Randomized Algorithms

1
Randomized Algorithms

• A deterministic algorithm with the capability of
  making random choices

Quicksort
Worst-case ?(n2)
Expected running-time O( n log n)
Randomized Quicksort
Both algorithms work correctly
2
Primarily Testing

• PRIME given a positive integer n in the binary
  form, determine whether n is a prime number
• GOAL a randomized algorithm that solves PRIME in
  polynomial time, with an arbitrarily small error
  probability ?
• Set Zn, for any n gt 0
• Zn x ? N ?0 lt x lt n, gcd(x, n) 1
• If a number p is prime, then
• Zp 1, 2, , p - 1

3

• For any prime p and x ? Zp
• the Legendre symbol
• 1 if x is a quadratic residue
• modulo p
• -1 otherwise
• a number x is a quadratic residue modulo p if for
  any x ? Zp
• x ? y2 mod p for some y ? Zp
• Fact. Legendre symbols can be computed in
  polynomial time
• Lemma Euler Criterion
• For all primes p gt 2 and x ? Zp

4

• Randomized Algorithm for PRIME
• Input n, ?
• k ? - ?log ??
• for i ? 1 to k do
• choose random x in the range
• 1, , n - 1
• if gcd(x, n) ? 1 then
• output NO and halt
• if
  then
• output NO and halt
• output YES and halt

5

• The randomized algorithm for PRIME outputs NO
  with probability ? 1 - ?, thanks to
• Lemma
• If n is an odd, composite number,
• A x ? Zn ?gcd(x, n) 1 and
• 
  
• is a proper subgroup of Zn and, hence, has size
  ?A? ? (n - 1) / 2