction of approximate capacity guaranteed by a single relay in an N-relay half-duplex diamond network 1 r 1 ℓ 1 C ( 𝒪 i ) C ( 𝒪 ) r 2 2 ℓ 2 S D S D r i r i ℓ i ℓ i max i C ( 𝒪 i ) i i r N ℓ N C ( 𝒪 ) 𝒪 i = {( ℓ i , r i )} N max i C ( 𝒪 i ) min ℓ i r i C ( 𝒪 ) 𝒪 C ( 𝒪 i ) = 𝒪 = {( ℓ j , r j ), j ∈ [1 : N ]} ℓ i + r i Approximate Capacity = C ( 𝒪 ) 20
Theorem 1: Minimal Ratio: max i C ( 𝒪 i ) C ( 𝒪 ) For any half-duplex diamond N-relay network 𝒪 with approximate capacity C ( 𝒪 ), the best relay has an approximate capacity such that max i C ( 𝒪 i ) 1 ≥ 2 + 2 cos ( N + 2 ) C ( 𝒪 ) 2 π Tightness: Moreover, this bound is tight, i.e., for any positive integer N , there exist Gaussian half-duplex diamond N − relay networks 𝒪 for which the bound is tight. 21
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