Interactive Channel Capacity

1. Interactive Channel Capacity Ran Raz Weizmann Institute Joint work with Gillat Kol Technion

2. [Shannon 48]: • A Mathematical Theory of Communication • An exact formula for the channel • capacity of any noisy channel

3. -noisy channel: • Each bit is flipped with prob • Alice wants to send bits to Bob. They • only have access to an -noisy channel. • How many bits Alice needs to send, so • that Bob can retrieve the original bits, • with prob ? 1- 0 0 1 1 1-

4. Channel Capacity [Shannon 48]: • 1) are sufficient • (using error correcting codes) • 2)are needed • channel capacity:

5. Communication Complexity [Yao 79]: • Player gets . Player gets • They need to compute • ( is publicly known) • How many bits they need to communicate? • deterministic CC of • (for worst case ) • probabilistic CC of • (with negligible error for every ) • (with shared random string)

6. CC over the -noisy channel: • Assume: • How many communication bits are needed • to compute over the -noisy channel? • deterministic CC of • CC of over -noisy channel • (with negligible error for every ) • (with shared random string)

7. Interactive Channel Capacity: • deterministic CC of • CC of over -noisy channel • (note: is not the input size)

8. Interactive Channel Capacity: • deterministic CC of • CC of over -noisy channel • Can use instead of • All the results hold for both

9. Interactive Channel Capacity: • deterministic CC of • CC of over -noisy channel • Assumption:Order of communication in • all protocols is pre-determined • (for simplicity) • Justification: Otherwise both players • may try to send bits at the same time

10. Types of Channels: • 1)Synchronous: At each time step • exactly one player sends a bit • 2)Alternating: The players alternate • in sending bits • 3)Asynchronous: If both send bits at • the same time these bits are lost • 4)Two channels: Each player sends a • bit whenever she wants

11. Previous Work: • [Schulman 92]: • Hence, • [Sch,BR,B,GMS,BK,BN]: • Simulation of any CC protocol in the • presence of adversarial noise • [Shannon 48]: • [Schulman 92]: Is ?

12. Our Results: • Upper Bound: • In particular, for small enough , • (with strict inequality) • Lower Bound: • in the case of alternating channel

13. Upper Bound: • We give a function that proves this • We prove a lower bound on

14. deg= • Pointer Jumping Game: • ary tree, depth • , • owns odd layers • owns even layers • Each player gets an edge going out of • every node that she owns • Goal: Find the leaf reached depth=

15. deg= • Pointer Jumping Game: • Our main result: • Hence, depth=

16. deg= • High Level Idea: • starts by sending • the first edge ( bits) • With one • of these bits was flipped • Case I: sends the next edge ( bits) • With these bits are wasted (since had the wrong first edge) • In expectation: wasted bits depth=

17. deg= • High Level Idea: • starts by sending • the first edge ( bits) • With one • of these bits was flipped • Case II: sends additional bits, to • correct the first edge. • Needs to send bits to correct one error depth=

18. deg= • High Level Idea: • starts by sending • the first edge ( bits) • With one • of these bits was flipped • In both cases bits were wasted (in expectation). • was chosen to be to balance the • losses in the two cases depth=

19. Lower Bound: • Given a communication protocol , we • simulate over the -noisy channel

20. The Basic Step: • Fix. • run steps of and observe the transcripts , , resp. • run a Consistency Check.If an • inconsistency was found they start over bits bits consistency check inconsistency

21. , • run steps of and observe the transcripts , , resp. • Consistency Check: choose random functions: . • sends 100 times each • takes majority vote of each and compares to . • sends 100 times each • takes majority vote of eachand compares to . • A player that finds inconsistency starts over

22. , • run steps of and observe the transcripts , • Consistency Check: choose random functions: . • sends 100 times each. takes majority vote of each and compares to . • sends 100 times each. takes majority vote of eachand compares to . • A player that finds inconsistency starts over bits bits consistency check inconsistency

23. Good: No player starts over • Bad: Both players start over • Very Bad: One player starts over bits bits consistency check inconsistency

24. bits consistency check • times • Inductive Protocol: • Consistency check: • Done with random functions, sent • times each () • In the protocol: • random functions, sent times each inconsistency

25. bits consistency check • times • Analysis: • If an error occurred or the players went • out of split, eventually they will fix it, since the consistency check is done with larger and larger parameters. • Thus, the final protocol simulates with • probability close to . • How many communication bits are wasted? inconsistency

26. Analysis of Wastes in the Basic Step: • Length of consistency check: bits • Probability to start over: • Total waste (in expectation): • bits • Fraction of bits wasted: bits bits consistency check inconsistency

27. bits consistency check • times • Wastes in First Inductive Step: • Length of consistency check: • Probability to start over: • Total waste (in expectation): • Fraction of bits wasted: • (negligible compared to the basic step) inconsistency

28. Bound on the Channel Capacity:

29. Thank You!