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Randomized Algorithms

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The BPWM algorithm is a Las Vegas algorithm for the BPWM problem with expected ... Devise a Las Vegas or a deterministic algorithm for min-cuts with running ... – PowerPoint PPT presentation

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Title: Randomized Algorithms


1
Randomized Algorithms
  • Morteza ZadiMoghaddam
  • Amin Sayedi

2
Types of Randomized algorithms
  • Las Vegas
  • Monte Carlo

3
Las Vegas
  • Always gives the true answer.
  • Running time is random.
  • Running time is bounded.
  • Quick sort is a Las Vegas algorithm.

4
Monte Carlo
  • It may produce incorrect answer!
  • We are able to bound its probability.
  • By running it many times on independent random
    variables, we can make the failure probability
    arbitrarily small at the expense of running time.

5
Monte Carlo Example
  • Suppose we want to find a number among n given
    numbers which is larger than or equal to the
    median.

6
Monte Carlo Example
  • Suppose A1 lt lt An.
  • We want Ai, such that i n/2.
  • Its obvious that the best deterministic
    algorithm needs O(n) time to produce the answer.
  • n may be very large!
  • Suppose n is 100,000,000,000 !

7
Monte Carlo Example
  • Choose 100 of the numbers with equal probability.
  • find the maximum among these numbers.
  • Return the maximum.

8
Monte Carlo Example
  • The running time of the given algorithm is O(1).
  • The probability of Failure is 1/(2100).
  • Consider that the algorithm may return a wrong
    answer but the probability is very smaller than
    the hardware failure or even an earthquake!

9
Monte Carlo
  • Suppose the output is Yes or No.
  • One sided error.
  • Two sided error.

10
RP Class( randomized polynomial )
  • Bounded polynomial time in the worst case.
  • If the answer is Yes Pr return Yes gt ½.
  • If the answer is No Pr return Yes 0.
  • ½ is not actually important.

11
PP Class( probabilistic polynomial )
  • Bounded polynomial time in worst case.
  • If the answer is Yes Pr return Yes gt ½.
  • If the answer is No Pr return Yes lt ½.
  • Unfortunately the definition is weak because the
    distance to ½ is important but is not considered.

12
Routing Problem
  • There are n computers.
  • Each computer has a packet.
  • Each packet has a destination D(i).
  • Packets can not follow the same edge
    simultaneously.
  • An oblivious algorithm is required.

13
Routing Problem
  • For any deterministic oblivious algorithm on a
    network of N nodes each of outdegree d, there is
    an instance of permutation routing requiring
    (N/d) ½.

14
Routing Problem
  • Pick random intermediate destination.
  • Packet i first travels to the intermediate
    destination and then to the final destination.
  • With probability at least 1-(1/N), every packet
    reaches its destination in 14n of fewer steps in
    Qn.
  • The expected number of steps is 15n.

15
Maximum Satisfiability
  • You have m clauses and n boolean variables.
  • Each clause contains some of variables or some of
    complements.
  • A clause is satisfied if at least one of its
    variables are satisfied.
  • We want to set the variables such that the number
    of satisfied clauses is maximized.

16
Example for Maximum Sat
  • There are 3 variables A, B and C.
  • M1 (A) or (B)
  • M2 (A) or (not B) or (not C)
  • M3 (C)
  • M4 (B) or (not C)
  • M5 (not C)

17
Example of Maximum Sat
  • Set A True
  • Set B True
  • Set C False
  • Four of the clauses are satisfied.

18
Maximum Sat
  • This problem is a famous problem which has no
    polynomial time algorithm yet. Its NP-hard.

19
Maximum Sat
  • For any set of m clauses, there is truth
    assignment for the variables that satisfies at
    least m/2 clauses.

20
Maximum Sat
  • Let Zi 1 if the i-th clause is satisfied and 0
    otherwise.
  • Set the variables in a random way.
  • The probability of a clause with k variables to
    be true is 1- (1/(2k)) gt ½.
  • So EZ1EZm gt ½.
  • Thus there exist at least one assignment such
    that Z1Zm gt ½.

21
Maximum Sat algorithm
  • This problem is NP-hard so we seek for
    approximation algorithms.
  • We have an algorithm that produces an answer
    which is at least ½ of the best answer.
  • If all clauses have at least 2 literals then we
    have an algorithm that produces an answer which
    is at least ¾ of the best one.

22
Maximum Sat algorithm
  • We want to maximize Z1Zm.
  • We have some inequalities
  • ? yi (if Xi is in Zj and is uncomplemented)
  • ? (1-yi) (if Xi is in Zj and is complemented).
  • This inequality must be hold
  • ? yi ? (1-yi) gt Zj
  • This problem could be solved using integer linear
    programming.
  • We have to use linear programming.

23
Maximum Sat
  • Solve the problem using linear programming.
  • You get a real number for each yi or zi.
  • Assign Xi true with the probability yi.
  • The expected number of clauses that are satisfied
    is (1- 1/e) of the best answer.

24
Maximum Sat algorithm
  • Using both algorithms and choosing the better
    answer gives us an answer which is at least ¾ of
    the best answer!!! Which is better than ½ and 1-
    1/e.

25
2-Sat
  • Every clause has at most 2 literals.
  • We want to check if all clauses can be satisfied.
  • It has polynomial algorithm.
  • Assign random values to the variables.
  • If all clauses are satisfied we are finished.
  • If there is an unsatisfied clause, the value of
    one of its literals is different from the best
    answer.
  • Change the value of one of the variables in this
    clause. You may make a good change or bad one.

26
2-Sat
  • You are walking on a path.
  • If you are on 0 you go to 1.
  • If you are on i you go to i1 or i-1 with equal
    probability.
  • The expected number of steps to reach the end of
    the path is O(n2).
  • So the given algorithm is O(n3).

