Randomized algorithms
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The document discusses using randomized algorithms to check if two large numbers or files are equal using only a small amount of communication. It explains how communicating the remainder of one number modulo a randomly selected prime number allows checking equality with very high probability while only communicating a number of bits proportional to the logarithm of the file size. Randomized algorithms can also be applied to problems like principal component analysis and checking matrix products to reduce computation time.
Randomized algorithms
- 1. The Power of Randomization
- 2. Example 1: Checking Equality • Two large files at two different locations. • Are they identical? – By communicating only a small amount of information!
- 3. Checking Equality The Challenge • Two large numbers N1 and N2 , n bits each • Communication allowed: m<<n bits • Possible?
- 4. Checking Equality Impossibility • 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)
- 5. 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
- 6. Checking Equality Analysis • 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
- 7. Checking Equality Analysis • 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)
- 8. Checking Equality Use 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)
- 9. Checking Equality The 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
- 10. Checking Equality Reducing Error Prob • Repeat k times • Communication is klog n bits • Error prob < (½)k
- 11. Checking Equality Example 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
- 12. Another Example PCA • Fit a line thru 0 to a collection of points so as to maximize sum of squares of projections
- 13. PCA Random Sampling • Too many points? • Pick a random sample – The fitting line doesn’t change too much?
- 14. PCA Random Sampling • How should you sample here?
- 15. Puzzle Checking 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?
- 16. Puzzle Checking 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?