1. Communication Complexity Guy Feigenblat Based on lecture by Dr. Ely Porat Some slides where adapted from various sources Complexity course Computer science department, Bar-Ilan university January 2008

2. The Model 2 Computers : Alice (A),Bob (B) All calculations for A are free All calculations for B are free Algorithm costs are measured by cost of communications. Cost is measured per bits Motivation – Distributed models

3. The Model Communication Complexity 14, 29,53,28,284,34839, 67,98,22,35,253 (48) 01110110 (98) 00100110 A B

4. Our goal Minimize the communication between A,B while calculating functions on their inputs

5. First Problem Input A has array of n numbers B has array of n numbers Output median of all 2n numbers Numbers are O(log n) bits long

6. Naive Solution 1. A sends all of his numbers to B 2. B calculates median of all 2n numbers Cost Each number is O(log n) bits n numbers are sent Total cost is O(n*log n) bits

7. Naive Solution Communication Complexity 14, 29,53,28,284,34839, 67,98,22,35,253 (284) 00011111 (348) 001110101 (53) 101011 (29) 10111 (14) 001110 A B Total cost is O(n*log n) bits

8. 8 Better algorithm 1. A sorts his array and sends his median ( ) to B 2. B sorts his array and sends his median ( ) to A. Exercise : define : r = real median b = MAX{ } s = MIN { } prove :

9. Better algorithm Communication Complexity 14, 28,29, 53,284,348,50022, 35,39, 67,98,253,300 A B B sort his array and find his median A sort his array and find his median

10. Better algorithm Communication Complexity 14, 28,29, 53,284,348,50022, 35,39, 67,98,253,300 A B 67 53 B send his median to A A send his median to B

11. Better algorithm 4. If = then return ( = ) 5. If > then A throws top (n/2) elements B throws low (n/2) elements 6. Otherwise, vice versa We reduces the size of the problem by half 7. Back to step 1, until size of arrays = 1

12. Better algorithm Communication Complexity, 53,284,348,500 A B 67>53 Then B throws the big half of his array 53<67 Then A throws the small half of his array 14, 28,29 22, 35,39, 67,98,253,30 0

13. Better algorithm Communication Complexity, 53,284,348,500 A B 14, 28,29 22, 35,39, 67,98,253,300 We will repeat this algorithm until the size of the array will be 1, while every loop the array is cut in half, and log n bits transferred Total cost is CC mid = O (log 2 n) bits

14. Even Better algorithm Exercise: Try reducing the communication complexity to O(log n) bits

15. EQ The previous subject talked about problem of finding median of a divided array. Now we consider a new problem : Each side has a number and we want to know if the numbers are equal. Communication Complexity A B X Y ? X=Y

16. Deterministic Algorithm Send all the data We can’t avoid it !! Think why ?

17. Probabilistic Algorithm

18. Analysis Like Co-RP

19. How can we lower the failure probability ? Run the experience few times Use larger “q” (i.e. q = n 10 )

20. G – Communication Complexity Till now we had to send the Random number. Consider a model in which A,B use a third party in order to synchronize random numbers. A,B use exactly the same random “Alise on the moon, Bob on earth, both take random from the sun” Hard to “understand” – Remember this is just a model!!

21. Protocol

22. Analysis Completeness – Perfect Completeness If A=B then all i times (a,r) = (b,r) Soundness – If A≠B the probability that all i times (a,r) = (b,r) is 2 -i Communication Complexity

23. Exercise Define LE to be : LE(x,y) = 1 x ≤ y 0 otherwise Show that

24. Theorem: We prove that we can give up synchronized random with an overhead of O(logn) communication bits. We will choose the same random numbers (n 2 ) using deterministic machine for both A,B

25. Protocol A choose and send it to B

26. If algorithm exists for L, we will run it 3 times and use it.

27. As for the proof… We have perfect completeness in But, we need to prove soundness For the chosen i

28. It is equivalent to argue: By union bound:

31. We have proved that there exist a group of There is a deterministic algorithm that can find them. Remember, both A and B have unlimited computational power.