Fundamental Limits of Wireless Caching Under Mixed Cacheable and - - PowerPoint PPT Presentation

fundamental limits of wireless caching under mixed
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Fundamental Limits of Wireless Caching Under Mixed Cacheable and - - PowerPoint PPT Presentation

Fundamental Limits of Wireless Caching Under Mixed Cacheable and Uncacheable Traffic Hamdi Joudeh , E. Lampiris and G. Caire P. Elia IEEE ISIT 2020 (virtual) Los Angeles, CA Supported by the European Research Council (ERC) under the ERC


slide-1
SLIDE 1

Fundamental Limits of Wireless Caching Under Mixed Cacheable and Uncacheable Traffic

IEEE ISIT 2020 (virtual) Los Angeles, CA

Hamdi Joudeh, E. Lampiris and G. Caire

  • P. Elia

1

Supported by the European Research Council (ERC) under the ERC Advanced Grant CARENET (Caire), and the ERC Consolidator Grant DUALITY (Elia)

slide-2
SLIDE 2

Content Library

New traffic model: cacheable content New network model: cache-aided

2

Content-Aware/Cache-Aided Wireless Networks

Cache-aided wireless network

= cache/memory

slide-3
SLIDE 3

Content Library

New traffic model: cacheable content New network model: cache-aided

3

Content-Aware/Cache-Aided Wireless Networks

Cache-aided wireless network

= cache/memory

Coded Multicasting

Maddah-Ali and Niesen “Fundamental limits

  • f caching" IEEE T-IT’14
slide-4
SLIDE 4

Content Library

New traffic model: cacheable content New network model: cache-aided

4

Content-Aware/Cache-Aided Wireless Networks

Cache-aided wireless network

= cache/memory

Not all traffic is cacheable content Non-content uncacheable ≈ 40%

Paschos et al. “Wireless caching: technical misconceptions and business barriers" IEEE Commun. Mag.’16

Coded Multicasting

Maddah-Ali and Niesen “Fundamental limits

  • f caching" IEEE T-IT’14
slide-5
SLIDE 5

5

Content-Aware/Cache-Aided Wireless Networks

Content Non-content

Unicast Multi-Multicast

Higher layer PHY

Service/traffic separator

Not all traffic is cacheable content Non-content uncacheable ≈ 40%

Paschos et al. “Wireless caching: technical misconceptions and business barriers" IEEE Commun. Mag.’16

Coded Multicasting

Maddah-Ali and Niesen “Fundamental limits

  • f caching" IEEE T-IT’14

Content Library

Content Non-content

Multi-Multicast + Unicast Layer separator

Higher layer PHY

slide-6
SLIDE 6

6

users signal levels noise floor

Multicasting, Channel Variability and Topological Holes

Multicast (Content) Unicast (Non-content) Various channel states (strengths)

slide-7
SLIDE 7

Problem Setting

𝑉! 𝐺

!

𝐺" ⋮ 𝐺# 𝑉$

⋮ ⋮

𝑉% 𝑋

!

𝑋

"

⋮ 𝑋

$

↓ content non-content cache

SNR! SNR" SNR#

Yk(t) = hkX(t) + Zk(t)

<latexit sha1_base64="IhA/VKG4viCFeTw9SHrEURjWrDY=">ACnicbZDLSgMxFIYz9VbrbdSlm2gRKkKZkYK6EIpuXFawF2HIZOmbWjmQnJGKEPXbnwVNy4UcesTuPNtzLSz0NYfAl/+cw7J+b1IcAW9W3kFhaXlfyq4W19Y3NLXN7p6HCWFJWp6EIZcsjigkesDpwEKwVSUZ8T7CmN7xK680HJhUPg1sYRczxST/gPU4JaMs19+/cZDguwRG+wIMUcSu9HOP7zC+4ZtEqWxPhebAzKJMNdf86nRDGvsACqIUm3bisBJiAROBRsXOrFiEaFD0mdtjQHxmXKSySpjfKidLu6FUp8A8MT9PZEQX6mR7+lOn8BAzdZS879aO4bemZPwIqBXT6UC8WGEKc5oK7XDIKYqSBUMn1XzEdEko6PTSEOzZlehcVK2K+Xzm0qxepnFkUd76ACVkI1OURVdoxqI4oe0TN6RW/Gk/FivBsf09ackc3soj8yPn8ASU6YGw=</latexit>

PHY: Degraded GBC

SNRk = |hk|2

<latexit sha1_base64="c8LK4CKkFrysh+Oo0eoypUvythw=">ACXicbZDLSsNAFIYnXmu9RV26GSyCq5KUgroQim5cSb30Am0Mk+mkHTqZhJmJUNJs3fgqblwo4tY3cOfbOEm70NYfBj7+cw5zu9FjEplWd/GwuLS8spqYa24vrG5tW3u7DZlGAtMGjhkoWh7SBJGOWkoqhpR4KgwGOk5Q0vsnrgQhJQ36nRhFxAtTn1KcYKW25JuwGSA2kn9xe3aRuMkzhGRwPMhjfJ5UFl2zZJWtXHAe7CmUwFR1/zq9kIcB4QrzJCUHduKlJMgoShmJC12Y0kihIeoTzoaOQqIdJL8khQeaqcH/VDoxXM3d8TCQqkHAWe7sz3nq1l5n+1Tqz8EyehPIoV4XjykR8zqEKYxQJ7VBCs2EgDwoLqXSEeIGw0uFlIdizJ89Ds1K2q+XT62qpdj6NowD2wQE4AjY4BjVwCeqgATB4BM/gFbwZT8aL8W58TFoXjOnMHvgj4/MH5syZ1A=</latexit>

SNR1 ≤ SNR2 ≤ · · · ≤ SNRK

<latexit sha1_base64="WxvPB63uOtYHX+40oBSx9PnDM=">ACNHicbVDLSsNAFJ34rPUVdelmsAiuSlIK6q7oRihIfQBTQiTyaQdOnk4MxFKyEe58UPciOBCEbd+g0kawT4ODBzOZe59gho0Jq2puytLyurZe2ihvbm3v7Kp7+x0RByTNg5YwHs2EoRn7QlYz0Qk6QZzPStUeXmd9JFzQwL+X45CYHhr41KUYyVSy1KbhITkUbnx3fZtYsZ5Ag5EHOK3W/lTsBFIsSjQTCMuWtGqWg4T/SCVECBlqW+GE6AI4/4EjMkRF/XQmnGiEuKGUnKRiRIiPAIDUg/pT7yiDj/OgEHqeKA92Ap8+XMFf/T8TIE2Ls2WkyX3XWy8RFXj+S7pkZUz+MJPHx5CM3YlAGMGsQOpQTLNk4JQhzmu4K8RBxhGXac1aCPnvyPOnUqnq9en5TrzQuijpK4BAcgROg1PQAFegBdoAgyfwCj7Ap/KsvCtfyvckuqQUMwdgCsrPLxyfqxM=</latexit>

N-file library

<latexit sha1_base64="xcOK4BVnJgTIqmgSl+h4oPuqChg=">ACAHicbVDJSgNBEO2JW4xb1IMHL42J4MUwIwH1FvTiSKYBZIQejo1SZOehe4aMQxz8Ve8eFDEq5/hzb+xsxw08UHB470quq5kRQabfvbyiwtr6yuZdzG5tb2zv53b26DmPFocZDGaqmyzRIEUANBUpoRgqY70pouMPrsd94AKVFGNzjKIKOz/qB8ARnaKRu/qCN8IhJ8bZ46gkJVApXMTVKu/mCXbInoIvEmZECmaHazX+1eyGPfQiQS6Z1y7Ej7CRMoeAS0lw71hAxPmR9aBkaMB90J5k8kNJjo/SoFypTAdKJ+nsiYb7WI981nT7DgZ73xuJ/XitG76KTiCKEQI+XeTFkmJIx2nQnlDAUY4MYVwJcyvlA6YR5NZzoTgzL+8SOpnJadcurwrFypXsziy5JAckRPikHNSITekSmqEk5Q8k1fyZj1ZL9a79TFtzVizmX3yB9bnDzJZli4=</latexit>

Normalize cache size µ , M/N

<latexit sha1_base64="TW8PKMKEpKGlMwiAdjtzugTfAj0=">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</latexit>

Caches U1, · · · , UK

<latexit sha1_base64="GOPuwU5MIyk+N87/AJ23JTXFzt4=">AC3icbVC7SgNBFJ31GeMramkzJBEsQtiVgNoF0wg2EcwDkhBmJzfJkNkHM3fFsGxv46/YWChi6w/Y+TdOHoUmHrhw5px7mXuPG0qh0ba/rZXVtfWNzdRWentnd28/c3BY10GkONR4IAPVdJkGKXyoUAJzVAB81wJDXdUmfiNe1BaBP4djkPoeGzgi7gDI3UzWTbCA8YVxgfgqb5Wjd2kKb9wLUBfO4SfJN5Ozi/YUdJk4c5Ijc1S7ma92L+CRBz5ybRuOXaInZgpFxCkm5HGkLGR2wALUN95oHuxNbEnpilB7tB8qUj3Sq/p6Imaf12HNp8dwqBe9ifif14qwf9GJhR9GCD6fdSPJMWAToKhPaGAoxwbwrgSZlfKh0wxjia+tAnBWTx5mdTPik6peHlbypWv5nGkyDHJklPikHNSJtekSmqEk0fyTF7Jm/VkvVjv1sesdcWazxyRP7A+fwAa7Zp4</latexit>

7

Each user caches M files (equiv.)

slide-8
SLIDE 8

Problem Setting

𝑉! 𝐺

!

𝐺" ⋮ 𝐺# 𝑉$

⋮ ⋮

𝑉%

Placement phase Ui = φi(F1, . . . , FN)

<latexit sha1_base64="f+y+eVTJvbF/cQ4LjiIvhfUzyIs=">ACnicbVDLSsNAFJ3UV62vqEs3o0WoUEoiBXUhFIXiSiqYtCEMJlM2qGTBzMToYSs3fgrblwo4tYvcOfOH0stPXAhcM593LvPV7CqJCG8a0VlpZXVteK6WNza3tHX13ry3ilGNi4ZjFvOshQRiNiCWpZKSbcIJCj5GON7we+50HwgWNo3s5SogTon5EA4qRVJKrH1puRnN4Ce1kQF1abqZmVdt5sdSVGHTvT1x9bJRMyaAi8SckTKYoeXqX7Yf4zQkcQMCdEzjUQ6GeKSYkbykp0KkiA8RH3SUzRCIRFONnklh8dK8WEQc1WRhBP190SGQiFGoac6QyQHYt4bi/95vVQG505GoySVJMLTRUHKoIzhOBfoU06wZCNFEOZU3QrxAHGEpUqvpEIw519eJO3TmlmvXdzVy42rWRxFcACOQAWY4Aw0wA1oAQtg8AiewSt40560F+1d+5i2FrTZzD74A+3zB7aGmQ=</latexit>

Yk(t) = hkX(t) + Zk(t)

<latexit sha1_base64="IhA/VKG4viCFeTw9SHrEURjWrDY=">ACnicbZDLSgMxFIYz9VbrbdSlm2gRKkKZkYK6EIpuXFawF2HIZOmbWjmQnJGKEPXbnwVNy4UcesTuPNtzLSz0NYfAl/+cw7J+b1IcAW9W3kFhaXlfyq4W19Y3NLXN7p6HCWFJWp6EIZcsjigkesDpwEKwVSUZ8T7CmN7xK680HJhUPg1sYRczxST/gPU4JaMs19+/cZDguwRG+wIMUcSu9HOP7zC+4ZtEqWxPhebAzKJMNdf86nRDGvsACqIUm3bisBJiAROBRsXOrFiEaFD0mdtjQHxmXKSySpjfKidLu6FUp8A8MT9PZEQX6mR7+lOn8BAzdZS879aO4bemZPwIqBXT6UC8WGEKc5oK7XDIKYqSBUMn1XzEdEko6PTSEOzZlehcVK2K+Xzm0qxepnFkUd76ACVkI1OURVdoxqI4oe0TN6RW/Gk/FivBsf09ackc3soj8yPn8ASU6YGw=</latexit>