27
Graph Connectivity
  • You want to check if two vertices u and v are in
    the same connected component.
  • Start a random walk from v.
  • Have a random walk of length 2n3.
  • If you havent visited u, the probability of u to
    be in this component is less than ½.
  • By repeating this algorithm, you can make the
    probability of failure arbitrarily small.

28
Graph Connectivity
  • Running time of algorithm is O(n3).
  • Required space is O(logn).

29
Diameter of a Point Set
  • You want to find the diameter of set of n points
    in the space.
  • Suppose I(x) is the convex body formed by the
    intersection of n sphere centered at n points
    with radius x.
  • F(p) is distance between p and the point in the
    set that is farthest from p.

30
Diameter of a Point Set
  • Consider I(x) when xF(p).
  • For any q in S, if q is in I(x) then F(q)ltF(p).
  • And if q is not in I(x) then F(p)ltF(q).

31
Diameter of a Point Set
  • Pick a point p in s at random. Computer F(p).
    O(n)
  • Set xF(p). Compute I(x). O(n logn)
  • Find the points outside I(x). Call this set T.
    O(n logn)
  • If T is empty return x as the answer, else
    continue on T.

32
Diameter of a Point Set
  • The running time of algorithm above is O(n log
    n).
  • In each step, all points that have smaller F(x)
    than the chosen point are removed.

33
All-Pairs Shortest Paths
  • Let G(V,E) be an undirected, connected graph with
    V1,,n and Em.
  • The adjacency matrix A is an n ? n 0-1 matrix
    with AijAji1 if and only if the edge (i,j) is
    present in E.
  • We are going to compute matrix D which Dij
    equals the length of a shortest path from vertex
    i to vertex j.

34
All-Pairs Distances
  • Z A2
  • Compute matrix A such that Aij1 if and only if
    i ? j and (Aij1 or Zijgt0)
  • If Aij1 for all i ? j then return D 2A-A.
  • Recursively compute the APD matrix D for the
    graph G with adjacency matrix A.
  • S AD
  • Return matrix D with Dij2Dij if SijDijZii ,
    otherwise Dij2Dij-1.

35
APSP
  • The APD algorithm computes the distance matrix
    for an n-vertex graph in time O(MM(n)log(n))
    using integer matrix multiplication algorithm.
  • Matrix multiplication algorithm running in time
    O(n2.376).

36
Boolean Product Witness Matrix
  • Suppose A and B are n?n boolean matrices and PAB
    is their product under Boolean matrix
    multiplication.
  • A witness for Pij is an index k ? 1,,n such
    that AikAkj1. Observe that Pij1 if and only if
    it has some witness k.

37
BPWM
  • W -AB
  • 2. for t0,, ?log(n)? do
  • 2.1. r 2t
  • 2.2. Repeat ?3.77log(n)? times
  • 2.2.1. Choose random R ? 1,,n with Rr .
  • 2.2.2. Compute AR and BR .
  • 2.2.3. Z ARBR .
  • 2.2.4. for all (i,j) do
  • if Wij lt 0 and Zij is witness then Wij Zij
  • 3. for all (i,j) do
  • if Wij lt 0 then find witness Wij by brute force.

38
BPWM
  • The BPWM algorithm is a Las Vegas algorithm for
    the BPWM problem with expected running time
    O(MM(n)log2(n)).
  • The probability that no witness is found for Pij
    before the end of Step 2 is at most
  • (1-1/2e)3.77log(n) ? 1/n.

39
Determining Shortest Path
  • A successor matrix S for an n-vertex graph G is
    an n ? n matrix such that Sij is the index of a
    neighbor of vertex i that lies on a shortest path
    from i to j.

40
APSP
  • Compute the distance matrix DAPD(A).
  • for s0,1,2 do
  • Compute 0-1 matrix Dkj(s)1 if and only if Dkj1
    s (mod 3)
  • Compute the witness matrix W(s)BPWM(A,D(s)).
  • Compute successor matrix S for G.

41
APSP
  • Algorithm APSP computes the successor matrix for
    an n-vertex graph G in expected time
    O(MM(n)log2(n)).

42
Algorithm contract
  • H G
  • While H has more than 2 vertices do
  • Choose an edge (x,y) uniformly at random from the
    edges in H.
  • F F ? (x,y).
  • H H / (x,y).
  • (C,V/C) the sets of vertices corresponding to
    the two meta-vertices in HG/F.

43
FastCut
  • n v
  • if n ? 6 then compute min-cut of G by
    brute-force enumeration else
  • t ?1n/?2?
  • Using Algorithm Contract, perform two independent
    contraction sequences to obtain graphs H1 and H2
    each with t vertices.
  • Recursively compute min-cuts in each of H1 and
    H2.
  • Return the smaller of the two min-cuts.

44
Fastcut
  • Algorithm Fastcut succeeds in finding a min-cut
    with probability ?(1/log(n)).
  • Algorithm Fastcut runs in O(n2log(n)) time and
    uses O(n2) space.

45
MST
  • Finding MST in a graph with n vertices and m
    edges has a Las Vegas algorithm which has the
    expected running time O(nm).
  • But We dont have enough time to Explain it !!!

46
Research problems
  • Devise an algorithm for the all-pairs shortest
    paths problem that does not use matrix
    multiplication and runs in time O(n3-?) for a
    positive constant ?.
  • Devise an algorithm for computing the diameter of
    an unweighted graph that does not use matrix
    multiplication and runs in time O(n3-?) for a
    positive constant ?.

47
Research problems
  • Devise a Las Vegas or a deterministic algorithm
    for min-cuts with running time close to O(n2).
  • Is there a randomized algorithm for min-cuts with
    expected running time close to O(m)?

48
Have a nice randomized life
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