The Karger's algorithm relates to graph theory where G=(V,E) is an undirected graph with |E| edges and |V| vertices. The objective is to find the minimum number of cuts in edges in order to separate G into two disjoint graphs. The algorithm is randomized and will, in some cases, give the minimum number of cuts. The more number of trials, the higher probability that the minimum number of cuts will be obtained.
The Karger's algorithm will succeed in finding the minimum cut if every edge contraction does not involve any of the edge set C of the minimum cut.
The probability of success, i.e. obtaining the minimum cut, can be shown to be ≥ 2/(n(n-1))=1/C(n,2), which roughly equals 2/n^2 given in the question.Given: EACH randomized trial using the Karger's algorithm has a success rate of P(success,1) ≥ 2/n^2.
This means that the probability of failure is P(F,1) ≤ (1-2/n^2) for each single trial.
We need to estimate the number of trials, t, such that the probability that all t trials fail is less than 1/n.
Using the multiplication rule in probability theory, this can be expressed as
P(F,t)= (1-2/n^2)^t < 1/n
We will use a tool derived from calculus that
Lim (1-1/x)^x as x->infinity = 1/e, and
(1-1/x)^x < 1/e for x finite.
Setting t=(1/2)n^2 trials, we have
P(F,n^2) = (1-2/n^2)^((1/2)n^2) < 1/e
Finally, if we set t=(1/2)n^2*log(n), [log(n) is log_e(n)]
P(F,(1/2)n^2*log(n))
= (P(F,(1/2)n^2))^log(n)
< (1/e)^log(n)
= 1/(e^log(n))
= 1/n
Therefore, the minimum number of trials, t, such that P(F,t)< 1/n is t=(1/2)(n^2)*log(n) [note: log(n) is natural log]
C. x³ - x²
In word problems, there terms which indicate the mathematical operation that needs to be performed.
difference = refers to the answer in subtraction
triple a number or double a number refers to the number of times a number is multiplied by itself.
in triple a number = a number is multiplied by itself three times.
in double a number = a number is multiplied by itself two times.
It would initially look like this: (x * x * x) - (x * x); to simplify the expression, a number to be multiplied is written down and the number of times it will multiply by itself is noted on the upper right side of the number when we look at it.
Thus: x³ - x² is the correct expression that best describes the word problem above.
Answer: 1. ∠A= 80.75° 2. 41.79 
Step-by-step explanation:
Since, According to the sines low,
Here, CB= 4.1 cm, AB = 3.3 and ∠ C = 52.6°
⇒ 
⇒ 
⇒ A = 80.75°
2. Since, the area of the given figure = Area of the rectangle having dimension 8.3 × 4.2 + Area of the half square of radius 2.1
=34.86 + 6.93
= 41.79 square cm
Answer:
Step-by-step explanation: see attachment below
Answer:
A≈530.93in²
Step-by-step explanation:
