Consider the following recurrence relation T(n) = 2T(n/2) + n lg n Can you solve that recurrence relation using the Master theorem? Justify your answer. Use the recurrence tree expansion method to find a tight asymptotic bound to the recurrence relation. For simplicity, assume that n is always a power of two and T(1) = c.

Answers

Answer 1

The asymptotic bound for the recurrence relation T(n) = 2T(n/2) + n lg n is Θ(n lg² n).

How to solve recurrence relation?

To determine the asymptotic bound for the recurrence relation T(n) = 2T(n/2) + n lg n using the Master theorem, we need to compare the function f(n) = n lg n to the function g(n) = [tex]n^log_b[/tex](a).

In this case, a = 2, b = 2, and f(n) = n lg n.

The Master theorem states that if f(n) = O(n[tex]^log_b[/tex](a - ε)) for some ε > 0, where a ≥ 1 and b > 1, then the solution to the recurrence relation is T(n) = Θ(n[tex]^log_b[/tex](a)).

Let's calculate the values:

n[tex]^log_b[/tex](a) = n[tex]^log_2[/tex](2) = n¹ = n

Since f(n) = n lg n and n¹ = n, we need to determine if f(n) satisfies the condition f(n) = O(n(¹ - ε)) for some ε > 0.

We can apply the limit test to check this condition:

lim (n->∞) [f(n) / (n(¹ - ε))] = lim (n->∞) [(n lg n) / (n(¹ - ε))]

= lim (n->∞) [lg n / [tex](n^ε)[/tex]]

Since the limit evaluates to 0, we can conclude that f(n) = O(n(¹ - ε)) for some ε > 0.

According to the Master theorem, the solution to the recurrence relation T(n) = 2T(n/2) + n lg n is T(n) = Θ(n lg n).

To find a tight asymptotic bound using the recurrence tree expansion method, we can visualize the expansion of the recurrence relation as a binary tree.

At each level of the tree, the cost of the nodes is n lg n.

The total number of levels in the tree is log n, since n is a power of two.

Therefore, the total cost of the recurrence relation can be calculated by multiplying the cost per level (n lg n) by the number of levels (log n):

Total cost = n lg n * log n = Θ(n lg² n)

Hence, a tight asymptotic bound for the recurrence relation T(n) = 2T(n/2) + n lg n is Θ(n lg² n).

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Related Questions

What is the volume of the rectangular prism below?

Answers

Answer:

V=140

Step-by-step explanation:

5*4=20

20*7=140

I need help, I don't just need to answer I need to know the steps on how to solve it asap

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Your goal when solving problems like these are to get V by itself. Meaning, on one side of the equal sign. So first you are to add -2 over to the side with the 4. Then multiple that by 11. To undo the division

Tyrone has 16 model airplanes on his shelf. He has 4 times as many model airplanes as model trains. How many model trains does Tyrone have?

Write multiplication and division equations to model and solve the problem. Use m for the unknown

Answers

Step-by-step explanation:

Write multiplication and division equations to model and solve the problem. Use m for the unknown

d) What would be the total cost of purchasing the number of shirts needed to use your coupon—after your coupon is applied and a 7.5% sales tax is charged on the purchase?

Answers

Answer: Multiply 7.5% by the price after subtracting the coupon's value.

Step-by-step explanation:

Let's assume that the cost of the shirts is x. So the coupon would subtract y from x so that's x-y which gives us z. Then to find z, on a calculator just put in 7.5% x z. I used variables since I don't know the value of the coupon or the original price of the shirts.

In ΔWXY, x = 680 inches, w = 900 inches and ∠W=157°. Find all possible values of ∠X, to the nearest degree.

Answers

Given:

In ΔWXY, x = 680 inches, w = 900 inches and ∠W=157°.

To find:

The all possible values of ∠X, to the nearest degree.

