Download - Interactive Channel Capacity Ran Raz Weizmann Institute Joint work with Gillat Kol Technion
Interactive Channel Capacity
Ran RazWeizmann Institute
Joint work withGillat KolTechnion
[Shannon 48]:A Mathematical Theory of
Communication
An exact formula for the channel
capacity of any noisy channel
-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-
1-
0
1 1
0
ππ
Channel Capacity [Shannon 48]: 1) are sufficient (using error correcting codes)2) are needed
channel capacity:
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)
CC over the -noisy channel: Assume: How many communication bits are
neededto compute over the -noisy channel?
deterministic CC of CC of over -noisy channel (with negligible error for
every ) (with shared random string)
Interactive Channel Capacity: deterministic CC of CC of over -noisy channel
(note: is not the input size)
Interactive Channel Capacity: deterministic CC of CC of over -noisy channel
Can use instead of All the results hold for both
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 playersmay try to send bits at the same
time
Types of Channels:
1) Synchronous: At each time stepexactly one player sends a bit2) Alternating: The players alternatein sending bits3) Asynchronous: If both send bits atthe same time these bits are lost4) Two channels: Each player sends abit whenever she wants
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 ?
πͺ (πΊ )=π₯π’π¦πββ
π¦π’π§{ π :πͺπͺ ( π )=π }( π
πͺπͺπΊ( π ))
Our Results:
Upper Bound: In particular, for small enough , (with strict inequality)
Lower Bound: in the case of alternating channel
πͺ (πΊ )=π₯π’π¦πββ
π¦π’π§{ π :πͺπͺ ( π )=π }( π
πͺπͺπΊ( π ))
Upper Bound:
We give a function that proves this
We prove a lower bound on
πͺ (πΊ )=π₯π’π¦πββ
π¦π’π§{ π :πͺπͺ ( π )=π }( π
πͺπͺπΊ( π ))
Pointer Jumping Game:
ary tree, depth , owns odd layers owns even layers
Each player gets an edge going out ofevery node that she ownsGoal: Find the leaf reached
deg=
depth=
Pointer Jumping Game:
Our main result:
Hence,
deg=
depth=
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
deg=
depth=
High Level Idea: starts by sending the first edge ( bits)With one of these bits was flipped
Case II: sends additional bits, tocorrect the first edge.
Needs to send bits to correct one error
deg=
depth=
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
deg=
depth=
Lower Bound:
Given a communication protocol , we
simulate over the -noisy channel
πͺ (πΊ )=π₯π’π¦πββ
π¦π’π§{ π :πͺπͺ ( π )=π }( π
πͺπͺπΊ( π ))
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 bitsconsistency checkinconsistenc
y
, 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 each and compares to .A player that finds inconsistency starts over
, 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 each and compares to .A player that finds inconsistency starts over
bits bitsconsistency checkinconsistenc
y
Good: No player starts over Bad: Both players start over Very Bad: One player starts over
bits bitsconsistency checkinconsistenc
y
Inductive Protocol:
Consistency check:Done with random functions, sent times each ()In the protocol: random functions, sent times each
times bitsconsistency checkinconsistenc
y
Analysis: If an error occurred or the players wentout of split, eventually they will fix it, since the consistency check is done with larger and larger parameters. Thus, the final protocol simulates withprobability close to .How many communication bits are wasted?
times bitsconsistency checkinconsistenc
y
Analysis of Wastes in the Basic Step:
Length of consistency check: bitsProbability to start over: Total waste (in expectation): bitsFraction of bits wasted:
bits bitsconsistency checkinconsistenc
y
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)
times bitsconsistency checkinconsistenc
y
Bound on the Channel Capacity:
Thank You!