Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower...
Transcript of Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower...
![Page 1: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/1.jpg)
Lower Bounds for Asymmetric CommunicationChannels and Distributed Source Coding
Micah Adler1 Erik D. Demaine2 Nicholas J. A. Harvey2
Mihai Patrascu2
1University of Massachusetts, Amherst
2MIT
SODA 2006
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 2: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/2.jpg)
Data Transmission
Client −→ Server
Send s ∈ 0, 1ns ← D, H(D) < n
Client sends ∼ H(D) bits
k clients −→ 1 server
Send s1, . . . , sk ∈ 0, 1n(s1, . . . , sk )← D (correlated!), H(D) < nk
Clients send ∼ H(D) bits in total
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 3: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/3.jpg)
Data Transmission
Client −→ Server
Send s ∈ 0, 1ns ← D, H(D) < n
Client sends ∼ H(D) bits
k clients −→ 1 server
Send s1, . . . , sk ∈ 0, 1n(s1, . . . , sk )← D (correlated!), H(D) < nk
Clients send ∼ H(D) bits in total
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 4: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/4.jpg)
What can be done?
1 client k clients
D fixed [Huffman] [Slepian-Wolf]client sends dH(D)e clients send dH(D)e
D known [Adler-Maggs]by server clients send O(H(D)) clients send O(H(D))
server sends O(n) server sends O(kn)expected O(1) roundsPr[t rounds] ≥ 2−O(t lg t) Ω( lg k
lg lg k ) needed
Cost of client not knowing D:1 communication by server – optimal2 rounds – quasioptimal [NEW]
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 5: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/5.jpg)
What can be done?
1 client k clients
D fixed [Huffman] [Slepian-Wolf]client sends dH(D)e clients send dH(D)e
D known [Adler-Maggs]by server clients send O(H(D)) clients send O(H(D))
server sends O(n) server sends O(kn)expected O(1) roundsPr[t rounds] ≥ 2−O(t lg t) Ω( lg k
lg lg k ) needed
Cost of client not knowing D:1 communication by server – optimal2 rounds – quasioptimal [NEW]
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 6: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/6.jpg)
What can be done?
1 client k clients
D fixed [Huffman] [Slepian-Wolf]client sends dH(D)e clients send dH(D)e
D known [Adler-Maggs]by server clients send O(H(D)) clients send O(H(D))
server sends O(n) server sends O(kn)expected O(1) roundsPr[t rounds] ≥ 2−O(t lg t) Ω( lg k
lg lg k ) needed
Cost of client not knowing D:1 communication by server – optimal2 rounds – quasioptimal [NEW]
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 7: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/7.jpg)
What can be done?
1 client k clients
D fixed [Huffman] [Slepian-Wolf]client sends dH(D)e clients send dH(D)e
D known [Adler-Maggs]by server clients send O(H(D)) clients send O(H(D))
server sends O(n) server sends O(kn)expected O(1) roundsPr[t rounds] ≥ 2−O(t lg t) Ω( lg k
lg lg k ) needed
Cost of client not knowing D:1 communication by server – optimal2 rounds – quasioptimal [NEW]
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 8: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/8.jpg)
What can be done?
1 client k clients
D fixed [Huffman] [Slepian-Wolf]client sends dH(D)e clients send dH(D)e
D known [Adler-Maggs]by server clients send O(H(D)) clients send O(H(D))
server sends O(n) server sends O(kn)expected O(1) roundsPr[t rounds] ≥ 2−O(t lg t) Ω( lg k
lg lg k ) needed
Cost of client not knowing D:1 communication by server – optimal2 rounds – quasioptimal [NEW]
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 9: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/9.jpg)
What can be done?
1 client k clients
D fixed [Huffman] [Slepian-Wolf]client sends dH(D)e clients send dH(D)e
D known [Adler-Maggs] [Adler-Maggs]by server clients send O(H(D)) clients send O(H(D))
server sends O(n) server sends O(kn)expected O(1) roundsPr[t rounds] ≥ 2−O(t lg t) Ω( lg k
lg lg k ) needed
Cost of client not knowing D:1 communication by server – optimal2 rounds – quasioptimal [NEW]
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 10: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/10.jpg)
What can be done?
1 client k clients
D fixed [Huffman] [Slepian-Wolf]client sends dH(D)e clients send dH(D)e
D known [Adler-Maggs] [Adler-Maggs]by server clients send O(H(D)) clients send O(H(D))
server sends O(n) server sends O(kn)expected O(1) rounds exp. O(k) roundsPr[t rounds] ≥ 2−O(t lg t) Ω( lg k
lg lg k ) needed
Cost of client not knowing D:1 communication by server – optimal2 rounds – quasioptimal [NEW]
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 11: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/11.jpg)
What can be done?