Solution:

Law of Sines:

[tex]\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}[/tex]

For ΔWXY,

[tex]\dfrac{w}{\sin W}=\dfrac{x}{\sin X}=\dfrac{y}{\sin Y}[/tex]

Now,

[tex]\dfrac{w}{\sin W}=\dfrac{x}{\sin X}[/tex]

[tex]\dfrac{900}{\sin (157^\circ)}=\dfrac{680}{\sin X}[/tex]

[tex]900\sin X=680\sin (157^\circ)[/tex]

[tex]\sin X=\dfrac{680}{900}\sin (157^\circ)[/tex]

[tex]\sin X=0.295219[/tex]

[tex]X=\sin^{-1}(0.295219)[/tex]

[tex]X=17.17067[/tex]

[tex]X\approx 17[/tex]

Therefore, the value of ∠X is 17 degrees.

plz I NEEEEED HELP NOWWWW PLZ PLZ PLZ

Answers

○=> Correct option :[tex]\color{hotpink}\bold{ (a) \: 130}[/tex]○=> Steps to derive correct option :

Angle (p+7)° and angle (3p+1)° are a linear pair so their sum will be equal to 180°.

Which means :

[tex] =\tt 3p + 1 + p + 7 = 180[/tex]

Let us solve that equation to find the value of p and the two angles :

[tex] =\tt 3p + 1 + p + 7 = 180[/tex]

[tex] =\tt 3p + p + 1 + 7 = 180[/tex]

[tex] =\tt 4p + 8 = 180[/tex]

[tex] = \tt4p = 180 - 8[/tex]

[tex] = \tt4p = 172[/tex]

[tex] =\tt p = \frac{172}{4} [/tex]

[tex]\hookrightarrow \tt \color{plum}p = 43[/tex]

Thus, value of p = 43

Measure of angle (p+7)° :

[tex] =\tt 43 + 7[/tex]

[tex]\tt\color{plum}angle \: (p + 7)° =\tt 50°[/tex]

Measure of angle (3p+1)° :

[tex] =\tt 43 \times 3 + 1[/tex]

[tex] = \tt129 + 1[/tex]

[tex]\tt\color{plum}angle \: (3p + 1)° = 130°[/tex]

Thus, the measure of the larger angle = 130°

Therefore, the correct option is (a) 130

It usually takes Mary 6 hours to do the cleaning that Jane can do in 5 hours and Ellen can do in 4 hours. If
Jane and Ellen agree to help Mary so that they can all go on a picnic, how long will it take them to do the
cleaning?

Answers

Answer:

It will take them about 5 hours to do the cleaning.

Step-by-step explanation:

Since all three of them are going to do it together, you must find the average amount of time between the three of them. To do this you add all of their cleaning times, 6, 5 and 4, and them divide by the amount of people there. 6+5+4= 15, 15/3=5.

Write Each Ratio as a Percent

1. 14 losses in 50 games

2. 4/5 of the students ride the bus

3. if AB represents 25% what is the length of a line segment that is 100%?

A________________B
5ft

Answers

Answer:

28%

80%

Step-by-step explanation:

hope this helps

Help me solve thank you will give 20 points

Answers

Answer:

Natalie bought 500 apples at $0.40 each, then she pays $0.40 500 times, this means that the total cost of the 500 apples is:

Cost = 500*$0.40 = $200

Now she threw away n apples from the 500 apples, then the number of apples that she has now is:

apples = 500 - n

And she sells the remaining apples for $0.70 each.

a) The amount that she gets by selling the apples is:

Revenue = (500 - n)*$0.70

b) We know that she did not make a loss, then the revenue must be larger than the cost, this means that:

cost ≤ revenue

$200 ≤ (500 - n)*$0.70

c) We need to solve the inequality for n.

$200 ≤ (500 - n)*$0.70

$200/$0.70 ≤ (500 - n)

285.7 ≤ 500 - n

n + 285.7 ≤ 500

n ≤ 500 - 285.7

n ≤ 214.3

Then the maximum value of n must be equal or smaller than 214.3

And n is a whole number, then we can conclude that the maximum number of rotten apples can be 214.

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Answer:

9 and -12

Step-by-step explanation:

This is the answer 9 and -12

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Answer:

It will be C

Step-by-step explanation:

This is because ecosystems worh high biodiversity tend to be more stable

Hope this Helped

Answer:

The awnser is c i took the test ( a high amount of bio deversety)

Step-by-step explanation:

there's 4 that describe the same function

Answers

Answer:

A,B,C,E

Step-by-step explanation:

for a everything is multiplied by 10

for c everything is divided by 6

for b everything is divided by 12

e is the only other one that fits

for a everything is multiplied by 10

for c everything is divided by 6

for b everything is divided by 12

e is the only other one that fits

ABCE is the answer to the question

−12 = −0.6k solve for k

Answers

Answer:

-20

Step-by-step explanation:

-12=-0.6k

divide both sides by -0.6

k=20

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what is the sum of 17 and a difference of 5

Answers

Answer:

The answer is 12.

