Asymptotic Notations: In asymptotic analysis of algorithms, mathematical tools are used to represent time complexity of algorithm . such mathematical tools are known as Asymptotic notations. Some of the asymptotic notations are Big-O. Big Theta Big-O: A function “g(n)” is O ( f ( n ) ) if there exists a constant c > 0 such that cf(n) grows faster than “g(n)” for all n > = n 0 , that is, “g(n)” is O ( f ( n ) ) iff for some constants “c” and " n 0 " , g ( n ) < = c f ( n ) for all n > = n 0 . Here, “f(n)” denotes the upper bound for “g(n)”. The above statement can be read as follows : “n” is size of data set “f(n)” denotes a function calculated while taking n as parameter O ( f ( n ) ) means that curve denoted by “f(n)” is an upper bound for a function’s need for resources. Big Theta: A function “g(n)” is Θ ( f ( n ) ) iff two positive constants “c 1 ” and “c 2 ” and a positive integer " n 0 " exists such that c 1 f ( n ) ≤ g ( n ) ≤ c 2 f ( n ) for all n > = n 0 , that is, “f(n)” denotes exact bound for “g(n)”, which denotes that growth rate of “g(n)” and “f(n)” are equal. The above statement can be read as follows : “n” is size of data set “f(n)” denotes a function that is been calculated taking “n” as parameter Θ ( f ( n ) ) means that curve denoted by “f(n)” is an exact bound for a function’s need for resources.

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Answer 1

Question

Chapter 2, Problem 6E

Program Plan Intro

Asymptotic Notations:

In asymptotic analysis of algorithms, mathematical tools are used to represent time complexity of algorithm.

such mathematical tools are known as Asymptotic notations. Some of the asymptotic notations are

Big-O.

Big Theta

Big-O:

A function “g(n)” is O(f(n)) if there exists a constant c > 0 such that cf(n) grows faster than “g(n)” for all n>=n0 , that is, “g(n)” is O(f(n)) iff for some constants “c” and "n0", g(n)<=cf(n) for all n>=n0. Here, “f(n)” denotes the upper bound for “g(n)”. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function calculated while taking n as parameter

O(f(n)) means that curve denoted by “f(n)” is an upper bound for a function’s need for resources.

Big Theta:

A function “g(n)” is Θ(f(n)) iff two positive constants “c₁” and “c₂” and a positive integer "n0" exists such that c1f(n)≤g(n)≤c2f(n) for all n>=n0, that is, “f(n)” denotes exact bound for “g(n)”, which denotes that growth rate of “g(n)” and “f(n)” are equal. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function that is been calculated taking “n” as parameter

Θ(f(n)) means that curve denoted by “f(n)” is an exact bound for a function’s need for resources.

Explanation of Solution

b) f(n) + g(n) is Θ(min(f(n),g(n)))

Explanation:

Take f(n) = n and g(n) = 1/n.

Now, f(n)+ g(n)&

Explanation of Solution

c) 2^na is O(2ⁿ)

Explanation:

The function 2^na is not O(2n), as there is no constant c ≥2na/2n= 2n(a-

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2. Consider the following pseudocode for partition: function partition (A,L,R) pivotkey = A [R] t = L for i L to R-1 inclusive: if A[i] A[i] t = t + 1 end if end for A [t] A[R] return t end function Suppose we call partition (A,0,5) on A=[10,1,9,2,8,5]. Show the state of the list at the indicated instances. Initial A After i=0 ends After 1 ends After i 2 ends After i = 3 ends After i = 4 ends After final swap 10 19 285 [12 pts]

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Answer 2

Question

Chapter 2, Problem 6E

Program Plan Intro

Asymptotic Notations:

In asymptotic analysis of algorithms, mathematical tools are used to represent time complexity of algorithm.

such mathematical tools are known as Asymptotic notations. Some of the asymptotic notations are

Big-O.

Big Theta

Big-O:

A function “g(n)” is O(f(n)) if there exists a constant c > 0 such that cf(n) grows faster than “g(n)” for all n>=n0 , that is, “g(n)” is O(f(n)) iff for some constants “c” and "n0", g(n)<=cf(n) for all n>=n0. Here, “f(n)” denotes the upper bound for “g(n)”. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function calculated while taking n as parameter

O(f(n)) means that curve denoted by “f(n)” is an upper bound for a function’s need for resources.

Big Theta:

A function “g(n)” is Θ(f(n)) iff two positive constants “c₁” and “c₂” and a positive integer "n0" exists such that c1f(n)≤g(n)≤c2f(n) for all n>=n0, that is, “f(n)” denotes exact bound for “g(n)”, which denotes that growth rate of “g(n)” and “f(n)” are equal. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function that is been calculated taking “n” as parameter

Θ(f(n)) means that curve denoted by “f(n)” is an exact bound for a function’s need for resources.

Explanation of Solution

b) f(n) + g(n) is Θ(min(f(n),g(n)))

Explanation:

Take f(n) = n and g(n) = 1/n.