1 client k clients
D fixed [Huffman] [Slepian-Wolf]client sends dH(D)e clients send dH(D)e
D known [Adler-Maggs] [Adler]by server clients send O(H(D)) clients send O(H(D))
server sends O(n) server sends O(kn)expected O(1) rounds exp. O(lg k) roundsPr[t rounds] ≥ 2−O(t lg t) Ω( lg k
lg lg k ) needed
Cost of client not knowing D:1 communication by server – optimal2 rounds – quasioptimal [NEW]
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 12: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/12.jpg)
What can be done?
1 client k clients
D fixed [Huffman] [Slepian-Wolf]client sends dH(D)e clients send dH(D)e
D known [Adler-Maggs] [Adler]by server clients send O(H(D)) clients send O(H(D))
server sends O(n) server sends O(kn)expected O(1) rounds exp. O(lg k) roundsPr[t rounds] ≥ 2−O(t lg t) Ω( lg k
lg lg k ) needed
Cost of client not knowing D:1 communication by server – optimal2 rounds – quasioptimal [NEW]
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 13: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/13.jpg)
What can be done?
1 client k clients
D fixed [Huffman] [Slepian-Wolf]client sends dH(D)e clients send dH(D)e
D known [Adler-Maggs] [Adler]by server clients send O(H(D)) clients send O(H(D))
server sends O(n) server sends O(kn)expected O(1) rounds exp. O(lg k) roundsPr[t rounds] ≥ 2−O(t lg t) Ω( lg k
lg lg k ) needed
Cost of client not knowing D:1 communication by server – optimal2 rounds – quasioptimal [NEW]
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 14: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/14.jpg)
What can be done?
1 client k clients
D fixed [Huffman] [Slepian-Wolf]client sends dH(D)e clients send dH(D)e
D known [Adler-Maggs] [Adler]by server clients send O(H(D)) clients send O(H(D))
server sends O(n) server sends O(kn)expected O(1) rounds exp. O(lg k) roundsPr[t rounds] ≥ 2−O(t lg t) Ω( lg k
lg lg k ) needed
Cost of client not knowing D:1 communication by server – optimal2 rounds – quasioptimal [NEW]
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 15: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/15.jpg)
What can be done?
1 client k clients
D fixed [Huffman] [Slepian-Wolf]client sends dH(D)e clients send dH(D)e
D known [Adler-Maggs] [Adler]by server clients send O(H(D)) clients send O(H(D))
server sends O(n) server sends O(kn)expected O(1) rounds exp. O(lg k) roundsPr[t rounds] ≥ 2−O(t lg t) Ω( lg k
lg lg k ) needed
Cost of client not knowing D:1 communication by server – optimal2 rounds – quasioptimal [NEW]
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 16: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/16.jpg)
The class of hard distributions D
Sample
T
Layer 1
T T
T T T
Layer 2
Layer 3
Depth n
Vestigial
Child
T
T
Non-vestigial
Child
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 17: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/17.jpg)
Intuition for hardness
Let h = height of one layerLet p = Pr[vestigial child]
=⇒ H(D) = ph + (1− p)ph + (1− p)2ph + . . .
H(D) is small=⇒ one client message cannot talkabout many layers for many samples
Random choice of vestigial child (left / right)=⇒ don’t know which samples need many layers
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 18: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/18.jpg)
Intuition for hardness
Let h = height of one layerLet p = Pr[vestigial child]
=⇒ H(D) = ph + (1− p)ph + (1− p)2ph + . . .
H(D) is small=⇒ one client message cannot talkabout many layers for many samples
Random choice of vestigial child (left / right)=⇒ don’t know which samples need many layers
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 19: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/19.jpg)
Intuition for hardness
Let h = height of one layerLet p = Pr[vestigial child]
=⇒ H(D) = ph + (1− p)ph + (1− p)2ph + . . .