~Hoped this helped~

Answer:

please mark me as brainliest answer

Step-by-step explanation:

The sum of two numbers is 17 and their difference is 5. What are the two numbers? Let's start by calling the two numbers we are looking for x and y.

The sum of x and y is 17. In other words, x plus y equals 17 and can be written as equation A:

x + y = 17

The difference between x and y is 5. In other words, x minus y equals 5 and can be written as equation B:

x - y = 5

Now solve equation B for x to get the revised equation B:

x - y = 5

x = 5 + y

Then substitute x in equation A from the revised equation B and then solve for y:

x + y = 17

5 + y + y = 17

5 + 2y = 17

2y = 12

y = 6

Now we know y is 6. Which means that we can substitute y for 6 in equation A and solve for x:

x + y = 17

x + 6 = 17

X = 11

Summary: The sum of two numbers is 17 and their difference is 5. What are the two numbers? Answer: 11 and 6 as proven here:

Sum: 11 + 6 = 17

Difference: 11 - 6 = 5

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Answer:

2πx - 4

Step-by-step explanation:

C = 2πr

Answers

Answer:

x = - 2 is not a zero

Step-by-step explanation:

If x = - 2 is a zero of the polynomial then f(- 2) = 0

f(x) = 3x² +4x + 32

f(- 2) = 3(- 2)² + 4(- 2) + 32 = 3(4) - 8 + 32 = 12 - 8 + 32 = 36

Since f(- 2) ≠ 0 , then x = - 2 is not a zero of the polynomial

The table shows how many hours Sara spent at several activities one Saturday.
Activity "Hours
Soccer practice 2
Birthday party 3
Science project 1
What is the ratio of hours spent at soccer practice to hours spent at a birthday party?
Choose 1 answer:
1 for every 2
B
2 for every 1
2 for every 3
3 for every 2

Answers

Answer:

Its2 for every 3

Step-by-step explanation:

PLEASE HELP! DUE SOON!

Answers

It’s 1.5. The question asking for the amount of seconds (x axis) have passed when the bird’s height (y axis) equals 0. Basically the x value of the point where the line touched the x axis

I need help, please solve this math problem. If you don't know the answer please don't answer. You will get reported.

Answers

Answer:

I believe it's A 2¹²

Step-by-step explanation:

The answer lies in the question itself.

In the equation, we have

3x - y = 12.

This means 3x = 12 + y.

We have to find 8^x/2^y.

8 can be written as 2^3. Therefore, 8^x = 2^3x.

But 3x = 12 + y, hence can be replaced.

Now we have, in the fraction, 2^(12 + y)/2^y.

This can be written as 2^(12 + y - y) = 2^12.

A 1,000 mL solution of acetic acid contains 26 mL of pure acetic acid. Find the percent concentration of this solution.

Answers

Given:

Total solution : 1000mL

Pure acetic acid = 26mL

To find:

The percent concentration of given solution.

Solution:

We know that,

[tex]\text{Percent concentration}=\dfrac{\text{Pure acetic acid}}{\text{Total solution}}\times 100[/tex]

Putting the given values, we get

[tex]\text{Percent concentration}=\dfrac{26}{1000}\times 100[/tex]

[tex]\text{Percent concentration}=\dfrac{26}{10}[/tex]

[tex]\text{Percent concentration}=2.6[/tex]

Therefore, the percent concentration of this solution is 2.6%.

Find an ordered triple that represents 65 = (2, 6, 8) and Ž = (1, -2, 6).

Answers

Answer:

< 40, 28, 72 >

Step-by-step explanation:

Given

y = < 2, 6, 8 > and z = < 7, - 2, 6 > , then

6y + 4z

= 6 < 2, 6, 8 > + 4 < 7, - 2, 6 >

Multiply each component by the scalar quantity

= < 6(2), 6(6), 6(8) > + < 4(7), 4(- 2), 4(6) >

= < 12, 36, 48 > + < 28, - 8, 24 >

Add corresponding components

= < 12 + 28, 36 - 8, 48 + 24 >

= < 40, 28, 72 >

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