PHY: Degraded GBC

SNRk = |hk|2

<latexit sha1_base64="c8LK4CKkFrysh+Oo0eoypUvythw=">ACXicbZDLSsNAFIYnXmu9RV26GSyCq5KUgroQim5cSb30Am0Mk+mkHTqZhJmJUNJs3fgqblwo4tY3cOfbOEm70NYfBj7+cw5zu9FjEplWd/GwuLS8spqYa24vrG5tW3u7DZlGAtMGjhkoWh7SBJGOWkoqhpR4KgwGOk5Q0vsnrgQhJQ36nRhFxAtTn1KcYKW25JuwGSA2kn9xe3aRuMkzhGRwPMhjfJ5UFl2zZJWtXHAe7CmUwFR1/zq9kIcB4QrzJCUHduKlJMgoShmJC12Y0kihIeoTzoaOQqIdJL8khQeaqcH/VDoxXM3d8TCQqkHAWe7sz3nq1l5n+1Tqz8EyehPIoV4XjykR8zqEKYxQJ7VBCs2EgDwoLqXSEeIGw0uFlIdizJ89Ds1K2q+XT62qpdj6NowD2wQE4AjY4BjVwCeqgATB4BM/gFbwZT8aL8W58TFoXjOnMHvgj4/MH5syZ1A=</latexit>

SNR1 ≤ SNR2 ≤ · · · ≤ SNRK

<latexit sha1_base64="WxvPB63uOtYHX+40oBSx9PnDM=">ACNHicbVDLSsNAFJ34rPUVdelmsAiuSlIK6q7oRihIfQBTQiTyaQdOnk4MxFKyEe58UPciOBCEbd+g0kawT4ODBzOZe59gho0Jq2puytLyurZe2ihvbm3v7Kp7+x0RByTNg5YwHs2EoRn7QlYz0Qk6QZzPStUeXmd9JFzQwL+X45CYHhr41KUYyVSy1KbhITkUbnx3fZtYsZ5Ag5EHOK3W/lTsBFIsSjQTCMuWtGqWg4T/SCVECBlqW+GE6AI4/4EjMkRF/XQmnGiEuKGUnKRiRIiPAIDUg/pT7yiDj/OgEHqeKA92Ap8+XMFf/T8TIE2Ls2WkyX3XWy8RFXj+S7pkZUz+MJPHx5CM3YlAGMGsQOpQTLNk4JQhzmu4K8RBxhGXac1aCPnvyPOnUqnq9en5TrzQuijpK4BAcgROg1PQAFegBdoAgyfwCj7Ap/KsvCtfyvckuqQUMwdgCsrPLxyfqxM=</latexit>

Normalize cache size µ , M/N

<latexit sha1_base64="TW8PKMKEpKGlMwiAdjtzugTfAj0=">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</latexit>

Caches U1, · · · , UK

<latexit sha1_base64="GOPuwU5MIyk+N87/AJ23JTXFzt4=">AC3icbVC7SgNBFJ31GeMramkzJBEsQtiVgNoF0wg2EcwDkhBmJzfJkNkHM3fFsGxv46/YWChi6w/Y+TdOHoUmHrhw5px7mXuPG0qh0ba/rZXVtfWNzdRWentnd28/c3BY10GkONR4IAPVdJkGKXyoUAJzVAB81wJDXdUmfiNe1BaBP4djkPoeGzgi7gDI3UzWTbCA8YVxgfgqb5Wjd2kKb9wLUBfO4SfJN5Ozi/YUdJk4c5Ijc1S7ma92L+CRBz5ybRuOXaInZgpFxCkm5HGkLGR2wALUN95oHuxNbEnpilB7tB8qUj3Sq/p6Imaf12HNp8dwqBe9ifif14qwf9GJhR9GCD6fdSPJMWAToKhPaGAoxwbwrgSZlfKh0wxjia+tAnBWTx5mdTPik6peHlbypWv5nGkyDHJklPikHNSJtekSmqEk0fyTF7Jm/VkvVjv1sesdcWazxyRP7A+fwAa7Zp4</latexit>
  • Independent of demands
  • Non-content messages are not born yet

8

content

Each user caches M files (equiv.)

slide-9
SLIDE 9

Problem Setting

Placement phase Ui = φi(F1, . . . , FN)

<latexit sha1_base64="f+y+eVTJvbF/cQ4LjiIvhfUzyIs=">ACnicbVDLSsNAFJ3UV62vqEs3o0WoUEoiBXUhFIXiSiqYtCEMJlM2qGTBzMToYSs3fgrblwo4tYvcOfOH0stPXAhcM593LvPV7CqJCG8a0VlpZXVteK6WNza3tHX13ry3ilGNi4ZjFvOshQRiNiCWpZKSbcIJCj5GON7we+50HwgWNo3s5SogTon5EA4qRVJKrH1puRnN4Ce1kQF1abqZmVdt5sdSVGHTvT1x9bJRMyaAi8SckTKYoeXqX7Yf4zQkcQMCdEzjUQ6GeKSYkbykp0KkiA8RH3SUzRCIRFONnklh8dK8WEQc1WRhBP190SGQiFGoac6QyQHYt4bi/95vVQG505GoySVJMLTRUHKoIzhOBfoU06wZCNFEOZU3QrxAHGEpUqvpEIw519eJO3TmlmvXdzVy42rWRxFcACOQAWY4Aw0wA1oAQtg8AiewSt40560F+1d+5i2FrTZzD74A+3zB7aGmQ=</latexit>

Yk(t) = hkX(t) + Zk(t)

<latexit sha1_base64="IhA/VKG4viCFeTw9SHrEURjWrDY=">ACnicbZDLSgMxFIYz9VbrbdSlm2gRKkKZkYK6EIpuXFawF2HIZOmbWjmQnJGKEPXbnwVNy4UcesTuPNtzLSz0NYfAl/+cw7J+b1IcAW9W3kFhaXlfyq4W19Y3NLXN7p6HCWFJWp6EIZcsjigkesDpwEKwVSUZ8T7CmN7xK680HJhUPg1sYRczxST/gPU4JaMs19+/cZDguwRG+wIMUcSu9HOP7zC+4ZtEqWxPhebAzKJMNdf86nRDGvsACqIUm3bisBJiAROBRsXOrFiEaFD0mdtjQHxmXKSySpjfKidLu6FUp8A8MT9PZEQX6mR7+lOn8BAzdZS879aO4bemZPwIqBXT6UC8WGEKc5oK7XDIKYqSBUMn1XzEdEko6PTSEOzZlehcVK2K+Xzm0qxepnFkUd76ACVkI1OURVdoxqI4oe0TN6RW/Gk/FivBsf09ackc3soj8yPn8ASU6YGw=</latexit>

PHY: Degraded GBC

SNRk = |hk|2

<latexit sha1_base64="c8LK4CKkFrysh+Oo0eoypUvythw=">ACXicbZDLSsNAFIYnXmu9RV26GSyCq5KUgroQim5cSb30Am0Mk+mkHTqZhJmJUNJs3fgqblwo4tY3cOfbOEm70NYfBj7+cw5zu9FjEplWd/GwuLS8spqYa24vrG5tW3u7DZlGAtMGjhkoWh7SBJGOWkoqhpR4KgwGOk5Q0vsnrgQhJQ36nRhFxAtTn1KcYKW25JuwGSA2kn9xe3aRuMkzhGRwPMhjfJ5UFl2zZJWtXHAe7CmUwFR1/zq9kIcB4QrzJCUHduKlJMgoShmJC12Y0kihIeoTzoaOQqIdJL8khQeaqcH/VDoxXM3d8TCQqkHAWe7sz3nq1l5n+1Tqz8EyehPIoV4XjykR8zqEKYxQJ7VBCs2EgDwoLqXSEeIGw0uFlIdizJ89Ds1K2q+XT62qpdj6NowD2wQE4AjY4BjVwCeqgATB4BM/gFbwZT8aL8W58TFoXjOnMHvgj4/MH5syZ1A=</latexit>

SNR1 ≤ SNR2 ≤ · · · ≤ SNRK

<latexit sha1_base64="WxvPB63uOtYHX+40oBSx9PnDM=">ACNHicbVDLSsNAFJ34rPUVdelmsAiuSlIK6q7oRihIfQBTQiTyaQdOnk4MxFKyEe58UPciOBCEbd+g0kawT4ODBzOZe59gho0Jq2puytLyurZe2ihvbm3v7Kp7+x0RByTNg5YwHs2EoRn7QlYz0Qk6QZzPStUeXmd9JFzQwL+X45CYHhr41KUYyVSy1KbhITkUbnx3fZtYsZ5Ag5EHOK3W/lTsBFIsSjQTCMuWtGqWg4T/SCVECBlqW+GE6AI4/4EjMkRF/XQmnGiEuKGUnKRiRIiPAIDUg/pT7yiDj/OgEHqeKA92Ap8+XMFf/T8TIE2Ls2WkyX3XWy8RFXj+S7pkZUz+MJPHx5CM3YlAGMGsQOpQTLNk4JQhzmu4K8RBxhGXac1aCPnvyPOnUqnq9en5TrzQuijpK4BAcgROg1PQAFegBdoAgyfwCj7Ap/KsvCtfyvckuqQUMwdgCsrPLxyfqxM=</latexit>

Normalize cache size µ , M/N

<latexit sha1_base64="TW8PKMKEpKGlMwiAdjtzugTfAj0=">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</latexit>

𝑉! 𝐺

!

𝐺" ⋮ 𝐺#

→ % 𝐺

$!, (

𝑋

!

→ % 𝐺

$", (

𝑋

"

𝑉$

→ % 𝐺

$#, (

𝑋

#

⋮ ⋮

𝑉% 𝑋

!

𝑋

"

⋮ 𝑋

$

𝑍

!

𝑍

"

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#

Caches U1, · · · , UK

<latexit sha1_base64="GOPuwU5MIyk+N87/AJ23JTXFzt4=">AC3icbVC7SgNBFJ31GeMramkzJBEsQtiVgNoF0wg2EcwDkhBmJzfJkNkHM3fFsGxv46/YWChi6w/Y+TdOHoUmHrhw5px7mXuPG0qh0ba/rZXVtfWNzdRWentnd28/c3BY10GkONR4IAPVdJkGKXyoUAJzVAB81wJDXdUmfiNe1BaBP4djkPoeGzgi7gDI3UzWTbCA8YVxgfgqb5Wjd2kKb9wLUBfO4SfJN5Ozi/YUdJk4c5Ijc1S7ma92L+CRBz5ybRuOXaInZgpFxCkm5HGkLGR2wALUN95oHuxNbEnpilB7tB8qUj3Sq/p6Imaf12HNp8dwqBe9ifif14qwf9GJhR9GCD6fdSPJMWAToKhPaGAoxwbwrgSZlfKh0wxjia+tAnBWTx5mdTPik6peHlbypWv5nGkyDHJklPikHNSJtekSmqEk0fyTF7Jm/VkvVjv1sesdcWazxyRP7A+fwAa7Zp4</latexit>
  • Users demand content
  • Non-content messages are born

Delivery phase

demands: d , (d1, . . . , dK)

<latexit sha1_base64="BZ/8avk4y7wLVWCAqU21RsxnRfY=">ACInicbZDLSgMxFIYz3q23qks3wSIoSJkRwctKdCO4UbAqdMqQSc7UYCYzJmfEMvRZ3Pgqblwo6krwYUxrF95+CHz85xyzh/nSlr0/XdvaHhkdGx8YrIyNT0zO1edXzizWE4NHimMnMRMwtKamigRAUXuQGWxgrO46uDXv38BoyVmT7FTg6tlLW1TCRn6KyouhMi3GIpIGVa2F3apWHK8DJOSuEQjWS6reCaroOuhEhnadSqio7WoWvPrfl/0LwQDqJGBjqPqaygyXqSgkStmbTPwc2yVzKDkCrqVsLCQM37F2tB0qFkKtlX2T+zSFecImTGPY20736fKFlqbSeNXWdvf/u71jP/qzULTLZbpdR5gaD510dJoShmtJcXFdIAR9VxwLiRblfKL5lhHF2qFRdC8Pvkv3C2UQ826zsnm7W9/UEcE2SJLJNVEpAtskcOyTFpE7uyAN5Is/evfovXhvX61D3mBmkfyQ9/EJHXWjWg=</latexit>

