The Power of Randomization

Example 1: Checking Equality

• Two large files at two different locations.

• Are they identical?– By communicating only a small amount of

information!

Checking EqualityThe Challenge

• Two large numbers N1 and N2 , n bits each

• Communication allowed: m<<n bits

• Possible?

Checking EqualityImpossibility

• Suppose the communication is based on N1 alone

• m<<n, – Two different N1’s will have the same m-bit

communication pattern– Switch N2 from one to another (YES->NO)

Checking Equality Randomized Algorithms

• Communicate N1 mod M for some number M

• If N1 = N2 then you always get YES

• If N1 != N2 then you get YES if M divides N1 - N2

Checking EqualityAnalysis

• Probability N1 != N2 but M divides N1 - N2 ?

• Probability over what?• M and not N1,N2

• Choose M at random in the range 1..2m

Checking EqualityAnalysis

• How many factors does N1 - N2 have?

– N1 - N2 <= 2n, so (2n)1/log n

• If we choose M randomly in the range 1..2 (2n)1/log

n

– Probability N1 != N2 but M divides N1 - N2 <= 1/2

– So m is ~ n/log n bits (minor gains)

Checking EqualityUse Prime Numbers

• How many prime factors does N1 - N2 have?

– N1 - N2 <= 2n, so 2n/log n

• If we choose M to be a random prime in 1..4n

– There are at least 4n/log 4n > 4n/log(4n) primes

– Probability N1 != N2 but M divides N1 - N2 <= ~ 1/2

– So m is ~ log n bits (major gains)

Checking EqualityThe Solution

• Two large numbers N1 and N2 , n bits each

• log n bits of communication – Remainder w.r.t random prime in range 1..4n

• Error Prob < 1/2

Checking EqualityReducing Error Prob

• Repeat k times

• Communication is klog n bits

• Error prob < (½)k

Checking EqualityExample Numbers

• 10GB file, n=1010

• Desired Error Prob 10-30

• Communication 99 * 33 = 3267 bits = 400 bytes

If 10 billion people do 10 billion checks a day, the prob that even one of the checks is erroneous is 1/10 billion

Another ExamplePCA

• Fit a line thru 0 to a collection of points so as to maximize sum of squares of projections

PCARandom Sampling

• Too many points?

• Pick a random sample– The fitting line doesn’t

change too much?

PCARandom Sampling

• How should you sample here?

PuzzleChecking Matrix Products

• Given three matrices A and BC, check if A=BC?– mod p for simplicity

• Matrices are n*n

• Easy to do in n3 time

• Can you do better?

PuzzleChecking Matrix Products

• Given three matrices A and BC, check if A=BC?

• Matrices are n*n

• Easy to do in n3 time

• Can you do better?