Now, f(n)+ g(n)&

Explanation of Solution

c) 2^na is O(2ⁿ)

Explanation:

The function 2^na is not O(2n), as there is no constant c ≥2na/2n= 2n(a-

Expert Solution & Answer

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Check out a sample textbook solution

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Answer 3

Question

Chapter 2, Problem 6E

Program Plan Intro

Asymptotic Notations:

In asymptotic analysis of algorithms, mathematical tools are used to represent time complexity of algorithm.

such mathematical tools are known as Asymptotic notations. Some of the asymptotic notations are

Big-O.

Big Theta

Big-O:

A function “g(n)” is O(f(n)) if there exists a constant c > 0 such that cf(n) grows faster than “g(n)” for all n>=n0 , that is, “g(n)” is O(f(n)) iff for some constants “c” and "n0", g(n)<=cf(n) for all n>=n0. Here, “f(n)” denotes the upper bound for “g(n)”. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function calculated while taking n as parameter

O(f(n)) means that curve denoted by “f(n)” is an upper bound for a function’s need for resources.

Big Theta:

A function “g(n)” is Θ(f(n)) iff two positive constants “c₁” and “c₂” and a positive integer "n0" exists such that c1f(n)≤g(n)≤c2f(n) for all n>=n0, that is, “f(n)” denotes exact bound for “g(n)”, which denotes that growth rate of “g(n)” and “f(n)” are equal. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function that is been calculated taking “n” as parameter

Θ(f(n)) means that curve denoted by “f(n)” is an exact bound for a function’s need for resources.

Explanation of Solution

b) f(n) + g(n) is Θ(min(f(n),g(n)))

Explanation:

Take f(n) = n and g(n) = 1/n.

Now, f(n)+ g(n)&

Explanation of Solution

c) 2^na is O(2ⁿ)

Explanation:

The function 2^na is not O(2n), as there is no constant c ≥2na/2n= 2n(a-

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 4

Question

Chapter 2, Problem 6E

Program Plan Intro

Asymptotic Notations:

In asymptotic analysis of algorithms, mathematical tools are used to represent time complexity of algorithm.

such mathematical tools are known as Asymptotic notations. Some of the asymptotic notations are

Big-O.

Big Theta

Big-O:

A function “g(n)” is O(f(n)) if there exists a constant c > 0 such that cf(n) grows faster than “g(n)” for all n>=n0 , that is, “g(n)” is O(f(n)) iff for some constants “c” and "n0", g(n)<=cf(n) for all n>=n0. Here, “f(n)” denotes the upper bound for “g(n)”. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function calculated while taking n as parameter

O(f(n)) means that curve denoted by “f(n)” is an upper bound for a function’s need for resources.

Big Theta:

A function “g(n)” is Θ(f(n)) iff two positive constants “c₁” and “c₂” and a positive integer "n0" exists such that c1f(n)≤g(n)≤c2f(n) for all n>=n0, that is, “f(n)” denotes exact bound for “g(n)”, which denotes that growth rate of “g(n)” and “f(n)” are equal. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function that is been calculated taking “n” as parameter

Θ(f(n)) means that curve denoted by “f(n)” is an exact bound for a function’s need for resources.

Explanation of Solution

b) f(n) + g(n) is Θ(min(f(n),g(n)))

Explanation:

Take f(n) = n and g(n) = 1/n.

Now, f(n)+ g(n)&

Explanation of Solution

c) 2^na is O(2ⁿ)

Explanation:

The function 2^na is not O(2n), as there is no constant c ≥2na/2n= 2n(a-

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 5

Chapter 2, Problem 6E

Program Plan Intro

Asymptotic Notations:

In asymptotic analysis of algorithms, mathematical tools are used to represent time complexity of algorithm.

such mathematical tools are known as Asymptotic notations. Some of the asymptotic notations are

Big-O.

Big Theta

Big-O:

A function “g(n)” is O(f(n)) if there exists a constant c > 0 such that cf(n) grows faster than “g(n)” for all n>=n0 , that is, “g(n)” is O(f(n)) iff for some constants “c” and "n0", g(n)<=cf(n) for all n>=n0. Here, “f(n)” denotes the upper bound for “g(n)”. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function calculated while taking n as parameter

O(f(n)) means that curve denoted by “f(n)” is an upper bound for a function’s need for resources.

Big Theta:

A function “g(n)” is Θ(f(n)) iff two positive constants “c₁” and “c₂” and a positive integer "n0" exists such that c1f(n)≤g(n)≤c2f(n) for all n>=n0, that is, “f(n)” denotes exact bound for “g(n)”, which denotes that growth rate of “g(n)” and “f(n)” are equal. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function that is been calculated taking “n” as parameter

Θ(f(n)) means that curve denoted by “f(n)” is an exact bound for a function’s need for resources.

Explanation of Solution

b) f(n) + g(n) is Θ(min(f(n),g(n)))

Explanation:

Take f(n) = n and g(n) = 1/n.

Now, f(n)+ g(n)&

Explanation of Solution

c) 2^na is O(2ⁿ)

Explanation:

The function 2^na is not O(2n), as there is no constant c ≥2na/2n= 2n(a-

Answer 6

Chapter 2, Problem 6E

Program Plan Intro

Asymptotic Notations:

In asymptotic analysis of algorithms, mathematical tools are used to represent time complexity of algorithm.

such mathematical tools are known as Asymptotic notations. Some of the asymptotic notations are

Big-O.