9

PHY codeword: XT

<latexit sha1_base64="FOcZIPCnKcmrCkNpafuwnvDMlU=">ACBnicbVDJSgNBEO1xjXGLehShMQiewowEXE5BLzlGyCZJD09laRJz/TQXaOGIScv/oXD4p49Ru8+TdOloMmPih4vFdFVT03lMKgbX9bC4tLyurqbX0+sbm1nZmZ7dqVKQ5VLiStdZkCKACoUEI91MB8V0LN7V+N/NodaCNUMZBC2fdQPREZxhIrUzB02EB4xLxRvKlQf3SnsXdFi/jctDStPtTNbO2WPQeJMSZMUWpnvpqe4pEPAXLJjGk4doitmGkUXMIw3YwMhIz3WRcaCQ2YD6YVj98Y0qNE8WhH6aQCpGP190TMfGMGvpt0+gx7ZtYbif95jQg7Z61YBGEPDJok4kKSo6yoR6QgNHOUgI41okt1LeY5pxTJIbheDMvjxPqic5J587v85nC5fTOFJknxySY+KQU1IgRVIiFcLJI3kmr+TNerJerHfrY9K6YE1n9sgfWJ8/t9qYA=</latexit>

user i estimates: ( ˆ Fdi, ˆ Wi)

<latexit sha1_base64="/oNkvODz31LjUBd2yIPVy1DFUKw=">ACJHicbVDJSgNBEO1xjXGLevTSGAUFCTMScLsEBfEYwSRCEoaeTsU06VnorhHDMB/jxV/x4sEFD178FnuSHDT6oOH1e1VU1fMiKTa9qc1NT0zOzefW8gvLi2vrBbW1us6jBWHGg9lqG48pkGKAGoUMJNpID5noSG1z/P/MYdKC3C4BoHEbR9dhuIruAMjeQWTlsI95jEGhTdFtsUNAqfIegTmu62egyTi9RNOm4i0pTu06HSLPvXt4tFO2SPQT9S5wxKZIxqm7hrdUJexDgFwyrZuOHWE7YQoFl5DmW2aNiPE+u4WmoQHzQbeT4ZEp3TFKh3ZDZV6AdKj+7EiYr/XA90ylOaCnJ71M/M9rxtg9aiciGKEgI8GdWNJMaRZYrQjFHCUA0MYV8LsSnmPKcbR5JqF4Eye/JfUD0pOuXR8VS5WzsZx5Mgm2SK7xCGHpEIuSZXUCcP5Im8kFfr0Xq23q2PUemUNe7ZIL9gfX0Dfi6kuA=</latexit>

content non-content

slide-10
SLIDE 10

Problem Setting

Placement phase Ui = φi(F1, . . . , FN)

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Yk(t) = hkX(t) + Zk(t)

<latexit sha1_base64="IhA/VKG4viCFeTw9SHrEURjWrDY=">ACnicbZDLSgMxFIYz9VbrbdSlm2gRKkKZkYK6EIpuXFawF2HIZOmbWjmQnJGKEPXbnwVNy4UcesTuPNtzLSz0NYfAl/+cw7J+b1IcAW9W3kFhaXlfyq4W19Y3NLXN7p6HCWFJWp6EIZcsjigkesDpwEKwVSUZ8T7CmN7xK680HJhUPg1sYRczxST/gPU4JaMs19+/cZDguwRG+wIMUcSu9HOP7zC+4ZtEqWxPhebAzKJMNdf86nRDGvsACqIUm3bisBJiAROBRsXOrFiEaFD0mdtjQHxmXKSySpjfKidLu6FUp8A8MT9PZEQX6mR7+lOn8BAzdZS879aO4bemZPwIqBXT6UC8WGEKc5oK7XDIKYqSBUMn1XzEdEko6PTSEOzZlehcVK2K+Xzm0qxepnFkUd76ACVkI1OURVdoxqI4oe0TN6RW/Gk/FivBsf09ackc3soj8yPn8ASU6YGw=</latexit>

PHY: Degraded GBC

SNRk = |hk|2

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SNR1 ≤ SNR2 ≤ · · · ≤ SNRK

<latexit sha1_base64="WxvPB63uOtYHX+40oBSx9PnDM=">ACNHicbVDLSsNAFJ34rPUVdelmsAiuSlIK6q7oRihIfQBTQiTyaQdOnk4MxFKyEe58UPciOBCEbd+g0kawT4ODBzOZe59gho0Jq2puytLyurZe2ihvbm3v7Kp7+x0RByTNg5YwHs2EoRn7QlYz0Qk6QZzPStUeXmd9JFzQwL+X45CYHhr41KUYyVSy1KbhITkUbnx3fZtYsZ5Ag5EHOK3W/lTsBFIsSjQTCMuWtGqWg4T/SCVECBlqW+GE6AI4/4EjMkRF/XQmnGiEuKGUnKRiRIiPAIDUg/pT7yiDj/OgEHqeKA92Ap8+XMFf/T8TIE2Ls2WkyX3XWy8RFXj+S7pkZUz+MJPHx5CM3YlAGMGsQOpQTLNk4JQhzmu4K8RBxhGXac1aCPnvyPOnUqnq9en5TrzQuijpK4BAcgROg1PQAFegBdoAgyfwCj7Ap/KsvCtfyvckuqQUMwdgCsrPLxyfqxM=</latexit>

Normalize cache size µ , M/N

<latexit sha1_base64="TW8PKMKEpKGlMwiAdjtzugTfAj0=">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</latexit>

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Delivery phase

demands: d , (d1, . . . , dK)

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T , 1/RF and R , (R1, . . . , RK)

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Delay (ch. uses) Caches U1, · · · , UK

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  • ptimize both

Non-content Rates

Optimal trade-off

10

PHY codeword: XT

<latexit sha1_base64="FOcZIPCnKcmrCkNpafuwnvDMlU=">ACBnicbVDJSgNBEO1xjXGLehShMQiewowEXE5BLzlGyCZJD09laRJz/TQXaOGIScv/oXD4p49Ru8+TdOloMmPih4vFdFVT03lMKgbX9bC4tLyurqbX0+sbm1nZmZ7dqVKQ5VLiStdZkCKACoUEI91MB8V0LN7V+N/NodaCNUMZBC2fdQPREZxhIrUzB02EB4xLxRvKlQf3SnsXdFi/jctDStPtTNbO2WPQeJMSZMUWpnvpqe4pEPAXLJjGk4doitmGkUXMIw3YwMhIz3WRcaCQ2YD6YVj98Y0qNE8WhH6aQCpGP190TMfGMGvpt0+gx7ZtYbif95jQg7Z61YBGEPDJok4kKSo6yoR6QgNHOUgI41okt1LeY5pxTJIbheDMvjxPqic5J587v85nC5fTOFJknxySY+KQU1IgRVIiFcLJI3kmr+TNerJerHfrY9K6YE1n9sgfWJ8/t9qYA=</latexit>

user i estimates: ( ˆ Fdi, ˆ Wi)

<latexit sha1_base64="/oNkvODz31LjUBd2yIPVy1DFUKw=">ACJHicbVDJSgNBEO1xjXGLevTSGAUFCTMScLsEBfEYwSRCEoaeTsU06VnorhHDMB/jxV/x4sEFD178FnuSHDT6oOH1e1VU1fMiKTa9qc1NT0zOzefW8gvLi2vrBbW1us6jBWHGg9lqG48pkGKAGoUMJNpID5noSG1z/P/MYdKC3C4BoHEbR9dhuIruAMjeQWTlsI95jEGhTdFtsUNAqfIegTmu62egyTi9RNOm4i0pTu06HSLPvXt4tFO2SPQT9S5wxKZIxqm7hrdUJexDgFwyrZuOHWE7YQoFl5DmW2aNiPE+u4WmoQHzQbeT4ZEp3TFKh3ZDZV6AdKj+7EiYr/XA90ylOaCnJ71M/M9rxtg9aiciGKEgI8GdWNJMaRZYrQjFHCUA0MYV8LsSnmPKcbR5JqF4Eye/JfUD0pOuXR8VS5WzsZx5Mgm2SK7xCGHpEIuSZXUCcP5Im8kFfr0Xq23q2PUemUNe7ZIL9gfX0Dfi6kuA=</latexit>

as T → ∞

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T ?(R; µ)

<latexit sha1_base64="ZQEr+wTYoISuF6uZONc/+oujwRw=">ACDXicbVDLSsNAFJ3UV62vqEs3g1Wom5JIQcVN0Y3LKn1BE8tkOmHziRhZiKUkB9w46+4caGIW/fu/BsnaRfaeuDC4Zx7ufceL2JUKsv6NgpLyura8X10sbm1vaOubvXlmEsMGnhkIWi6yFJGA1IS1HFSDcSBHGPkY43vs78zgMRkoZBU0i4nI0DKhPMVJa6ptHDkdqhBFLmul94kiFRFrJNc9P7tJL6PD4pG+WraqVAy4Se0bKYIZG3/xyBiGOQkUZkjKnm1Fyk2QUBQzkpacWJI4TEakp6mAeJEukn+TQqPtTKAfih0BQrm6u+JBHEpJ9zTndmdct7LxP+8Xqz8czehQRQrEuDpIj9mUIUwiwYOqCBYsYkmCAuqb4V4hATCSgdY0iHY8y8vkvZp1a5VL25r5frVLI4iOACHoAJscAbq4AY0QAtg8AiewSt4M56MF+Pd+Ji2FozZzD74A+PzB91TnA8=</latexit>
slide-11
SLIDE 11

𝑉! 𝐺

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Maddah-Ali and Niesen (MAN) scheme MAN placement {φi : i ∈ [K]}

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Symmetric Setting: Content Only

SNRi = SNR, ∀i ∈ [K]

<latexit sha1_base64="XmNUSJMSom8Kx4cCrx1ybED2zqw=">ACF3icbVDLSsNAFJ3UV62vqks3g0VwISGRgroQim4EQeqjD2hCmEwn7dDJMxMhBL6F278FTcuFHGrO/GaZpFbT0wcDjnXubc48eMSmVZP0ZhYXFpeaW4Wlpb39jcKm/vNGWUCEwaOGKRaPtIEkY5aSiqGnHgqDQZ6TlDy7HfuRCEkj/qCGMXFD1OM0oBgpLXl0wmR6sgvb+5G3kUnsNp4cgJIoEYg9ShHauXa9csUwrA5wndk4qIEfdK3873QgnIeEKMyRlx7Zi5aZIKIoZGZWcRJIY4QHqkY6mHIVEuml21wgeaKULdQT9uIKZOr2RolDKYejrySz0rDcW/M6iQpO3ZTyOFGE48lHQcKgiuC4JNilgmDFhpogLKjOCnEfCYSVrKkS7BnT54nzWPTrpnt9VK7SKvowj2wD4BDY4ATVwBeqgATB4Ai/gDbwbz8ar8WF8TkYLRr6zC/7A+PoFGgOfQw=</latexit>

11 Maddah-Ali and Niesen “Fundamental limits of caching" IEEE T-IT’14

Coded multicast messages:

  • WS : S ⊆ [K], |S| = σ

, σ = Kµ + 1

<latexit sha1_base64="gQ4VgXV4CF/9gt1QtPvudGDaU=">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</latexit>

Content demands: Fd1, . . . , FdK

<latexit sha1_base64="E28zXO7dYE0CqDym0iniGxYkanY=">ACH3icbVBNS8NAEN3Ur1q/qh69LBbFg5REil8nURDBi4KthSaEzWZrl26yYXcilpB/4sW/4sWDIuLNf+O2zUGrDwYeb2aYeS9IBNdg219WaWp6ZnauPF9ZWFxaXqmurW0TBVlTSqFVO2AaCZ4zJrAQbB2ohiJAsFug/7ZsH97z5TmMr6BQcK8iNzFvMspASP51f1tF9gDZGcyBhYDlE4lAf4xy7bmX73M9C38l3sStCXoXj4TL3K/W7Lo9Av5LnILUIErv/rphpKmkblBdG649gJeBlRwKlgecVNUsI7ZM71jE0JhHTXjbyl+Mto4S4K5Up8+NI/bmRkUjrQRSYyYhAT0/2huJ/vU4K3UMv43GSGvN0fKibCgwSD8PCIVeMghgYQqji5ldMe0QRCibSignBmbT8l7T26k6jfnTdqJ2cFnGU0QbaRDvIQfoBF2gK9REFD2iZ/SK3qwn68V6tz7GoyWr2FlHv2B9fQPfA6Gh</latexit>
slide-12
SLIDE 12

Symmetric Setting: Content Only

𝑉! 𝐺

!

