Study Group Randomized Algorithms

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Study GroupRandomized Algorithms

• Jun 7, 2003
• Jun 14, 2003

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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
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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.

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Question

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

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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.

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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).

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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).

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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
  iltj if and only if s(i) or s(j) is the first one
  among 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
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Analysis of Randomized QS

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

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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.

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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
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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)

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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.

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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?

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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)

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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 ?

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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?

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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.)

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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 ?

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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.

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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 . )

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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 ?

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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?