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On the Memory Requirements of Block Interleaver for Batched Network Codes Hoover H. F. Yin, Ka Hei Ng, Xishi Wang, Qi Cao, and Lucien K. L. Ng The Chinese University of Hong Kong June, 2020 Hoover H. F. Yin (CUHK) Mem. Req. of Block

  1. On the Memory Requirements of Block Interleaver for Batched Network Codes Hoover H. F. Yin, Ka Hei Ng, Xishi Wang, Qi Cao, and Lucien K. L. Ng The Chinese University of Hong Kong June, 2020 Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 1 / 13

  2. Background Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 2 / 13

  3. Background Packet loss Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 2 / 13

  4. Background Packet loss No feedback Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 2 / 13

  5. Batched Network Coding Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 3 / 13

  6. Batched Network Coding Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 3 / 13

  7. Batched Network Coding Burst loss? Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 3 / 13

  8. Batched Network Coding Burst loss? Depth of interleaver = # of batches in a block Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 3 / 13

  9. Block Interleaved Batched Network Coding batches 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 blocks 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 interleaved 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 blocks Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 4 / 13

  10. Block Interleaved Batched Network Coding batches 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 blocks 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 interleaved 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 blocks Reinterleave Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 4 / 13

  11. Block Interleaved Batched Network Coding batches 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 blocks 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 interleaved 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 blocks Reinterleave Temporary storage for the batches Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 4 / 13

  12. Block Interleaved Batched Network Coding batches 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 blocks 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 interleaved 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 blocks Reinterleave Temporary storage for the batches Lower bound on the memory requirement? Tight bound? Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 4 / 13

  13. Batch Buffer change in buffer size + L 1 + L 1 + L 1 + L 1 + L 1 incoming blocks T 1 T 1 T 1 T 1 T 1 · · · outgoing blocks T 2 T 2 · · · − L 2 − L 2 change in buffer size Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 5 / 13

  14. Batch Buffer change in buffer size + L 1 + L 1 + L 1 + L 1 + L 1 incoming blocks T 1 T 1 T 1 T 1 T 1 · · · outgoing blocks T 2 T 2 · · · − L 2 − L 2 change in buffer size Enqueue event : put L 1 batches (after deinterleaving) Dequeue event : fetch L 2 batches (for recoding and reinterleaving) Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 5 / 13

  15. Batch Buffer change in buffer size + L 1 + L 1 + L 1 + L 1 + L 1 incoming blocks T 1 T 1 T 1 T 1 T 1 · · · outgoing blocks T 2 T 2 · · · − L 2 − L 2 change in buffer size Enqueue event : put L 1 batches (after deinterleaving) Dequeue event : fetch L 2 batches (for recoding and reinterleaving) More terminologies: Total delay Delay-bounded enqueuer Valid dequeuing scheme Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 5 / 13

  16. Batch Buffer change in buffer size + L 1 + L 1 + L 1 + L 1 + L 1 incoming blocks T 1 T 1 T 1 T 1 T 1 · · · outgoing blocks T 2 T 2 · · · − L 2 − L 2 change in buffer size Enqueue event : put L 1 batches (after deinterleaving) Dequeue event : fetch L 2 batches (for recoding and reinterleaving) More terminologies: Total delay Delay-bounded enqueuer Valid dequeuing scheme Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 5 / 13

  17. Batch Buffer change in buffer size + L 1 + L 1 + L 1 + L 1 + L 1 incoming blocks T 1 T 1 T 1 T 1 T 1 · · · outgoing blocks T 2 T 2 · · · − L 2 − L 2 change in buffer size Enqueue event : put L 1 batches (after deinterleaving) Dequeue event : fetch L 2 batches (for recoding and reinterleaving) More terminologies: Total delay Delay-bounded enqueuer Valid dequeuing scheme Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 5 / 13

  18. Buffer Sizes Put or fetch first? + L 1 − L 2 Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 6 / 13

  19. Buffer Sizes Put or fetch first? Put first + L 1 − L 2 Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 6 / 13

  20. Buffer Sizes Put or fetch first? Put first Biqueue event + L 1 − L 2 Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 6 / 13

  21. Buffer Sizes Put or fetch first? Put first Biqueue event + L 1 − L 2 Before fetching: prior buffer size After fetching: buffer size Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 6 / 13

  22. Buffer Sizes Put or fetch first? Put first Biqueue event + L 1 − L 2 Before fetching: prior buffer size Memory required After fetching: buffer size Causality Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 6 / 13

  23. Bounded Buffer Size and Finite Total Delay Incoming rate > Outgoing rate ? Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 7 / 13

  24. Bounded Buffer Size and Finite Total Delay Incoming rate > Outgoing rate ? Outgoing rate > Incoming rate ? Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 7 / 13

  25. Bounded Buffer Size and Finite Total Delay Incoming rate > Outgoing rate ? Outgoing rate > Incoming rate ? L 1 / T 1 = L 2 / T 2 Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 7 / 13

  26. Bounded Buffer Size and Finite Total Delay Incoming rate > Outgoing rate ? Outgoing rate > Incoming rate ? L 1 / T 1 = L 2 / T 2 T 1 = L 1 ω and T 2 = L 2 ω Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 7 / 13

  27. Bounded Buffer Size and Finite Total Delay Incoming rate > Outgoing rate ? Outgoing rate > Incoming rate ? L 1 / T 1 = L 2 / T 2 T 1 = L 1 ω and T 2 = L 2 ω d timeslots = total delay d ′ timeslots = total incoming delay Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 7 / 13

  28. Bounded Buffer Size and Finite Total Delay Incoming rate > Outgoing rate ? Outgoing rate > Incoming rate ? L 1 / T 1 = L 2 / T 2 T 1 = L 1 ω and T 2 = L 2 ω d timeslots = total delay d ′ timeslots = total incoming delay Theorem The minimum total delay among all valid dequeuing schemes is L 1 + L 2 − gcd( L 1 , L 2 ) + d ′ timeslots. Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 7 / 13

  29. Periodic Structure: L 1 = 8 , L 2 = 6 in s1 s2 s1 prior buffer size 16 s2 12 10 t 0 52 76 100 124 148 Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 8 / 13

  30. Periodic Structure: L 1 = 8 , L 2 = 6 in s1 s2 s1 prior buffer size 16 s2 12 10 t 0 52 76 100 124 148 Exists scheme with periods Exists scheme achieving min total delay and lowest memory requirement Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 8 / 13

  31. � Periodic Structure: L 1 = 8 , L 2 = 6 in s1 s2 s1 prior buffer size 16 s2 12 10 t 0 52 76 100 124 148 Exists scheme with periods Exists scheme achieving min total delay and lowest memory requirement Same total delay = ⇒ same memory requirement Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 8 / 13

  32. Periodic Structure: L 1 = 8 , L 2 = 6 in s3 s4 s3 prior buffer size 16 s4 12 t 0 52 76 100 124 148 Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 9 / 13

  33. � Periodic Structure: L 1 = 8 , L 2 = 6 in s3 s4 s3 prior buffer size 16 s4 12 t 0 52 76 100 124 148 Exists scheme with no period and no biqueue event in a long run Lowest memory requirement = ⇒ min total delay Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 9 / 13

  34. � � Periodic Structure: L 1 = 8 , L 2 = 6 in s3 s4 s3 prior buffer size 16 s4 12 t 0 52 76 100 124 148 Exists scheme with no period and no biqueue event in a long run Lowest memory requirement = ⇒ min total delay Exists period = ⇒ exists biqueue event in the period Peak is not always unique Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 9 / 13

  35. Periodic Structure: L 1 = 8 , L 2 = 6 in s5 s6 s5 prior buffer size 16 s6 14 12 2 t 0 52 76 100 124 148 Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 10 / 13

  36. � Periodic Structure: L 1 = 8 , L 2 = 6 in s5 s6 s5 prior buffer size 16 s6 14 12 2 t 0 52 76 100 124 148 Same total delay = ⇒ same lower bound on the memory requirement Hoover H. F. Yin (CUHK) Mem. Req. of Block Interleaver for BNC June, 2020 10 / 13

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