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]
Answer: Orange = 5 rupee
Apple = 6 rupees
Step-by-step explanation:
Let the cost of an orange be x
Let the cost of an apple be y
We can form an equation from the question given which will be:
5x + 3y = 43 ......... i
2x + 4y = 34 ......... ii
Multiply equation i by 2
Multiply equation ii by 5
10x + 6y = 86 ........ iii
10x + 20y = 170 ..... iv
Subtract iii from iv
14y = 84
y = 84/14
y = 6
An apple cost 6 rupees
Since from equation ii
2x + 4y = 34
2x + 4(6) = 34
2x + 24 = 34
2x = 34 - 24
2x = 10
x = 10/2
x = 5
An orange cost 5 rupee
Answer:
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Step-by-step explanation:
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u can see in pictures
It shows that the values of the mileages are close to 24 mpg. Hence statistics refes 24 as the mean.
<h3>Mean of a data.</h3>
Mean is one of the measure of dispersion and is the avearage of a set of data.
According to the question, the magazine reports that the most common gas mileage was 24 mpg. This shows that reports have published several gas mileages for cars and truck and the average of this mileage is 24mpg
It shows that the values of the mileages are close to 24 mpg. Hence statistics refes 24 as the mean.
Learn more on mean here: brainly.com/question/14532771
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