Download presentation

Presentation is loading. Please wait.

Published byIsrael Mossman Modified over 2 years ago

1
1 Information complexity and exact communication bounds April 26, 2013 Mark Braverman Princeton University Based on joint work with Ankit Garg, Denis Pankratov, and Omri Weinstein

2
Overview: information complexity Information complexity :: communication complexity as Shannon’s entropy :: transmission cost 2

3
Background – information theory Shannon (1948) introduced information theory as a tool for studying the communication cost of transmission tasks. 3 communication channel Alice Bob

4
Shannon’s entropy 4 communication channel X

5
Shannon’s noiseless coding 5

6
Shannon’s entropy – cont’d communication channel X Y

7
A simple example 7 Easy and complete!

8
Communication complexity [Yao] Focus on the two party randomized setting. 8 A B X Y F(X,Y) Meanwhile, in a galaxy far far away… Shared randomness R

9
Communication complexity A B X Y F(X,Y) m 1 (X,R) m 2 (Y,m 1,R) m 3 (X,m 1,m 2,R) Communication cost = #of bits exchanged. Shared randomness R

10
Communication complexity Numerous applications/potential applications (streaming, data structures, circuits lower bounds…) Considerably more difficult to obtain lower bounds than transmission (still much easier than other models of computation). Many lower-bound techniques exists. Exact bounds?? 10

11
Communication complexity 11

12
Set disjointness and intersection

13
Information complexity 13

14
Basic definition 1: The information cost of a protocol A B X Y Protocol π what Alice learns about Y + what Bob learns about X

15
Mutual information 15 H(A) H(B) I(A,B)

16
Basic definition 1: The information cost of a protocol A B X Y Protocol π what Alice learns about Y + what Bob learns about X

17
Example A B X Y what Alice learns about Y + what Bob learns about X MD5(X) [128 bits] X=Y? [1 bit]

18
Information complexity 18

19
Prior-free information complexity 19

20
Connection to privacy 20

21
Information equals amortized communication 21

22
Without priors 22

23
Intersection 23

24
The two-bit AND 24

25
The optimal protocol for AND A B X {0,1} Y {0,1} If X=1, A=1 If X=0, A=U [0,1] If Y=1, B=1 If Y=0, B=U [0,1] 0 1 “Raise your hand when your number is reached”

26
The optimal protocol for AND A B If X=1, A=1 If X=0, A=U [0,1] If Y=1, B=1 If Y=0, B=U [0,1] 0 1 “Raise your hand when your number is reached” X {0,1} Y {0,1}

27
Analysis 27

28
The analytical view A message is just a mapping from the current prior to a distribution of posteriors (new priors). Ex: 28 Y=0Y=1 X=00.40.2 X=10.30.1 Y=0Y=1 X=02/31/3 X=100 Y=0Y=1 X=000 X=10.750.25 Alice sends her bit “0”: 0.6 “1”: 0.4

29
The analytical view 29 Y=0Y=1 X=00.40.2 X=10.30.1 Y=0Y=1 X=00.5450.273 X=10.1360.045 Y=0Y=1 X=02/91/9 X=11/21/6 Alice sends her bit w.p ½ and unif. random bit w.p ½. “0”: 0.55 “1”: 0.45

30
Analytical view – cont’d 30

31
IC of AND 31

32
*Not a real protocol 32

33
Previous numerical evidence [Ma,Ishwar’09] – numerical calculation results. 33

34
Applications: communication complexity of intersection 34

35
Applications 2: set disjointness 35

36
A hard distribution? 36 001101000100111101100 101001110011101011000 Y=0Y=1 X=01/4 X=11/4 Very easy!

37
A hard distribution 37 000101000100110101100 101000110011100010000 Y=0Y=1 X=01/3 X=11/3 At most one (1,1) location!

38
Communication complexity of Disjointness 38

39
Small-set Disjointness 39

40
Using information complexity Y=0Y=1 X=01-2k/nk/n X=1k/n

41
Overview: information complexity Information complexity :: communication complexity as Shannon’s entropy :: transmission cost Today: focused on exact bounds using IC. 41

42
Selected open problems 1

43
Interactive compression? 43

44
Interactive compression? 44

45
Selected open problems 2 45

46
External information cost A B X Y Protocol π C

47
External information complexity 47

48
48 Thank You!

Similar presentations

OK

Mutual Information and Channel Capacity Multimedia Security.

Mutual Information and Channel Capacity Multimedia Security.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on weapons of mass destruction wow Download ppt on school management system Ppt on national education policy 1986 world Free ppt on moving coil galvanometer experiment Ppt on op amp circuits filter Ppt on computer graphics applications Ppt on e waste recycling Ppt on index numbers definition Marketing mix ppt on nokia pc Ppt on network graph theory