𝐺" ⋮ 𝐺# 𝑉$

𝑉%

Maddah-Ali and Niesen (MAN) scheme MAN placement {φi : i ∈ [K]}

<latexit sha1_base64="OBRPwSc7FV43TsknpgM/YGFrc=">ACFXicbVDLSgMxFM3UV62vqks3wSK4kDIjBR+rqhtBlAr2ATNDyaS3bWjmQZIRyzA/4cZfceNCEbeCO/GtJ2Fth4IHM659+be40WcSWa30Zubn5hcSm/XFhZXVvfKG5uNWQYCwp1GvJQtDwigbMA6opDq1IAPE9Dk1vcDHym/cgJAuDOzWMwPVJL2BdRonSUrt4Ch4UMn12Q2OKHgQ6Bw6iTYifqszfApZg4LsH3lYidtF0tm2RwDzxIrIyWUodYufjmdkMajoZQTKW3LjJSbEKEY5ZAWnFhCROiA9MDWNCA+SDcZX5XiPa10cDcU+umlxurvjoT4Ug59T1f6RPXltDcS/PsWHWP3YQFUawgoJOPujHKsSjiHCHCaCKDzUhVDC9K6Z9IghVOsiCDsGaPnmWNA7LVqV8clspVc+zOPJoB+2ifWShI1RFl6iG6oiR/SMXtGb8WS8GO/Gx6Q0Z2Q92+gPjM8fvdKd6w=</latexit>

Coded multicast messages:

  • WS : S ⊆ [K], |S| = σ

, σ = Kµ + 1

<latexit sha1_base64="gQ4VgXV4CF/9gt1QtPvudGDaU=">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</latexit>

T = 1 log(1 + SNR) · K(1 − µ) 1 + Kµ

<latexit sha1_base64="59WGpnRXH4z7ZVIbwpRPV7Uk3cQ=">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</latexit>

Content demands: Fd1, . . . , FdK

<latexit sha1_base64="E28zXO7dYE0CqDym0iniGxYkanY=">ACH3icbVBNS8NAEN3Ur1q/qh69LBbFg5REil8nURDBi4KthSaEzWZrl26yYXcilpB/4sW/4sWDIuLNf+O2zUGrDwYeb2aYeS9IBNdg219WaWp6ZnauPF9ZWFxaXqmurW0TBVlTSqFVO2AaCZ4zJrAQbB2ohiJAsFug/7ZsH97z5TmMr6BQcK8iNzFvMspASP51f1tF9gDZGcyBhYDlE4lAf4xy7bmX73M9C38l3sStCXoXj4TL3K/W7Lo9Av5LnILUIErv/rphpKmkblBdG649gJeBlRwKlgecVNUsI7ZM71jE0JhHTXjbyl+Mto4S4K5Up8+NI/bmRkUjrQRSYyYhAT0/2huJ/vU4K3UMv43GSGvN0fKibCgwSD8PCIVeMghgYQqji5ldMe0QRCibSignBmbT8l7T26k6jfnTdqJ2cFnGU0QbaRDvIQfoBF2gK9REFD2iZ/SK3qwn68V6tz7GoyWr2FlHv2B9fQPfA6Gh</latexit>

SNRi = SNR, ∀i ∈ [K]

<latexit sha1_base64="XmNUSJMSom8Kx4cCrx1ybED2zqw=">ACF3icbVDLSsNAFJ3UV62vqks3g0VwISGRgroQim4EQeqjD2hCmEwn7dDJMxMhBL6F278FTcuFHGrO/GaZpFbT0wcDjnXubc48eMSmVZP0ZhYXFpeaW4Wlpb39jcKm/vNGWUCEwaOGKRaPtIEkY5aSiqGnHgqDQZ6TlDy7HfuRCEkj/qCGMXFD1OM0oBgpLXl0wmR6sgvb+5G3kUnsNp4cgJIoEYg9ShHauXa9csUwrA5wndk4qIEfdK3873QgnIeEKMyRlx7Zi5aZIKIoZGZWcRJIY4QHqkY6mHIVEuml21wgeaKULdQT9uIKZOr2RolDKYejrySz0rDcW/M6iQpO3ZTyOFGE48lHQcKgiuC4JNilgmDFhpogLKjOCnEfCYSVrKkS7BnT54nzWPTrpnt9VK7SKvowj2wD4BDY4ATVwBeqgATB4Ai/gDbwbz8ar8WF8TkYLRr6zC/7A+PoFGgOfQw=</latexit>

12 Maddah-Ali and Niesen “Fundamental limits of caching" IEEE T-IT’14

Achievable delivery times (w. ch. coding):

<latexit sha1_base64="5p1bf6F+Eel3Zwi/KN14TXheUYs=">ACGnicbVDJSgNBEO2JW4xb1KOXxiDoJcyI4HKevEYwSyQhNDTU0ma9PQM3TVqGPIdXvwVLx4U8SZe/Bs7y0GNDwoer+p1dT0/lsKg6345mbn5hcWl7HJuZXVtfSO/uVU1UaI5VHgkI13mQEpFRQoIR6rIGFvoSa378c9Wu3oI2I1A0OYmiFrKtER3CGVmrnvSbCPabnvCfglkXDexb1jGgKEIwdP+uSHnPVhQI1T04G7bzBbfojkFniTclBTJFuZ3/aAYRT0JQyCUzpuG5MbZSplFwCcNcMzEQM95nXWhYqphd20rHpw3pnlUC2om0LYV0rP50pCw0ZhD6djJk2DN/eyPxv14jwc5JKxUqThAUnyzqJiREc50UBo4CgHljCuhf2rjYFpxtGmbMheH9PniXVw6J3VDy9PiqULqZxZMkO2SX7xCPHpESuSJlUCcP5Im8kFfn0Xl23pz3yWjGmXq2yS84n9MfKBh</latexit>

local/global gains

slide-13
SLIDE 13

𝑉! 𝐺

!

𝐺" ⋮ 𝐺# 𝑉$

𝑉%

Maddah-Ali and Niesen (MAN) scheme MAN placement {φi : i ∈ [K]}

<latexit sha1_base64="OBRPwSc7FV43TsknpgM/YGFrc=">ACFXicbVDLSgMxFM3UV62vqks3wSK4kDIjBR+rqhtBlAr2ATNDyaS3bWjmQZIRyzA/4cZfceNCEbeCO/GtJ2Fth4IHM659+be40WcSWa30Zubn5hcSm/XFhZXVvfKG5uNWQYCwp1GvJQtDwigbMA6opDq1IAPE9Dk1vcDHym/cgJAuDOzWMwPVJL2BdRonSUrt4Ch4UMn12Q2OKHgQ6Bw6iTYifqszfApZg4LsH3lYidtF0tm2RwDzxIrIyWUodYufjmdkMajoZQTKW3LjJSbEKEY5ZAWnFhCROiA9MDWNCA+SDcZX5XiPa10cDcU+umlxurvjoT4Ug59T1f6RPXltDcS/PsWHWP3YQFUawgoJOPujHKsSjiHCHCaCKDzUhVDC9K6Z9IghVOsiCDsGaPnmWNA7LVqV8clspVc+zOPJoB+2ifWShI1RFl6iG6oiR/SMXtGb8WS8GO/Gx6Q0Z2Q92+gPjM8fvdKd6w=</latexit>

Coded multicast messages:

  • WS : S ⊆ [K], |S| = σ

, σ = Kµ + 1

<latexit sha1_base64="gQ4VgXV4CF/9gt1QtPvudGDaU=">ACdXicbVFdaxNBFJ1dv2r8ivoiHBtNAiWuCsFtVAo9kXoS0XTFDJLmJ29SYfO7K4zd8Sw3X/gr/PNv+GLr07SCLH1wsCZc+69M+fevNbKUZL8jOIrV69dv7Fxs3Pr9p2797r3Hxy5yluJQ1npyh7nwqFWJQ5Jkcbj2qIwucZRfrq/0Edf0TpVlZ9pXmNmxKxUyUFBWrS/d7nhN+o2a8KLMB4TUFyBAadEzN0Oy3nT7P1Yw3o0nDjaATKXTzqW1hB9auwJ3PHRJ+gfFBtgVna9oZ7AZzYwAWLZqt4D/ZXbhIPTx8BLSbeXDJlwGWQrkCPreJw0v3Bi0p6gyVJLZwbp0lNWSNscKGx7XDvsBbyNDgZB1iKYCtrlNr4XlgCphWNpySYMmuVzTCODc3echcWHEXtQX5P23safo2a1RZe8JSnj809RqogsUKoFAWJel5AEJaFf4K8kRYISksqhOGkF60fBkcvR6k24N3H7d7e+9X49hgj9kme8FS9obtsQ/skA2ZL+iR9HTaDP6HT+Jn8X989Q4WtU8ZP9E/OoPa7u8yw=</latexit>

𝑋

!

𝑋

"

⋮ 𝑋

$

Symmetric Setting: Content and Non-Content

T (R) = 1

  • log(1 + SNR) − P

i∈[K] Ri

· K(1 − µ) 1 + Kµ

<latexit sha1_base64="BoqD3ZVY7hd8IOShncs1PdmrM=">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</latexit>

X

i∈[K]

Ri ≤ log(1 + SNR)

<latexit sha1_base64="JEM1NnRVvhZg2Zhp8dJoIn27V8o=">ACG3icbVDLSgMxFM34rOr6tJNsAgVocyUgroruhEqdU+oDMmThmYyY5IRyjD/4cZfceNCEVeC/G9LHQ1gMJ+fcS+49fsyoVJb1bSwsLi2vrObWzPWNza3t/M5uU0aJwKSBIxaJto8kYZSThqKkXYsCAp9Rlr+4GLktx6IkDTid2oYEzdEPU4DipHSkpcvOzIJvZRCh3LYuXIzWPf0g5F7fUW9og2PoRMi1ZdBentdz45M08sXrJI1Bpwn9pQUwBQ1L/pdCOchIQrzJCUHduKlZsioShmJDOdRJIY4QHqkY6mHIVEul4tweaqULg0jowxUcq787UhRKOQx9XTkec9Ybif95nUQFp25KeZwowvHkoyBhUEVwFBTsUkGwYkNEBZUzwpxHwmElY5zFI9u/I8aZLdqV0dlMpVM+nceTAPjgARWCDE1AFl6AGgCDR/AMXsGb8WS8GO/Gx6R0wZj27IE/ML5+AE/tnxM=</latexit>

With non-content:

<latexit sha1_base64="qOtvY+qGqel9YeEZwM1MaN+7DA=">ACAXicbVDLSgMxFM34rPU16kZwEyCG8uMFHysim5cVrAPaIeSdM2NJMyR2xDHXjr7hxoYhb/8Kdf2PazkJbDwQO59zDzT1hLgBz/t2FhaXldWc2v59Y3NrW13Z7dmVKIpq1IlG6ExDBJasCB8EasWYkCgWrh4PrsV+/Z9pwJe9gGLMgIj3Ju5wSsFLb3W8Be4C0zqGPpZInVElgEi5HbfgFb0J8DzxM1JAGSpt96vVUTSJbJoKYkzT92IUqKBU8FG+VZiWEzogPRY01JImaCdHLBCB9ZpYO7StsnAU/U34mURMYMo9BORgT6ZtYbi/95zQS650HKZzYs+h0UTcRGBQe14E7XDMKYmgJoZrbv2LaJ5pQsKXlbQn+7MnzpHZa9EvFi9tSoXyV1ZFDB+gQHSMfnaEyukEVEUPaJn9IrenCfnxXl3PqajC06W2UN/4Hz+ANfylyk=</latexit>

SNRi = SNR, ∀i ∈ [K]

