a full presentation on randomized algorithm

1. Randomized
Algorithm
By
Shoaib Khan
IMCA/4511/12
3rd Semester

2. • It is defined as an algorithm that is
allowed to access a source of
independent, unbaised random bits.
• The "worst case" is typically so unlikely to
occur that it cannot be ignored .

3. Why randomized
algorithms ?
• Simplicity: This is the first reason for
using randomized algorithms. There are
numerous example where an easy
randomized algorithm can match the
performance of a deterministic algorithm.
• Speed: The best known randomized
algorithm are faster than the best known
deterministic algorithm.

4. Las Vegas Algorithm
• A Las Vegas algorithm is a randomized
algorithm that always gives correct results,
that is, it always produces the correct result or it
informs about the failure. In other words, a Las
Vegas algorithm does not gamble with the
correctness of the result, it gambles only with the
resources used for the computation. A simple
example is randomized quicksort.
• Randomized quick sort :randomized quicksort algorithm makes random
choices during execution.

5. Randomized quick sort algo :Function RANDOMIZED-QUICKSORT(A, p, q)
1. if (p >=q) then
2. return
3. else
4. Choose a number, say r, uniformly and at random from the set
{p,p+1,...,q}.
5. Swap A[p] and A[r].
6. j = PARTITION(A, p, q).
7. Quicksort(A, p, j-1).
8. Quicksort(A, j+1, q).
9. end
The complexity of randomized quick sort algorithm is O(n2).

6. Monte carlo algorithm
• A randomized algorithm may produce incorrect
results, but with bounded error probability.
• For decision problems there are two kind of
monte carlo algorithms:
Two side error : If there is non zero probability
error.when its output is either yes or no.
One side error : When the probability that its
error is zero for atleast one of possible output.

7. Min cut algorithm
Let G = (V,E) be a connected , undirected multigraph with n vertices.A
multigragh may contain multiple edges between any pair ofvertices.A cut
in G is a set of edges whose removal results in G being broken into two or
more components. A min-cut is a cut of minimum cardinality. For instance,
say this graph represents a network, and we want to know how ”robust” it
is in the sense of the the minimum number of links whose failure causes
the network to become disconnected.The minimum cut problem is to find
a cut of minimum size.
Algorithm Min-Cut
Input: Connected, undirected multigragh G(V,E).
Output: Size of the min-cut.
1. while |V| >= 2 do
2. Pick an edge e randomly and contract it.
3. Remove Self Loops.
4. end while
5. Return

8. Thank you