.

CLASSES RP AND ZPP

By:

SARIKA PAMMI

CONTENTS:

INTRODUCTION

RP

FACTS ABOUT RP

MONTE CARLO ALGORITHM

CO-RP

ZPP

FACTS ABOUT ZPP

RELATION BETWEEN RP AND ZPP

RELATION BETWEEN P AND ZPP

RELATION BETWEEN RP AND NP

RP(Randomized Polynomial time)

A class of problems that will run in polynomial time on a probabilistic TM with the following properties:

If the correct answer is

• no, always return no

• yes, return yes with probability at least ½

Otherwise, returns no

Formally

The class of languages for which membership can be determined in polynomial time by a probabilistic TM with no false acceptances and less than half of the rejections are false rejections

Facts about RP:

If the algorithm returns a yes answer, then yes is the correct answer

If the algorithm returns a no answer, then it may or may not be correct

The ½ in the definition is arbitrary

Like running the algorithm addition repetitions will decrease the chance of the algorithm giving the wrong answer

Often referred to as a Monte-Carlo Algorithm (or Monte-Carlo Turing Machine)

Monte carlo algorithm:

A numerical Monte Carlo method used to find solutions to problems that cannot easily to solved using standard numerical methods

Often relies on random (or pseudo-random) numbers

Is stochastic or nondeterministic in some manner

Co RP:

A class of problems that will run in polynomial time on a probabilistic TM with the following properties: If the correct answer is

yes, always return yes

no, return no with probability at least ½

Otherwise, returns a yes

In other words:

If the algorithm returns a no answer, then no is the correct answer

If the algorithm returns a yes answer, then it may or may not be correct

ZPP:

Zero-error Probabilistic Polynomial

The class of languages for which a probabilistic TM halts in polynomial time with no false acceptances or rejections, but sometimes gives an “I don’t know” answer

In other words:

It always returns a guaranteed correct yes or no answer

It might return an “I don’t know” answer

Facts about zpp:

The running time is unbounded

But it is polynomial on average (for any input)

It is expected to halt in polynomial time

Similar to definition of P except:

ZPP allows the TM to have “randomness”

The expected running time is measured (instead of the worst-case)

Often referred to as a Las-Vegas algorithm (or Las-Vegas Turning Machine)

Relations between RP and ZPP:

ZPP = RP co-RP

Proof Part 1: RP co-RP is in ZPP

Let L be a language recognized by RP algorithm A and co-RP algorithm B

Let w be in L

Run w on A. If A returns yes, the answer must be yes. If A returns no, run w on B. If B returns no, then the answer must be no. Otherwise, repeat.

Only one of the algorithms can ever give a wrong answer. The chance of an algorithm giving the wrong answer is 50%.

The chance of having the kth repetition shrinks exponentially. Therefore, the expected running time is polynomial

Hence, RP intersect co-RP is contained in ZPP

Continue …

ZPP = RP co-RP

Proof Part 2: ZPP is contained in RP co-RP

Let C be an algorithm in ZPP

Construct the RP algorithm using C:

Run C for (at least) double its expected running time.

If it gives an answer, that must be the answer

If it doesn’t given an answer before the algorithm stops, then the answer is no

The chance that algorithm C produces an answer before it is stopped is ½ (and hence fitting the definition of an RP algorithm)

The co-RP algorithm is almost identical, but it gives a yes answer if C does produce an answer.

Therefore, we can conclude that ZPP is contained in RP co-RP

As we conclude :

As seen in the proof of ZPP = RP co-RP we can conclude that

ZPP RP

ZPP co-RP

Relations between P and ZPP:

P ZPP

Proof

Any deterministic, polynomial time bounded TM is also a probabilistic TM that ignores its special feature that allows it to make random choices

Relation between RP and NP:

Proof continued

Let w be in L

M1 has a 50% probability of accepting w.

There must be some sequence of bits on the random tape that leads to the acceptance of w

M2 will choose that sequence of bits and accepts when the choice is made. Thus, w is in the language of M2

If w is not in L, then there is no sequence of random bits that will make M1 accept. Therefore, M2 cannot choose a sequence of bits that leads to acceptance. Thus, w is not in the language

Diagram Showing Relationship of Problem Classes

NP

Co-NP

Co-RP

RP

ZPP

P

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THANK YOU

REFERENCES:

http://en.wikipedia.org/wiki/RP_(complexity)

http://en.wikipedia.org/wiki/ZPP_(complexity)

https://www.cs.duke.edu/~reif/courses/complectures/books/T/Ch10.pdf

http://www.cs.kent.edu/~dragan/ThComp/RP-ZPP-cl.pdf