<latexit sha1_base64="XmNUSJMSom8Kx4cCrx1ybED2zqw=">ACF3icbVDLSsNAFJ3UV62vqks3g0VwISGRgroQim4EQeqjD2hCmEwn7dDJMxMhBL6F278FTcuFHGrO/GaZpFbT0wcDjnXubc48eMSmVZP0ZhYXFpeaW4Wlpb39jcKm/vNGWUCEwaOGKRaPtIEkY5aSiqGnHgqDQZ6TlDy7HfuRCEkj/qCGMXFD1OM0oBgpLXl0wmR6sgvb+5G3kUnsNp4cgJIoEYg9ShHauXa9csUwrA5wndk4qIEfdK3873QgnIeEKMyRlx7Zi5aZIKIoZGZWcRJIY4QHqkY6mHIVEuml21wgeaKULdQT9uIKZOr2RolDKYejrySz0rDcW/M6iQpO3ZTyOFGE48lHQcKgiuC4JNilgmDFhpogLKjOCnEfCYSVrKkS7BnT54nzWPTrpnt9VK7SKvowj2wD4BDY4ATVwBeqgATB4Ai/gDbwbz8ar8WF8TkYLRr6zC/7A+PoFGgOfQw=</latexit>

Non-content: W1, . . . , WK

<latexit sha1_base64="+C+iU+Zi5M67HBNTQZcThp/Kr70=">ACE3icbVDLSgMxFM34tr6qLt0Ei0VEy4wIPlZFN4IgFawtdIYhk6Y2NJMyR2xDP0HN/6KGxeKuHXjzr8xrbPwdSBwOfeJOdEieAGXPfDGRufmJyanpktzM0vLC4Vl1eujEo1ZXWqhNLNiBgmuGR14CBYM9GMxJFgjah3MvQbN0wbruQl9BMWxORa8g6nBKwUFrfKPrBbyM6V3KFKApNwhAe+j8uNMPMG275oKzDbjfAMh8WSW3FHwH+Jl5MSylELi+9+W9E0tndSQYxpeW4CQUY0cCrYoOCnhiWE9sg1a1kqScxMkI0yDfCGVdq4o7Q9EvBI/b6RkdiYfhzZyZhA1/z2huJ/XiuFzkGQcZmkNiz9eqiTCgwKDwvCba4ZBdG3hFDN7V8x7RJNKNgaC7YE73fkv+Rqt+LtVQ4v9krV47yOGbSG1tEm8tA+qJTVEN1RNEdekBP6Nm5dx6dF+f1a3TMyXdW0Q84b5869J0f</latexit>

Content demands: Fd1, . . . , FdK

<latexit sha1_base64="E28zXO7dYE0CqDym0iniGxYkanY=">ACH3icbVBNS8NAEN3Ur1q/qh69LBbFg5REil8nURDBi4KthSaEzWZrl26yYXcilpB/4sW/4sWDIuLNf+O2zUGrDwYeb2aYeS9IBNdg219WaWp6ZnauPF9ZWFxaXqmurW0TBVlTSqFVO2AaCZ4zJrAQbB2ohiJAsFug/7ZsH97z5TmMr6BQcK8iNzFvMspASP51f1tF9gDZGcyBhYDlE4lAf4xy7bmX73M9C38l3sStCXoXj4TL3K/W7Lo9Av5LnILUIErv/rphpKmkblBdG649gJeBlRwKlgecVNUsI7ZM71jE0JhHTXjbyl+Mto4S4K5Up8+NI/bmRkUjrQRSYyYhAT0/2huJ/vU4K3UMv43GSGvN0fKibCgwSD8PCIVeMghgYQqji5ldMe0QRCibSignBmbT8l7T26k6jfnTdqJ2cFnGU0QbaRDvIQfoBF2gK9REFD2iZ/SK3qwn68V6tz7GoyWr2FlHv2B9fQPfA6Gh</latexit>

Bring non-content into the picture

13

T = 1 log(1 + SNR) · K(1 − µ) 1 + Kµ

<latexit sha1_base64="59WGpnRXH4z7ZVIbwpRPV7Uk3cQ=">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</latexit>

Achievable delivery times (w. ch. coding):

<latexit sha1_base64="5p1bf6F+Eel3Zwi/KN14TXheUYs=">ACGnicbVDJSgNBEO2JW4xb1KOXxiDoJcyI4HKevEYwSyQhNDTU0ma9PQM3TVqGPIdXvwVLx4U8SZe/Bs7y0GNDwoer+p1dT0/lsKg6345mbn5hcWl7HJuZXVtfSO/uVU1UaI5VHgkI13mQEpFRQoIR6rIGFvoSa378c9Wu3oI2I1A0OYmiFrKtER3CGVmrnvSbCPabnvCfglkXDexb1jGgKEIwdP+uSHnPVhQI1T04G7bzBbfojkFniTclBTJFuZ3/aAYRT0JQyCUzpuG5MbZSplFwCcNcMzEQM95nXWhYqphd20rHpw3pnlUC2om0LYV0rP50pCw0ZhD6djJk2DN/eyPxv14jwc5JKxUqThAUnyzqJiREc50UBo4CgHljCuhf2rjYFpxtGmbMheH9PniXVw6J3VDy9PiqULqZxZMkO2SX7xCPHpESuSJlUCcP5Im8kFfn0Xl23pz3yWjGmXq2yS84n9MfKBh</latexit>
slide-14
SLIDE 14

𝑉! 𝐺

!

𝐺" ⋮ 𝐺# 𝑉$

𝑉%

Maddah-Ali and Niesen (MAN) scheme MAN placement {φi : i ∈ [K]}

<latexit sha1_base64="OBRPwSc7FV43TsknpgM/YGFrc=">ACFXicbVDLSgMxFM3UV62vqks3wSK4kDIjBR+rqhtBlAr2ATNDyaS3bWjmQZIRyzA/4cZfceNCEbeCO/GtJ2Fth4IHM659+be40WcSWa30Zubn5hcSm/XFhZXVvfKG5uNWQYCwp1GvJQtDwigbMA6opDq1IAPE9Dk1vcDHym/cgJAuDOzWMwPVJL2BdRonSUrt4Ch4UMn12Q2OKHgQ6Bw6iTYifqszfApZg4LsH3lYidtF0tm2RwDzxIrIyWUodYufjmdkMajoZQTKW3LjJSbEKEY5ZAWnFhCROiA9MDWNCA+SDcZX5XiPa10cDcU+umlxurvjoT4Ug59T1f6RPXltDcS/PsWHWP3YQFUawgoJOPujHKsSjiHCHCaCKDzUhVDC9K6Z9IghVOsiCDsGaPnmWNA7LVqV8clspVc+zOPJoB+2ifWShI1RFl6iG6oiR/SMXtGb8WS8GO/Gx6Q0Z2Q92+gPjM8fvdKd6w=</latexit>

Coded multicast messages:

  • WS : S ⊆ [K], |S| = σ

, σ = Kµ + 1

<latexit sha1_base64="gQ4VgXV4CF/9gt1QtPvudGDaU=">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</latexit>

𝑋

!

𝑋

"

⋮ 𝑋

$

Symmetric Setting: Content and Non-Content

T (R) = 1

  • log(1 + SNR) − P

i∈[K] Ri

· K(1 − µ) 1 + Kµ

<latexit sha1_base64="BoqD3ZVY7hd8IOShncs1PdmrM=">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</latexit>

X

i∈[K]

Ri ≤ log(1 + SNR)

<latexit sha1_base64="JEM1NnRVvhZg2Zhp8dJoIn27V8o=">ACG3icbVDLSgMxFM34rOr6tJNsAgVocyUgroruhEqdU+oDMmThmYyY5IRyjD/4cZfceNCEVeC/G9LHQ1gMJ+fcS+49fsyoVJb1bSwsLi2vrObWzPWNza3t/M5uU0aJwKSBIxaJto8kYZSThqKkXYsCAp9Rlr+4GLktx6IkDTid2oYEzdEPU4DipHSkpcvOzIJvZRCh3LYuXIzWPf0g5F7fUW9og2PoRMi1ZdBentdz45M08sXrJI1Bpwn9pQUwBQ1L/pdCOchIQrzJCUHduKlZsioShmJDOdRJIY4QHqkY6mHIVEul4tweaqULg0jowxUcq787UhRKOQx9XTkec9Ybif95nUQFp25KeZwowvHkoyBhUEVwFBTsUkGwYkNEBZUzwpxHwmElY5zFI9u/I8aZLdqV0dlMpVM+nceTAPjgARWCDE1AFl6AGgCDR/AMXsGb8WS8GO/Gx6R0wZj27IE/ML5+AE/tnxM=</latexit>

With non-content:

<latexit sha1_base64="qOtvY+qGqel9YeEZwM1MaN+7DA=">ACAXicbVDLSgMxFM34rPU16kZwEyCG8uMFHysim5cVrAPaIeSdM2NJMyR2xDHXjr7hxoYhb/8Kdf2PazkJbDwQO59zDzT1hLgBz/t2FhaXldWc2v59Y3NrW13Z7dmVKIpq1IlG6ExDBJasCB8EasWYkCgWrh4PrsV+/Z9pwJe9gGLMgIj3Ju5wSsFLb3W8Be4C0zqGPpZInVElgEi5HbfgFb0J8DzxM1JAGSpt96vVUTSJbJoKYkzT92IUqKBU8FG+VZiWEzogPRY01JImaCdHLBCB9ZpYO7StsnAU/U34mURMYMo9BORgT6ZtYbi/95zQS650HKZzYs+h0UTcRGBQe14E7XDMKYmgJoZrbv2LaJ5pQsKXlbQn+7MnzpHZa9EvFi9tSoXyV1ZFDB+gQHSMfnaEyukEVEUPaJn9IrenCfnxXl3PqajC06W2UN/4Hz+ANfylyk=</latexit>

SNRi = SNR, ∀i ∈ [K]

<latexit sha1_base64="XmNUSJMSom8Kx4cCrx1ybED2zqw=">ACF3icbVDLSsNAFJ3UV62vqks3g0VwISGRgroQim4EQeqjD2hCmEwn7dDJMxMhBL6F278FTcuFHGrO/GaZpFbT0wcDjnXubc48eMSmVZP0ZhYXFpeaW4Wlpb39jcKm/vNGWUCEwaOGKRaPtIEkY5aSiqGnHgqDQZ6TlDy7HfuRCEkj/qCGMXFD1OM0oBgpLXl0wmR6sgvb+5G3kUnsNp4cgJIoEYg9ShHauXa9csUwrA5wndk4qIEfdK3873QgnIeEKMyRlx7Zi5aZIKIoZGZWcRJIY4QHqkY6mHIVEuml21wgeaKULdQT9uIKZOr2RolDKYejrySz0rDcW/M6iQpO3ZTyOFGE48lHQcKgiuC4JNilgmDFhpogLKjOCnEfCYSVrKkS7BnT54nzWPTrpnt9VK7SKvowj2wD4BDY4ATVwBeqgATB4Ai/gDbwbz8ar8WF8TkYLRr6zC/7A+PoFGgOfQw=</latexit>

Non-content: W1, . . . , WK

<latexit sha1_base64="+C+iU+Zi5M67HBNTQZcThp/Kr70=">ACE3icbVDLSgMxFM34tr6qLt0Ei0VEy4wIPlZFN4IgFawtdIYhk6Y2NJMyR2xDP0HN/6KGxeKuHXjzr8xrbPwdSBwOfeJOdEieAGXPfDGRufmJyanpktzM0vLC4Vl1eujEo1ZXWqhNLNiBgmuGR14CBYM9GMxJFgjah3MvQbN0wbruQl9BMWxORa8g6nBKwUFrfKPrBbyM6V3KFKApNwhAe+j8uNMPMG275oKzDbjfAMh8WSW3FHwH+Jl5MSylELi+9+W9E0tndSQYxpeW4CQUY0cCrYoOCnhiWE9sg1a1kqScxMkI0yDfCGVdq4o7Q9EvBI/b6RkdiYfhzZyZhA1/z2huJ/XiuFzkGQcZmkNiz9eqiTCgwKDwvCba4ZBdG3hFDN7V8x7RJNKNgaC7YE73fkv+Rqt+LtVQ4v9krV47yOGbSG1tEm8tA+qJTVEN1RNEdekBP6Nm5dx6dF+f1a3TMyXdW0Q84b5869J0f</latexit>

