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:
divide
Step-by-step explanation:
divide the first one by the second that might help
Answer : 2.5
Step - by - step explanation :
X / Y = COP
5 / 2 = 2.5 or 2 1/2
Y = 2.5x
Answer:
The ball will be 84 feet above the ground 1.125 seconds and 4.5 seconds after launch.
Step-by-step explanation:
Statement is incorrect. Correct form is presented below:
<em>The height </em>
<em> of an ball that is thrown straight upward from an initial position 3 feet off the ground with initial velocity of 90 feet per second is given by equation </em>
<em>, where </em>
<em> is time in seconds. After how many seconds will the ball be 84 feet above the ground. </em>
We equalize the kinematic formula to 84 feet and solve the resulting second-order polynomial by Quadratic Formula to determine the instants associated with such height:
(1)
By Quadratic Formula:
, 
The ball will be 84 feet above the ground 1.125 seconds and 4.5 seconds after launch.
Answer:
2 gallons of pure alcohol
Step-by-step explanation:
Let x represent the number of gallons of pure alcohol needed, and create an equation:
4(0.25) + x(1) = (x + 4)0.5
1 + x = 0.5x + 2
x = 0.5x + 1
0.5x = 1
x = 2
So, 2 gallons of pure alcohol should be added.
