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Study GroupRandomized Algorithms

- Jun 7, 2003
- Jun 14, 2003

Randomized Algorithms

- A randomized algorithm is defined as an algorithm

that is allowed to access a source of

independent, unbiased random bits, and it is then

allowed to use these random bits to influence its

computation.

Output

Input

Algorithm

Random bits

Monte Carlo and Las Vegas

- There are two kinds of randomized algorithms
- Monte Carlo A Monte Carlo algorithm runs for a

fixed number of steps for each input and produces

an answer that is correct with a bounded

probability - Las Vegas A Las Vegas algorithm always produces

the correct answer, but its runtime for each

input is a random variable whose expectation is

bounded.

Question

- Is the max-cut algorithm that we discussed

previously a Monte Carlo or Las Vegas algorithm? - We will see two other examples today.

Randomized Quick Sort

- In traditional Quick Sort, we will always pick

the first element as the pivot for partitioning. - The worst case runtime is O(n2) while the

expected runtime is O(nlogn) over the set of all

input. - Therefore, some input are born to have long

runtime, e.g., an inversely sorted list.

Randomized Quick Sort

- In randomized Quick Sort, we will pick randomly

an element as the pivot for partitioning. - The expected runtime of any input is O(nlogn).

Analysis of Randomized QS

- Let s(i) be the ith smallest element in the input

list S. - Xij is a random variable such that Xij 1 if

s(i) is compared with s(j) Xij 0 otherwise. - Expected runtime t of randomized QS is
- EXij is the expected value of Xij over the set

of all random choices of the pivots, which is

equal to the probability pij that s(i) will be

compared with s(j).

Analysis of Randomized QS

- We can represent the whole sorting process by a

binary tree T - Notice that s(i) will be compared with s(j) where

iamong the set s(i), s(i1), , s(j) to be

selected as the pivot. - Note that pij 2/(j-i1). Why?

1st pivot

5

2nd pivot

3rd pivot

7

2

4th pivot

5th pivot

4

1

Analysis of Randomized QS

- Therefore, the expected runtime t
- Note that
- Randomized QS is a Las Vegas algorithm.

Randomized Min-cut

- Given an undirected, connected multi-graph G(V,E)

, we want to find a cut (V1,V2) such that the

number of edges between V1 and V2 is minimum. - This problem can be solved optimally by applying

the max-flow min-cut algorithm O(n2) time by

trying all pairs of source and destination.

Randomized Min-cut

- In randomized Min-cut, we repeatedly do the

following - Pick randomly an edge e(u,v). Merge u and v, and

remove all the edges between u and v. For

example - until there are only 2 vertices left. We will

report the cut between these 2 vertices as the

min-cut.

y

y

x

x

u

v

u,v

z

z

Analysis of Randomized Min-cut

- Let k be the min-cut of the given graph G(E,V)

where Vn. - Then E kn/2.
- The probability q1 of picking one of those k

edges in the first merging step 2/n - The probability p1 of not picking any of those k

edges in the first merging step (1-2/n) - Repeat the same argument for the first n-2

merging steps. - Probability p of not picking any of those k edges

in all the merging steps (1-2/n)(1-2/(n-1))(1-2/

(n-2))(1-2/3)

Analysis of Randomized Min-cut

- Therefore, the probability of finding the

min-cut - If we repeat the whole procedure n2/2 times, the

probability of not finding the min-cut is at most - Randomized Min-cut is a Monte Carlo Algorithm.

Question

- What will happen if we apply a similar approach

to find the max-cut instead? Will it be better or

worse than the previous method of random

assignment?

Complexity Classes

- There are some interesting complexity classes

involving randomized algorithms - Randomized Polynomial time (RP)
- Zero-error Probabilistic Polynomial time (ZPP)
- Probabilistic Polynomial time (PP)
- Bounded-error Probabilistic Polynomial time (BPP)

RP

- Definition The class RP consists of all

languages L that have a randomized algorithm A

running in worst-case polynomial time such that

for any input x in ?

RP

- Independent repetitions of the algorithms can be

used to reduce the probability of error to

exponentially small. - Notice that the success probability can be

changed to an inverse polynomial function of the

input size without affecting the definition of

RP. Why?

ZPP

- Definition The class ZPP is the class of

languages which have Las Vegas algorithms running

in expected polynomial time. - ZPP RP n co-RP. Why?
- (Note that a language L is in co-X where X is a

complexity class if and only if its complement

?-L is in X.)

PP

- Definition The class PP consists of all

languages L that have a randomized algorithm A

running in worst-case polynomial time such that

for any input x in ?

PP

- To reduce the error probability, we can repeat

the algorithm several times on the same input and

produce the output which occurs in the majority

of those trials. - However, the definition of PP is quite weak since

we have no bound on how far from ½ the

probabilities are. It may not be possible to use

a small number (e.g., polynomial no.) of

repetitions to obtain a significantly small error

probability.

Question

- Consider a randomized algorithm with 2-sided

error as in the definition of PP. Show that a

polynomial no. of independent repetitions of this

algorithm needs not suffice to reduce the error

probability to ¼. (Hint Consider the case where

the error probability is ½ - ½n . )

BPP

- Definition The class BPP consists of all

languages L that have a randomized algorithm A

running in worst-case polynomial time such that

for any input x in ?

BPP

- For this class of algorithms, the error

probability can be reduced to ½n with only a

polynomial number of iterations. - In fact, the probability bounds ¾ and ¼ can be

changed to ½ 1/p(n) and ½ -1/p(n) respectively

where p(n) is a polynomial function of the input

size n without affecting the definition of BPP.

Why?