Content demands: Fd1, . . . , FdK

<latexit sha1_base64="E28zXO7dYE0CqDym0iniGxYkanY=">ACH3icbVBNS8NAEN3Ur1q/qh69LBbFg5REil8nURDBi4KthSaEzWZrl26yYXcilpB/4sW/4sWDIuLNf+O2zUGrDwYeb2aYeS9IBNdg219WaWp6ZnauPF9ZWFxaXqmurW0TBVlTSqFVO2AaCZ4zJrAQbB2ohiJAsFug/7ZsH97z5TmMr6BQcK8iNzFvMspASP51f1tF9gDZGcyBhYDlE4lAf4xy7bmX73M9C38l3sStCXoXj4TL3K/W7Lo9Av5LnILUIErv/rphpKmkblBdG649gJeBlRwKlgecVNUsI7ZM71jE0JhHTXjbyl+Mto4S4K5Up8+NI/bmRkUjrQRSYyYhAT0/2huJ/vU4K3UMv43GSGvN0fKibCgwSD8PCIVeMghgYQqji5ldMe0QRCibSignBmbT8l7T26k6jfnTdqJ2cFnGU0QbaRDvIQfoBF2gK9REFD2iZ/SK3qwn68V6tz7GoyWr2FlHv2B9fQPfA6Gh</latexit>

Bring non-content into the picture

14

T = 1 log(1 + SNR) · K(1 − µ) 1 + Kµ

<latexit sha1_base64="59WGpnRXH4z7ZVIbwpRPV7Uk3cQ=">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</latexit>

Achievable delivery times (w. ch. coding):

<latexit sha1_base64="5p1bf6F+Eel3Zwi/KN14TXheUYs=">ACGnicbVDJSgNBEO2JW4xb1KOXxiDoJcyI4HKevEYwSyQhNDTU0ma9PQM3TVqGPIdXvwVLx4U8SZe/Bs7y0GNDwoer+p1dT0/lsKg6345mbn5hcWl7HJuZXVtfSO/uVU1UaI5VHgkI13mQEpFRQoIR6rIGFvoSa378c9Wu3oI2I1A0OYmiFrKtER3CGVmrnvSbCPabnvCfglkXDexb1jGgKEIwdP+uSHnPVhQI1T04G7bzBbfojkFniTclBTJFuZ3/aAYRT0JQyCUzpuG5MbZSplFwCcNcMzEQM95nXWhYqphd20rHpw3pnlUC2om0LYV0rP50pCw0ZhD6djJk2DN/eyPxv14jwc5JKxUqThAUnyzqJiREc50UBo4CgHljCuhf2rjYFpxtGmbMheH9PniXVw6J3VDy9PiqULqZxZMkO2SX7xCPHpESuSJlUCcP5Im8kFfn0Xl23pz3yWjGmXq2yS84n9MfKBh</latexit>

Caching PHY

slide-15
SLIDE 15

Asymmetric Setting

𝑉! 𝐺

!

𝐺" ⋮ 𝐺# 𝑉$

⋮ ⋮

𝑉% 𝑋

!

𝑋

"

⋮ 𝑋

$

SNR! SNR" SNR#

SNR1 ≤ SNR2 ≤ · · · ≤ SNRK

<latexit sha1_base64="WxvPB63uOtYHX+40oBSx9PnDM=">ACNHicbVDLSsNAFJ34rPUVdelmsAiuSlIK6q7oRihIfQBTQiTyaQdOnk4MxFKyEe58UPciOBCEbd+g0kawT4ODBzOZe59gho0Jq2puytLyurZe2ihvbm3v7Kp7+x0RByTNg5YwHs2EoRn7QlYz0Qk6QZzPStUeXmd9JFzQwL+X45CYHhr41KUYyVSy1KbhITkUbnx3fZtYsZ5Ag5EHOK3W/lTsBFIsSjQTCMuWtGqWg4T/SCVECBlqW+GE6AI4/4EjMkRF/XQmnGiEuKGUnKRiRIiPAIDUg/pT7yiDj/OgEHqeKA92Ap8+XMFf/T8TIE2Ls2WkyX3XWy8RFXj+S7pkZUz+MJPHx5CM3YlAGMGsQOpQTLNk4JQhzmu4K8RBxhGXac1aCPnvyPOnUqnq9en5TrzQuijpK4BAcgROg1PQAFegBdoAgyfwCj7Ap/KsvCtfyvckuqQUMwdgCsrPLxyfqxM=</latexit>

15

R12 + R1 ≤ log ✓ 1 + qSNR1 1 + (1 − q)SNR1 ◆ R2 ≤ log

  • 1 + (1 − q)SNR2
  • <latexit sha1_base64="MIiWDtGjYXjyAhoFdNTCBbq4pqU=">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</latexit>

where 0 ≤ q ≤ 1

<latexit sha1_base64="H4QUe8jqiRJewO4umdvUhDu6MpM=">ACBnicbVDJSgNBEO2JW4zbqEcRGoPgKcxIQL0FvXiMYBbIhNDTqSRNepZ016hyMmLv+LFgyJe/QZv/o2d5aCJD4p6vFdFdz0/lkKj43xbmaXldW17HpuY3Nre8fe3avqKFEcKjySkar7TIMUIVRQoIR6rIAFvoSa378a+7U7UFpE4S0OY2gGrBuKjuAMjdSyDz2EB0zve6CAjqhDPQkDOpg2t2XnYIzAV0k7ozkyQzlv3ltSOeBAil0zrhuvE2EyZQsEljHJeoiFmvM+60DA0ZAHoZjo5Y0SPjdKmnUiZCpFO1N8bKQu0Hga+mQwY9vS8Nxb/8xoJds6bqQjBCHk04c6iaQY0XEmtC0UcJRDQxhXwvyV8h5TjKNJLmdCcOdPXiTV04JbLFzcFPOly1kcWXJAjsgJckZKZFrUiYVwskjeSav5M16sl6sd+tjOpqxZjv75A+szx9lJfb</latexit>

superposition coding + successive decoding

E.g. 2-User PHY capacity region

(σ = 2)

<latexit sha1_base64="4ZfVYI7CDHwfC+t8Bvt7HiwxuxU=">AB83icbVBNSwMxEJ2tX7V+VT16CRahXspuKagHoejFYwVrC92lZNsG5pklyQrlKV/w4sHRbz6Z7z5b0zbPWjrg4HezPMzAsTzrRx3W+nsLa+sblV3C7t7O7tH5QPjx51nCpC2yTmseqGWFPOJG0bZjtJopiEXLaCce3M7/zRJVmsXwk4QGAg8lixjBxkp+1dsKDC6RvXzfrni1tw50CrxclKBHK1+csfxCQVBrCsdY9z01MkGFlGOF0WvJTRNMxnhIe5ZKLKgOsvnNU3RmlQGKYmVLGjRXf09kWGg9EaHtFNiM9LI3E/zeqmJLoOMySQ1VJLFoijlyMRoFgAaMEWJ4RNLMFHM3orICtMjI2pZEPwl9eJY/1mteoXd03Ks2bPI4inMApVMGDC2jCHbSgDQSeIZXeHNS58V5dz4WrQUnzmGP3A+fwARoJBr</latexit>
slide-16
SLIDE 16

Asymmetric Setting: GDoF Limit

SNR1 ≤ SNR2 ≤ · · · ≤ SNRK

<latexit sha1_base64="WxvPB63uOtYHX+40oBSx9PnDM=">ACNHicbVDLSsNAFJ34rPUVdelmsAiuSlIK6q7oRihIfQBTQiTyaQdOnk4MxFKyEe58UPciOBCEbd+g0kawT4ODBzOZe59gho0Jq2puytLyurZe2ihvbm3v7Kp7+x0RByTNg5YwHs2EoRn7QlYz0Qk6QZzPStUeXmd9JFzQwL+X45CYHhr41KUYyVSy1KbhITkUbnx3fZtYsZ5Ag5EHOK3W/lTsBFIsSjQTCMuWtGqWg4T/SCVECBlqW+GE6AI4/4EjMkRF/XQmnGiEuKGUnKRiRIiPAIDUg/pT7yiDj/OgEHqeKA92Ap8+XMFf/T8TIE2Ls2WkyX3XWy8RFXj+S7pkZUz+MJPHx5CM3YlAGMGsQOpQTLNk4JQhzmu4K8RBxhGXac1aCPnvyPOnUqnq9en5TrzQuijpK4BAcgROg1PQAFegBdoAgyfwCj7Ap/KsvCtfyvckuqQUMwdgCsrPLxyfqxM=</latexit>

16

R12 + R1 ≤ log ✓ 1 + qSNR1 1 + (1 − q)SNR1 ◆ R2 ≤ log

  • 1 + (1 − q)SNR2
  • <latexit sha1_base64="MIiWDtGjYXjyAhoFdNTCBbq4pqU=">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</latexit>

where 0 ≤ q ≤ 1

<latexit sha1_base64="H4QUe8jqiRJewO4umdvUhDu6MpM=">ACBnicbVDJSgNBEO2JW4zbqEcRGoPgKcxIQL0FvXiMYBbIhNDTqSRNepZ016hyMmLv+LFgyJe/QZv/o2d5aCJD4p6vFdFdz0/lkKj43xbmaXldW17HpuY3Nre8fe3avqKFEcKjySkar7TIMUIVRQoIR6rIAFvoSa378a+7U7UFpE4S0OY2gGrBuKjuAMjdSyDz2EB0zve6CAjqhDPQkDOpg2t2XnYIzAV0k7ozkyQzlv3ltSOeBAil0zrhuvE2EyZQsEljHJeoiFmvM+60DA0ZAHoZjo5Y0SPjdKmnUiZCpFO1N8bKQu0Hga+mQwY9vS8Nxb/8xoJds6bqQjBCHk04c6iaQY0XEmtC0UcJRDQxhXwvyV8h5TjKNJLmdCcOdPXiTV04JbLFzcFPOly1kcWXJAjsgJckZKZFrUiYVwskjeSav5M16sl6sd+tjOpqxZjv75A+szx9lJfb</latexit>

superposition coding + successive decoding

E.g. 2-User PHY capacity region

(σ = 2)

<latexit sha1_base64="4ZfVYI7CDHwfC+t8Bvt7HiwxuxU=">AB83icbVBNSwMxEJ2tX7V+VT16CRahXspuKagHoejFYwVrC92lZNsG5pklyQrlKV/w4sHRbz6Z7z5b0zbPWjrg4HezPMzAsTzrRx3W+nsLa+sblV3C7t7O7tH5QPjx51nCpC2yTmseqGWFPOJG0bZjtJopiEXLaCce3M7/zRJVmsXwk4QGAg8lixjBxkp+1dsKDC6RvXzfrni1tw50CrxclKBHK1+csfxCQVBrCsdY9z01MkGFlGOF0WvJTRNMxnhIe5ZKLKgOsvnNU3RmlQGKYmVLGjRXf09kWGg9EaHtFNiM9LI3E/zeqmJLoOMySQ1VJLFoijlyMRoFgAaMEWJ4RNLMFHM3orICtMjI2pZEPwl9eJY/1mteoXd03Ks2bPI4inMApVMGDC2jCHbSgDQSeIZXeHNS58V5dz4WrQUnzmGP3A+fwARoJBr</latexit>
  • Approx. PHY capacity region (GDoF)

Allows eliminating power control variables Linear inequalities – Nice and simple

r12 + r1 ≤ α1 r2 ≤ α2

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GDoF: Etkin et al. “Gaussian interference channel capacity to within one bit” IEEE T-IT’08

R2

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R12 + R1

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in

  • ut

0 < α1 ≤ α2 ≤ · · · ≤ αK

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slide-17
SLIDE 17

Way Forward

𝑉! 𝐺

!

𝐺" ⋮ 𝐺# 𝑉$

⋮ ⋮

𝑉% 𝑋

!

