Kargers Algorithm Assignment Help

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What is kerger's algorithm?

kerger's algorithm
- for i=1 to 100n power2
at random select an edge (u, v)
contract u and
till 2 vertices tend to be left
ci <-- the number of edges between them
-output min ci

Analysis of karger's algorithm

Let K be the number of edges of minimum cut(S, V-S)

Step1: If we never picket a crossing edge in the algorithm, then the number of edge between two last vertices is the correct answer.

Step2: The probability that in step1 of an iteration a crossing edge is not picket = (|E|- k)/|E|.

Step3: By definition of minimum cut, and each vertex v has degree at least k. otherwise the cut ({v}, V-{v}) is lighter.

  Thus |E| >= n k/2 and (|E| - k) / |E| 1- k / |E| >= 1- 2/n

Step4: In step1, Pr[no –crossing_edge picked] >= 1-2/n

Step5: similarly, in step2, Pr[no crossing_edge picked] >= 1-2/(n-1)

Step6: in general, in step j, Pr[no crossing_edge picket] >= 1-2/(n-j+1)

Step7: Pr{the n-2 contractions never contract a crossing edge}

  1. Pr[step1 is good]
  2. Pr[step2 good after surviving step1]
  3. Pr[step3 good after surviving first two steps]
  4. ....
  5. Pr[(n-2)th step good after surviving first (n-3) steps]

          >= (1-2/n) (1-2/(n-1) ) ....(1-2/3)

          = [(n - 2)/n][(n - 3)(n-1)] ....[1/3]= 2/[n(n - 1)]=  Ω(1 /n power2)