Big Theta

Big-O:

A function “g(n)” is O(f(n)) if there exists a constant c > 0 such that cf(n) grows faster than “g(n)” for all n>=n0 , that is, “g(n)” is O(f(n)) iff for some constants “c” and "n0", g(n)<=cf(n) for all n>=n0. Here, “f(n)” denotes the upper bound for “g(n)”. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function calculated while taking n as parameter

O(f(n)) means that curve denoted by “f(n)” is an upper bound for a function’s need for resources.

Big Theta:

A function “g(n)” is Θ(f(n)) iff two positive constants “c₁” and “c₂” and a positive integer "n0" exists such that c1f(n)≤g(n)≤c2f(n) for all n>=n0, that is, “f(n)” denotes exact bound for “g(n)”, which denotes that growth rate of “g(n)” and “f(n)” are equal. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function that is been calculated taking “n” as parameter

Θ(f(n)) means that curve denoted by “f(n)” is an exact bound for a function’s need for resources.

Answer 7

Chapter 2, Problem 6E

Program Plan Intro

Asymptotic Notations:

In asymptotic analysis of algorithms, mathematical tools are used to represent time complexity of algorithm.

such mathematical tools are known as Asymptotic notations. Some of the asymptotic notations are

Big-O.

Big Theta

Big-O:

A function “g(n)” is O(f(n)) if there exists a constant c > 0 such that cf(n) grows faster than “g(n)” for all n>=n0 , that is, “g(n)” is O(f(n)) iff for some constants “c” and "n0", g(n)<=cf(n) for all n>=n0. Here, “f(n)” denotes the upper bound for “g(n)”. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function calculated while taking n as parameter

O(f(n)) means that curve denoted by “f(n)” is an upper bound for a function’s need for resources.

Big Theta:

A function “g(n)” is Θ(f(n)) iff two positive constants “c₁” and “c₂” and a positive integer "n0" exists such that c1f(n)≤g(n)≤c2f(n) for all n>=n0, that is, “f(n)” denotes exact bound for “g(n)”, which denotes that growth rate of “g(n)” and “f(n)” are equal. The above statement can be read as follows :

“n” is size of data set

“f(n)” denotes a function that is been calculated taking “n” as parameter

Θ(f(n)) means that curve denoted by “f(n)” is an exact bound for a function’s need for resources.

Answer 8

Explanation of Solution

b) f(n) + g(n) is Θ(min(f(n),g(n)))

Explanation:

Take f(n) = n and g(n) = 1/n.

Now, f(n)+ g(n)&

Answer 9

Explanation of Solution

b) f(n) + g(n) is Θ(min(f(n),g(n)))

Explanation:

Take f(n) = n and g(n) = 1/n.

Now, f(n)+ g(n)&

Answer 10

Explanation of Solution

c) 2^na is O(2ⁿ)

Explanation:

The function 2^na is not O(2n), as there is no constant c ≥2na/2n= 2n(a-

Answer 11

Explanation of Solution

c) 2^na is O(2ⁿ)

Explanation:

The function 2^na is not O(2n), as there is no constant c ≥2na/2n= 2n(a-

Answer 12

Expert Solution & Answer

Answer 13

Expert Solution & Answer

Answer 14

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Answer 15

Ch. 2 - Prob. 1E Ch. 2 - Prob. 2E Ch. 2 - Prob. 3E Ch. 2 - Prob. 4E Ch. 2 - Prob. 5E Ch. 2 - Prob. 6E Ch. 2 - Prob. 7E Ch. 2 - Prob. 8E Ch. 2 - Prob. 9E Ch. 2 - Prob. 10E

Solution Summary: The author explains how mathematical tools are used to represent time complexity of algorithms in asymptotic analysis.

Explanation of Solution

b) f(n) + g(n) is Θ(min(f(n),g(n)))

Explanation:

Take f(n) = n and g(n) = 1/n.

Now, f(n)+ g(n)&

Explanation of Solution

c) 2^na is O(2ⁿ)

Explanation:

The function 2^na is not O(2n), as there is no constant c ≥2na/2n= 2n(a-

Want to see the full answer?

Chapter 2 Solutions

Solution Summary: The author explains how mathematical tools are used to represent time complexity of algorithms in asymptotic analysis.

Explanation of Solution

b) f(n) + g(n) is Θ(min(f(n),g(n))) Explanation: Take f(n) = n and g(n) = 1/n. Now, f(n)+ g(n)&

Explanation of Solution

c) 2na is O(2n) Explanation: The function 2na is not O(2n), as there is no constant c ≥2na/2n= 2n(a-

Want to see the full answer?

Chapter 2 Solutions

b) f(n) + g(n) is Θ(min(f(n),g(n)))

Explanation:

Take f(n) = n and g(n) = 1/n.

Now, f(n)+ g(n)&

c) 2^na is O(2ⁿ)

Explanation:

The function 2^na is not O(2n), as there is no constant c ≥2na/2n= 2n(a-