𝑋

"

⋮ 𝑋

$

SNR! SNR" SNR#

GDoF region for unicast and multiple-multicast (group size 𝜏)

τ ub(r)

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Achievable delay-rate trade-off (GDoF limit)

17

PHY

Unicast + Multiple-Multicast

superposition coding + successive decoding

Higher Layer

  • Wi : i ∈ [K]
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  • WS : S ⊆ [K], |S| = σ
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Unicast Multiple-Multicast Separation

slide-18
SLIDE 18

PHY GDoF Region

Example: 3 users, σ = 2 Unicast: {W1, W2, W3} Multicast: {W12, W13, W23}

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Scheme based on superposition coding, power control, and successive decoding

18

slide-19
SLIDE 19

PHY GDoF Region

Example: 3 users, σ = 2 Unicast: {W1, W2, W3} Multicast: {W12, W13, W23}

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Scheme based on superposition coding, power control, and successive decoding

PHY GDoF region

Fourier-Motzkin elimination of power control variables

r1 + r12 + r13 ≤ α1 r1 + r2 + r12 + r13 + r23 ≤ α2 r1 + r2 + r3 + r12 + r13 + r23 ≤ α3

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19

slide-20
SLIDE 20

PHY GDoF Region

Example: 3 users, σ = 2 Unicast: {W1, W2, W3} Multicast: {W12, W13, W23}

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Scheme based on superposition coding, power control, and successive decoding

PHY GDoF region

Fourier-Motzkin elimination of power control variables

r1 + r12 + r13 ≤ α1 r1 + r2 + r12 + r13 + r23 ≤ α2 r1 + r2 + r3 + r12 + r13 + r23 ≤ α3

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20

r1 + 2 · rsym ≤ α1 r1 + r2 + 3 · rsym ≤ α2 r1 + r2 + r3 + 3 · rsym ≤ α3

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Symmetric multicast

r12 = r13 = r23 = rsym

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slide-21
SLIDE 21

Achievable Delivery Time

Example: 3 users, σ = 2 Unicast: {W1, W2, W3} Multicast: {W12, W13, W23}

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Scheme based on superposition coding, power control, and successive decoding

21

r1 + 2 · rsym ≤ α1 r1 + r2 + 3 · rsym ≤ α2 r1 + r2 + r3 + 3 · rsym ≤ α3

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Symmetric multicast

r12 = r13 = r23 = rsym

<latexit sha1_base64="4x9xYdPLazaulfdu9Gxu0bMjOo=">ACNnicbZDNSsNAEMc39avGr6hHL4tF8CAlqQX1IBS9eKxgP6ANYbPdtEt3k7C7EUrIS/g0vepbePEmXgVfwE2bg20dWPjNf2aY2b8fMyqVb8bpbX1jc2t8ra5s7u3f2AdHrVlAhMWjhikej6SBJGQ9JSVDHSjQVB3Gek4/v83rnmQhJo/BJTWLicjQMaUAxUlryrAvhpU4tg7cwh8sCagX0OVIjwVM54Zlpmp5Vsav2LOAqOAVUQBFNz/rpDyKcBIqzJCUPceOlZsioShmJDP7iSQxwmM0JD2NIeJEunsVxk808oABpHQL1Rwpv6dSBGX+jBfd+ZnyuVaLv5X6yUquHZTGsaJIiGeLwoSBlUEc4vgAqCFZtoQFhQfSvEIyQVtrIhS0+zxbzXNAuOcuerEK7VnXq1ZvHeqVxV/hVBifgFJwDB1yBngATdACGLyAKXgFb8bU+DA+ja95a8koZo7BQhjfv9DAqkY=</latexit>

rsym = min ⇢α1 − r1 2 , α2 − (r1 + r2) 3 , α3 − (r1 + r2 + r3) 3

  • <latexit sha1_base64="ydUOs7/K1kF78/zesj7sLascYA=">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</latexit>

multicast message (subpacket) size

τ lb(r) = 1 rsym · 1 3

<latexit sha1_base64="FeqzpkLoGHNrRhyDYrvaC2hDmA=">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</latexit>
slide-22
SLIDE 22

Achievable Delivery Time

Example: 3 users, σ = 2 Unicast: {W1, W2, W3} Multicast: {W12, W13, W23}

<latexit sha1_base64="eqa7GjOpXjlrqDwVwO49BJAIE2k=">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</latexit>

Scheme based on superposition coding, power control, and successive decoding

22

r1 + 2 · rsym ≤ α1 r1 + r2 + 3 · rsym ≤ α2 r1 + r2 + r3 + 3 · rsym ≤ α3

<latexit sha1_base64="lQTQX6B+18k7Os4saJCUzec1K9Q=">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</latexit>

Symmetric multicast

r12 = r13 = r23 = rsym

<latexit sha1_base64="4x9xYdPLazaulfdu9Gxu0bMjOo=">ACNnicbZDNSsNAEMc39avGr6hHL4tF8CAlqQX1IBS9eKxgP6ANYbPdtEt3k7C7EUrIS/g0vepbePEmXgVfwE2bg20dWPjNf2aY2b8fMyqVb8bpbX1jc2t8ra5s7u3f2AdHrVlAhMWjhikej6SBJGQ9JSVDHSjQVB3Gek4/v83rnmQhJo/BJTWLicjQMaUAxUlryrAvhpU4tg7cwh8sCagX0OVIjwVM54Zlpmp5Vsav2LOAqOAVUQBFNz/rpDyKcBIqzJCUPceOlZsioShmJDP7iSQxwmM0JD2NIeJEunsVxk808oABpHQL1Rwpv6dSBGX+jBfd+ZnyuVaLv5X6yUquHZTGsaJIiGeLwoSBlUEc4vgAqCFZtoQFhQfSvEIyQVtrIhS0+zxbzXNAuOcuerEK7VnXq1ZvHeqVxV/hVBifgFJwDB1yBngATdACGLyAKXgFb8bU+DA+ja95a8koZo7BQhjfv9DAqkY=</latexit>

rsym = min ⇢α1 − r1 2 , α2 − (r1 + r2) 3 , α3 − (r1 + r2 + r3) 3

  • <latexit sha1_base64="ydUOs7/K1kF78/zesj7sLascYA=">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</latexit>

Topological holes Free non-content rates multicast message (subpacket) size

τ lb(r) = 1 rsym · 1 3

<latexit sha1_base64="FeqzpkLoGHNrRhyDYrvaC2hDmA=">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</latexit>

max rsym, min τ lb(r)

<latexit sha1_base64="b3WmuhW5uecyXA/RHy34Vr7D1Sk=">ACU3icbVBNSwMxE3Xr/pRrXr0EixCBSm7UlBvohe9KVgVurVk02wNJtklmRXLkt/kr+nBi/4Ij17MtkWsdWDgzXszMyLUsEN+P5byZubX1hcKi+vrK5V1jeqm1u3Jsk0ZS2aiETfR8QwRVrAQfB7lPNiIwEu4uezgv97plpwxN1A4OUdSTpKx5zSsBR3eplCOwFcklesMW6m4eSwKOWuRlIaw/wWMWSKyeHQLKHnw4RWVsfFVGca7vfrdb8hj8KPAuCaihSVx1q59hL6GZAqoIMa0Az+FTk40cCqYXQkzw1JCn0iftR1URDLTyUcvW7znmB6OE+1SAR6xvydyIo37IXKdxYnmr1aQ/2ntDOLjTs5VmgFTdLwozgSGBf+4R7XjIYOECo5u5WTB+JhScy1NbImn64JwLgV/PZkFt4eNoNk4uW7WTs8mfpXRDtpFdRSgI3SKLtAVaiGKXtEQvaOP0rD05Xne/LjVK01mtFUeJVv4HG3Hw=</latexit>
slide-23
SLIDE 23

GDoF region with unicast and symmetric multiple-multicast

# of multicast messages decoded by at least one user in 1, ⋯ , 𝑘

X

i∈[k]

ri + ✓K σ ◆ − ✓K − k σ ◆ · rsym ≤ αk, ∀k ∈ [K]

<latexit sha1_base64="mqFTGtiCG7lCo3z+NCHnF26iNSM=">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</latexit>

23

where r1 + r2 + · · · + rk ≤ αk, ∀k ∈ [K]

<latexit sha1_base64="OpBbZXajFfoy41r5Dec/xkZjA=">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</latexit>

τ lb(r; µ, α) = max

k∈[K]

(

K Kµ+1

K−k

Kµ+1

  • αk − P

i∈[k] ri

  • )

· 1 K

  • <latexit sha1_base64="PdnLm641UetCE60WoGKvuqtUr6c=">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</latexit>

PHY

Achievable Delivery Time: General Case

multicast message (subpacket) size

slide-24
SLIDE 24
  • Theorem. Optimal trade-off τ ?(r; µ, α) satisfies:

1 2.01 · τ ub(r; µ, α) ≤ τ ?(r; µ, α) ≤ τ ub(r; µ, α)

<latexit sha1_base64="2YQBm62Dn4L9v4HQdGLBLb92G+g=">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</latexit>

Main Result

where r1 + r2 + · · · + rk ≤ αk, ∀k ∈ [K]

<latexit sha1_base64="OpBbZXajFfoy41r5Dec/xkZjA=">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</latexit>

Lampiris et al. "Fundamental Limits of Wireless Caching under Uneven-Capacity Channels" arXiv:1908.04036 (2019).

Recovers a previous result (with content only) in:

24

where r1 + r2 + · · · + rk ≤ αk, ∀k ∈ [K]

<latexit sha1_base64="OpBbZXajFfoy41r5Dec/xkZjA=">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</latexit>

τ lb(r; µ, α) = max

k∈[K]

(

K Kµ+1

K−k

Kµ+1

  • αk − P

i∈[k] ri

  • )

· 1 K

  • <latexit sha1_base64="PdnLm641UetCE60WoGKvuqtUr6c=">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</latexit>
slide-25
SLIDE 25
  • Theorem. Optimal trade-off τ ?(r; µ, α) satisfies:

1 2.01 · τ ub(r; µ, α) ≤ τ ?(r; µ, α) ≤ τ ub(r; µ, α)

<latexit sha1_base64="2YQBm62Dn4L9v4HQdGLBLb92G+g=">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</latexit>

Main Result

where r1 + r2 + · · · + rk ≤ αk, ∀k ∈ [K]

<latexit sha1_base64="OpBbZXajFfoy41r5Dec/xkZjA=">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</latexit>

Achievability Converse

Presented next

25

where r1 + r2 + · · · + rk ≤ αk, ∀k ∈ [K]

<latexit sha1_base64="OpBbZXajFfoy41r5Dec/xkZjA=">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</latexit>

τ lb(r; µ, α) = max

k∈[K]

(

K Kµ+1

K−k

Kµ+1

  • αk − P

i∈[k] ri

  • )

· 1 K

  • <latexit sha1_base64="PdnLm641UetCE60WoGKvuqtUr6c=">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</latexit>
slide-26
SLIDE 26

Converse

Enhance user k through Sk, independent of Wk

<latexit sha1_base64="KDN/MGRlcSwVBgwvcbx83FPpg=">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</latexit>

side inf.

TRk + TRF = H

  • Wk, Fdk|Uk, Sk
  • ≤ I
  • Wk, Fdk; Y T

k |Uk, Sk

  • + T✏T
<latexit sha1_base64="WbEsO8xqcgSN4AklhS0C3y+jmR8=">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</latexit>

Fano

X

k∈[s]

H

  • Fdk|Uk, Sk

X

k∈[s]

h I

  • Wk, Fdk; Y T

k |Uk, Sk

  • − T(Rk + ✏T )

i

<latexit sha1_base64="9G4pWVi/WiGajPTmHO1cruPsI=">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</latexit>

set Sk =

  • Wi, Fi, Ui : i ∈ [k − 1]
  • <latexit sha1_base64="6AtfOcQMfEqiDj5XNsyPGIe/1Cc=">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</latexit>

26

slide-27
SLIDE 27

Converse

Enhance user k through Sk, independent of Wk

<latexit sha1_base64="KDN/MGRlcSwVBgwvcbx83FPpg=">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</latexit>

TRk + TRF = H

  • Wk, Fdk|Uk, Sk
  • ≤ I
  • Wk, Fdk; Y T

k |Uk, Sk

  • + T✏T
<latexit sha1_base64="WbEsO8xqcgSN4AklhS0C3y+jmR8=">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</latexit>

X

k∈[s]

H

  • Fdk|Uk, Sk

X

k∈[s]

h I

  • Wk, Fdk; Y T

k |Uk, Sk

  • − T(Rk + ✏T )

i

<latexit sha1_base64="9G4pWVi/WiGajPTmHO1cruPsI=">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</latexit>

content PHY

set Sk =

  • Wi, Fi, Ui : i ∈ [k − 1]
  • <latexit sha1_base64="6AtfOcQMfEqiDj5XNsyPGIe/1Cc=">ACTHicbVBdSxtBFJ1N1dr40dg+9uXSICho2BWhtVAQC8VHS40RsyO7mJQ2Znl5m7pWHZ/+Ov8dX2d7RvRejsmgejHhjumXP5c6cJFfSku/9lovlpZXq6+aq+tb2y+7my9ubBZYQT2RaYyc5lwi0pq7JMkhZe5QZ4mCgfJ9EvdH/xAY2Wmz2mWY5TyiZjKTg5Ke6chIQ/qbRIUMH3uJxW8BkgTORkZxCXstqDr3WBvX5TPoGEUGoYTveDqLHtuNO1+/5DeApCeaky+Y4izt/wlEmihQ1CcWtHQZ+TlHJDUmhsGqHhcWciymf4NBRzVO0Udn8tYJtp4xgnBl3NEGjPpwoeWrtLE2cM+V0ZR/3avG53rCg8ceolDovCLW4XzQuFAGdXAwkgYFqZkjXBjp3grihsuyMW7sCVJq8V7LbiUgseZPCUXB73gsHf07bB7fDLPa5W9Y+/ZDgvYB3bMTtkZ6zPBrtkNu2W/vBvr/fPu7u3trz5zFu2gNbKf/m7sY8=</latexit>

Separation!

