Download Randomised-algorithms

Randomized Algorithms
Prof. Dr. Th. Ottmann
University of Freiburg
[email protected]
Classes of Randomised Algorithms
Las Vegas type
• Yield always a correct result.
• For a specific input: Performance (runtime) may be bad, but the
extected runtime is good!
• Example: Randomised version of Quicksort
Monte Carlo type (most correctly):
• May produce an incorrect result (with a certain error probability).
• For each specific input: (Worst case) runtime is good
• Example: Randomised primality test.
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Quicksort
Unsorted part A[l,r] in an array A
A[l … r-1]
A[l...m – 1]
Quicksort
p
p
A[m + 1...r]
Quicksort
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Quicksort
Algorithm: Quicksort
Input: unsorted part [l, r] of an array A
Output: sorted part [l, r] of the array A
1 if r > l
2
then choose pivot-element p = A[r]
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m = divide(A, l , r)
/* partition Awith respect to p:
A[l],....,A[m – 1]  p  A[m + 1],...,A[r]
*/
4 Quicksort(A, l , m - 1)
Quicksort (A, m + 1, r)
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Division of the Array
l
r
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Division
divide(A, l , r):
• Yields the index of the pivot elements in A
• Can be carried out in time O(r – l)
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Worst-Case-Input
n elements:
Runtime: (n-1) + (n-2) + … + 2 + 1 = n(n-1)/2
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Randomised Version of Quicksort
Algorithmus: Quicksort
Input: unsorted part [l, r] of an array A
Output: sorted part [l, r] of the array A
if r > l then ramdomly choose a pivot-element p = A[i] in the part [l, r]
of the array;
exchange A[ i] and A[r];
m = divide(A, l, r);
/* divide A with respect to p:
A[l],....,A[m – 1]  p  A[m + 1],...,A[r] */
Quicksort(A, l, m - 1);
Quicksort(A, m + 1, r)
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Primality Test
Definition:
The natural number p  2 is prime, iff a | p implies a = 1 or a = p.
Algorithm: Deterministic primality test (naive version)
Input: A natural number n  2
Output: Answer to the question: Is n prime?
if n = 2 then return true;
if n even then return false;
for i = 1 to n/2 do
if 2i + 1 divides n
then return false
return true
Runtime: (n)
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Primality Test
Goal:
Randomised algorithm
• With polynomial runtime
• If the algorithm yields the answer “not prime”, then n is definitely not
prime.
• If the algorithm yields the answer “prime”, then this answer is wrong
with a certain error probability p>0 , i.e. n is prime with a certain
probability (1- p) only.
k iterations of the algorithm: the algorithm yields the wrong answer with
probability pk only.
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Randomised Primality Test
Theorem 1: (Fermat‘s theorem)
Is p prim and 1 < a < p, then
ap-1 mod p = 1.
Theorem 2:
Is p prim and 0 < a < p, then the equation
a2 mod p = 1
Has exactly two solutions, namely a = 1 und a = p – 1.
Randomised algorithm:
Choose an a with 1 < a < p randomly and check whether it fulfills the test
of theorem 1; while computing ap-1 simultaneously check whether the
test of theorem 2 is fulfilled for all numbers occurring during the
computation of ap-1 using the fast exponentiation method.
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Randomisierter Primzahltest
Algorithmus: Randomisierter Primzahltest 1
1 Wähle a im Bereich [2, n-1] zufällig
2 Berechne an-1 mod n
3 if an-1 mod n = 1
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then n ist möglicherweise prim
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else n ist definitiv nicht prim
Prob(n ist nicht prim, aber an-1 mod n = 1 ) ?
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Randomised Primality Test
Theorem:
Is n not prime, then there are at most n – 4/ 9 numbers 0 < a < n, such
that the randomized algorithm for primality testing yields the wrong
result.
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