H(D) is small=⇒ one client message cannot talkabout many layers for many samples
Random choice of vestigial child (left / right)=⇒ don’t know which samples need many layers
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 20: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/20.jpg)
Communication Complexity Tools
Message switching
Alice sends a message of ≤ a bits⇒ eliminate, increasing Bob’s message by a factor of 2a
Round elimination lemma
Alice getsx1, . . . , xk
Bob getsy , i ∈ [k ]
−→ they computef (xi , y)
Alice sends a message of a k bits⇒ message irrelevant for average i ; eliminate
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 21: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/21.jpg)
Communication Complexity Tools
Message switching
Alice sends a message of ≤ a bits⇒ eliminate, increasing Bob’s message by a factor of 2a
Round elimination lemma
Alice getsx1, . . . , xk
Bob getsy , i ∈ [k ]
−→ they computef (xi , y)
Alice sends a message of a k bits⇒ message irrelevant for average i ; eliminate
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 22: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/22.jpg)
Communication Complexity Tools
Message switching
Alice sends a message of ≤ a bits⇒ eliminate, increasing Bob’s message by a factor of 2a
Round elimination lemma
Alice getsx1, . . . , xk
Bob getsy , i ∈ [k ]
−→ they computef (xi , y)
Alice sends a message of a k bits⇒ message irrelevant for average i ; eliminate
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 23: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/23.jpg)
Formal strategy
1 switch client’s messageNB: need hard upper bound on message size (Markov)
2 round elimination of server’s messagesubproblems: what is below each T leafprefix of client’s sample chooses subproblem
3 repeat, in the smaller probability space where the sampleis not vestigial at this level
Contradiction
Eliminated i rounds by introducing “small” errorWith no rounds, cannot solve better than random guessingSample is at level > i ⇒ nontrivial problem
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 24: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/24.jpg)
Formal strategy
1 switch client’s messageNB: need hard upper bound on message size (Markov)
2 round elimination of server’s messagesubproblems: what is below each T leafprefix of client’s sample chooses subproblem
3 repeat, in the smaller probability space where the sampleis not vestigial at this level
Contradiction
Eliminated i rounds by introducing “small” errorWith no rounds, cannot solve better than random guessingSample is at level > i ⇒ nontrivial problem
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 25: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/25.jpg)
Formal strategy
1 switch client’s messageNB: need hard upper bound on message size (Markov)
2 round elimination of server’s messagesubproblems: what is below each T leafprefix of client’s sample chooses subproblem
3 repeat, in the smaller probability space where the sampleis not vestigial at this level
Contradiction
Eliminated i rounds by introducing “small” errorWith no rounds, cannot solve better than random guessingSample is at level > i ⇒ nontrivial problem
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 26: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/26.jpg)
Formal strategy
1 switch client’s messageNB: need hard upper bound on message size (Markov)
2 round elimination of server’s messagesubproblems: what is below each T leafprefix of client’s sample chooses subproblem
3 repeat, in the smaller probability space where the sampleis not vestigial at this level
Contradiction
Eliminated i rounds by introducing “small” errorWith no rounds, cannot solve better than random guessingSample is at level > i ⇒ nontrivial problem
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 27: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/27.jpg)
Trouble in paradise
many complications and subtleties
innovative communication complexity analysis
Example
Obtaining a hard bound for the client’s messages:
Pr[sample is from level≥ i] = (1− p)i
error introduced must be small in this space
hard bound (by Markov) must be huge ∼ H(D)/(1− p)i
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 28: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/28.jpg)
Trouble in paradise
many complications and subtleties
innovative communication complexity analysis
Example
Obtaining a hard bound for the client’s messages:
Pr[sample is from level≥ i] = (1− p)i
error introduced must be small in this space
hard bound (by Markov) must be huge ∼ H(D)/(1− p)i
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 29: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/29.jpg)
Trouble in paradise
many complications and subtleties
innovative communication complexity analysis
Example
Obtaining a hard bound for the client’s messages:
Pr[sample is from level≥ i] = (1− p)i
error introduced must be small in this space
hard bound (by Markov) must be huge ∼ H(D)/(1− p)i
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 30: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/30.jpg)
Trouble in paradise
many complications and subtleties
innovative communication complexity analysis
Example
Obtaining a hard bound for the client’s messages:
Pr[sample is from level≥ i] = (1− p)i
error introduced must be small in this space
hard bound (by Markov) must be huge ∼ H(D)/(1− p)i
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 31: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/31.jpg)
Trouble in paradise
many complications and subtleties
innovative communication complexity analysis
Example
Obtaining a hard bound for the client’s messages:
Pr[sample is from level≥ i] = (1− p)i
error introduced must be small in this space
hard bound (by Markov) must be huge ∼ H(D)/(1− p)i
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 32: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/32.jpg)
Technical insight: Unilateral error
Regular error Unilateral error
Application
Markov on client’s message introduces unilateral error
conditioning the sample being from level ≥ i does notchange the marginal distribution on the client’s input
=⇒ much better Markov bound
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 33: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/33.jpg)
Technical insight: Unilateral error
Regular error Unilateral error
Application
Markov on client’s message introduces unilateral error
conditioning the sample being from level ≥ i does notchange the marginal distribution on the client’s input
=⇒ much better Markov bound
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 34: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/34.jpg)
Technical insight: Unilateral error
Regular error Unilateral error
Application
Markov on client’s message introduces unilateral error
conditioning the sample being from level ≥ i does notchange the marginal distribution on the client’s input
=⇒ much better Markov bound
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
![Page 35: Lower Bounds for Asymmetric Communication …people.csail.mit.edu/mip/papers/coding/talk.pdfLower Bounds for Asymmetric Communication Channels and Distributed Source Coding Micah Adler1](https://reader033.fdocuments.net/reader033/viewer/2022050523/5fa7195a55771270c757e312/html5/thumbnails/35.jpg)
Technical insight: Unilateral error
Regular error Unilateral error
Application
Markov on client’s message introduces unilateral error
conditioning the sample being from level ≥ i does notchange the marginal distribution on the client’s input
=⇒ much better Markov bound
Adler, Demaine, Harvey, P atrascu Distributed Source Coding
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Thank you
T HE END
Adler, Demaine, Harvey, P atrascu Distributed Source Coding