27

slide-28
SLIDE 28

Converse

Enhance user k through Sk, independent of Wk

<latexit sha1_base64="KDN/MGRlcSwVBgwvcbx83FPpg=">ACR3icbVBNSxtBGJ5N/ar1I7bHXgazgcJuxJoPQiFDxaoyQhGV28m52NmZeZdMSz7a/w1Xu2pv8KDID12EnMw2hdmeOZ5v54kIKi0Hwx2t8WFpeWV37uP5pY3Nru7nz+crq0nDoci21uY6ZBSkUdFGghOvCAMtjCb04O5vmezdgrNDqEicFDHM2ViIRnKGjoubxAOEWqx8qZYoDLS0Y6mc+xdTocpxS/1dUZbV/QIUaQHuUkh1Qv1elPl1GwF7WAW9D0I56BF5nERNZ8GI83L3E3hklnbD4MChxUzKLiEen3gBSMZ2wMfQcVy8EOq9k3a7rnmBFNtHqZixrzsqls7yWNXmTNM7dvclPxfrl9i8n1YCVWUCIq/LEpKSVHTqWd0JAxwlBMHGDfCaU8ZYZxdM4ubInzevE9JZxL4VtP3oOrw3bYaR/97LROTud+rZGvZJfsk5B8IyfknFyQLuHkjtyTB/Lbu/cevWfv70tpw5v3fCEL0fD+Ab3MsUE=</latexit>

TRk + TRF = H

  • Wk, Fdk|Uk, Sk
  • ≤ I
  • Wk, Fdk; Y T

k |Uk, Sk

  • + T✏T
<latexit sha1_base64="WbEsO8xqcgSN4AklhS0C3y+jmR8=">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</latexit>

X

k∈[s]

H

  • Fdk|Uk, Sk

X

k∈[s]

h I

  • Wk, Fdk; Y T

k |Uk, Sk

  • − T(Rk + ✏T )

i

<latexit sha1_base64="9G4pWVi/WiGajPTmHO1cruPsI=">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</latexit>

content PHY

set Sk =

  • Wi, Fi, Ui : i ∈ [k − 1]
  • <latexit sha1_base64="6AtfOcQMfEqiDj5XNsyPGIe/1Cc=">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</latexit>

Separation!

28

slide-29
SLIDE 29

Converse: PHY

  • Lemma. I
  • Wk, Fk; Y T

k |Uk, Sk

  • ≤ I
  • Wk, Fk; Y T

s |Uk, Sk

  • , ∀k ≤ s
<latexit sha1_base64="xqsC2fdbp63HKmE18XhWhZ83QA=">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</latexit>

X

k∈[s]

I

  • Wk, Fk; Y T

k |Uk, Sk

X

k∈[s]

I

  • Wk, Fk; Y T

s |Uk, Sk

X

k∈[s]

h

  • Y T

s |Uk, Sk

  • − h
  • Y T

s |Uk+1, Sk+1

  • = h
  • Y T

s |U1

  • − h
  • ZT

s

  • = I
  • XT ; Y T

s |U1

  • ≤ T log(1 + P αs)
<latexit sha1_base64="fAY6sj7bHbDOg6nj5vPte0GSQo=">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</latexit>

= ⇒

<latexit sha1_base64="kyOFkeghUAB9Z2gTGK+ayHt/5bU=">ACEHicbVDLSgNBEOyNrxhfUY9eFoPgKeyKoN6CXjxGMA9IljA76SRDZmaXmVkhLPmJXPVDvIlX/8Dv8AecTfZgEgsGqu7qZ4KY8608bxvp7CxubW9U9wt7e0fHB6Vj0+aOkoUxQaNeKTaIdHImcSGYZjO1ZIRMixFY4fsn7rBZVmkXw2kxgDQYaSDRglxkrtLhPWBXWvXPGq3hzuOvFzUoEc9V75p9uPaCJQGsqJ1h3fi02QEmUY5TgtdRONMaFjMsSOpZI1E6v3fqXlil7w4iZ807lz9u5ESofVEhHZSEDPSq71M/K/XSczgNkiZjBODki6MBgl3TeRmn3f7TCE1fGIJoYrZW106IopQYyNacgnFdLnOBJuSv5rJOmleVf3r6t3TdaV2n+dVhDM4h0vw4QZq8Ah1aAFDjN4hTdn5rw7H87nYrTg5DunsATn6xcjR53Z</latexit>

strongest user capacity

set Sk =

  • Wi, Fi, Ui : i ∈ [k − 1]
  • <latexit sha1_base64="6AtfOcQMfEqiDj5XNsyPGIe/1Cc=">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</latexit>

29

slide-30
SLIDE 30

Enhance user k through Sk, independent of Wk

<latexit sha1_base64="KDN/MGRlcSwVBgwvcbx83FPpg=">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</latexit>

TRk + TRF = H

  • Wk, Fdk|Uk, Sk
  • ≤ I
  • Wk, Fdk; Y T

k |Uk, Sk

  • + T✏T
<latexit sha1_base64="WbEsO8xqcgSN4AklhS0C3y+jmR8=">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</latexit>

X

k∈[s]

H

  • Fdk|Uk, Sk

X

k∈[s]

h I

  • Wk, Fdk; Y T

k |Uk, Sk

  • − T(Rk + ✏T )

i

<latexit sha1_base64="9G4pWVi/WiGajPTmHO1cruPsI=">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</latexit>

content

Converse: Content

set Sk =

  • Wi, Fi, Ui : i ∈ [k − 1]
  • <latexit sha1_base64="6AtfOcQMfEqiDj5XNsyPGIe/1Cc=">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</latexit>

≤ T log

  • 1 + P αs

− T X

k∈[s]

(Rk + ✏T )

<latexit sha1_base64="9vztQOu+PNANF8k6mhMJSiom8R8=">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</latexit>

30

slide-31
SLIDE 31

Enhance user k through Sk, independent of Wk

<latexit sha1_base64="KDN/MGRlcSwVBgwvcbx83FPpg=">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</latexit>

TRk + TRF = H

  • Wk, Fdk|Uk, Sk
  • ≤ I
  • Wk, Fdk; Y T

k |Uk, Sk

  • + T✏T
<latexit sha1_base64="WbEsO8xqcgSN4AklhS0C3y+jmR8=">ACqXicdVHbTtAEF275VK3hVAe+zJqBAIFRTZKVRBCQiAh+gYlIaFxsNabSVhlfWF3jRQZfwvf1e/oD3Rt8tBwGWk1Z86c2RnNhKngSrvuH8t+935hcWn5g/Px0+eV1dralyuVZJhyUikb2QKhQ8xo7mWmAvlUijUGA3nJyU+e49SsWTuK2nKQ4iOo75iDOqDRXUHtu/gnxSQAPaYNBpAZtwCGd+yMdb0C1TO3Aa5MPAaB6gU2l34LyUKq2wXjf2QRf4B3AT3i1FA4ArkvqJm+b6I2fGmCm8DFVXJjZjNJxglrdbqVwUvgzUCdzOw8qP31hwnLIow1E1SpvuemepBTqTkTWDh+pjClbELH2DcwphGqQV4tsoANwxhlEjzYg0V+39FTiOlplFolBHVt+p5riRfy/UzPdob5DxOM40xe2o0ygToBMqrwJBLZFpMDaBMcjMrsFsqKdPmdnNdwqiYj0vCbMl7vpOX4Gq36bWa+xet+tHxbF/L5Cv5RraIR36QI3JGzkmHMGvBalgt67vdsC/snv37SWpbs5p1Mmc2+wdXe8jw</latexit>

X

k∈[s]

H

  • Fdk|Uk, Sk

X

k∈[s]

h I

  • Wk, Fdk; Y T

k |Uk, Sk

  • − T(Rk + ✏T )

i

<latexit sha1_base64="9G4pWVi/WiGajPTmHO1cruPsI=">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</latexit>

content

Converse: Content

1 s!N! X

(p,q)

X

k∈[s]

H

  • Fq(k)|Up(k), Sp(k)
  • ≥ TRF

2.01 ·

  • K

Kµ+1

K−s

Kµ+1

  • K

  • <latexit sha1_base64="UWJthpLKIscluWs1LahoAtIwxRY=">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</latexit>

Yu et al. “The exact rate-memory tradeoff for caching with uncoded prefetching" IEEE T-IT’18 Yu et al. “Characterizing the rate-memory tradeoff in cache networks within a factor of 2” IEEE T-IT’19

Yu et al.

symmetrization

= ⇒ Sp(k) =

  • Wp(i), Fq(i), Up(i) : i ∈ [k − 1]
  • <latexit sha1_base64="1l96cqpBRkpMjel3jpKlBgrPWg0=">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</latexit>

set Sk =

  • Wi, Fi, Ui : i ∈ [k − 1]
  • <latexit sha1_base64="6AtfOcQMfEqiDj5XNsyPGIe/1Cc=">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</latexit>

≤ T log

  • 1 + P αs

− T X

k∈[s]

(Rk + ✏T )

<latexit sha1_base64="9vztQOu+PNANF8k6mhMJSiom8R8=">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</latexit>

31

slide-32
SLIDE 32

Converse: Combining Bounds

TRF 2.01 ·

  • K

Kµ+1

K−s

Kµ+1

  • K

  • ≤ T log
  • 1 + P αs

− T X

k∈[s]

(Rk + ✏T )

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1 2.01T ·

  • K

Kµ+1

K−s

Kµ+1

  • K

  • ≤ log
  • 1 + P αs

− X

k∈[s]

Rk

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2.01 · τ ≥ 1

  • αs − P

k∈[s] rk

·

  • K

Kµ+1

K−s

Kµ+1

  • K

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2.01 · τ ≥ 1

  • α1 − r1

·

  • K

Kµ+1

K−1

Kµ+1

  • K

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2.01 · τ ≥ 1

  • αK − P

k∈[K] rk

·

  • K

Kµ+1

  • K

  • <latexit sha1_base64="+vcL8zviNAFHwP+7wjeUchZV8A4=">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</latexit>

. . .

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. . .

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Shannon limit GDoF limit

Desired bounds !

Repeating for all s ∈ [K]

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32

slide-33
SLIDE 33

Summary

  • Mixed traffic model – cacheable and uncacheable
  • Degraded Gaussian Broadcast Channel (single-cell)
  • It is easier to approximate ! (GDoF limit)
  • Achievable trade-off. Optimal within a factor of 2.01
  • Topological holes – non-zero rates with minimum delay

Full version available on arXiv: Hamdi Joudeh, E. Lampiris, P. Elia, & G. Caire “Fundamental Limits of Wireless Caching Under Mixed Cacheable and Uncacheable Traffic” arXiv:2002.07691

Thank You

33

Supported by the European Research Council (ERC) under the ERC Advanced Grant CARENET (Caire), and the ERC Consolidator Grant DUALITY (Elia)