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:
-2.5
Step-by-step explanation:
Answer:
Step-by-step explanation:
Box plot is enclosed.
To find out 5 number summary
3.0, 3.5, 4.5, 5.5, 6.5, 6.5, 7.0, 7.0, 7.0, 7.0, 7.0, 7.5, 8.0, 9.0, 9.0, 9.0, 9.5, 9.5, 9.5, 9.5}
Minimum: 3.0
Quartile Q1: 6.5
Median: 7
Quartile Q3: 9
Maximum: 9.5
Besides we find that
Average (mean): μ=7.25
Absolute deviation: 30.5
Mean deviation: 1.525
Minimum: 3.0
Maximum: 9.5
Variance: 3.7125
Standard deviation σ=1.926
Answer:
yes... 13.3
Step-by-step explanation:
y=5x^2+7 is Non-Linear Functions
Option B is correct option.
Step-by-step explanation:
We need to identify Non-Linear Functions from the equations given.
First we will define Non-Linear Functions
<u>Linear Functions</u>
A function having exponent of variable equal to 1 or of the form y=c, where c is constant is called linear function.
<u>Non-Linear Functions</u>
A function that has variable having power greater than 1 (i.e 2 or above) is called non-linear function.
So, from all the options given, only Option B has power greater than 1 i.e 2. All remaining options are linear functions.
So, y=5x^2+7 is Non-Linear Functions
Option B is correct option.
Keywords: Linear and Non-Linear Functions
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