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Study Group Randomized Algorithms

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Monte Carlo and Las Vegas. There are two kinds of randomized ... Is the max-cut algorithm that we discussed previously a Monte Carlo or Las Vegas algorithm? ... – PowerPoint PPT presentation

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Title: Study Group Randomized Algorithms


1
Study GroupRandomized Algorithms
  • Jun 7, 2003
  • Jun 14, 2003

2
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
3
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.

4
Question
  • Is the max-cut algorithm that we discussed
    previously a Monte Carlo or Las Vegas algorithm?
  • We will see two other examples today.

5
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.

6
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).

7
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).

8
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
9
Analysis of Randomized QS
  • Therefore, the expected runtime t
  • Note that
  • Randomized QS is a Las Vegas algorithm.

10
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.

11
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
12
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)

13
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.

14
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?

15
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)

16
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 ?

17
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?

18
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.)

19
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 ?

20
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.

21
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 . )

22
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 ?

23
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?
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