Question 1

Prove that $$5 x ^ { 4 } + 2 x ^ { 3 } - 1 \text { is } \Theta \left( x ^ { 4 } \right)$$

Accepted Answer

$5 x ^ { 4 } + 2 x ^ { 3 } - 1$ is $O \left( x ^ { 4 } \right)$ since $\left| 5 x ^ { 4 } + 2 x ^ { 3 } - 1 \right| \leq \left| 5 x ^ { 4 } + 2 x ^ { 4 } \right| \leq 7 \left| x ^ { 4 } \right|$ (if $\left. x \geq 1 \right)$. Also, $x ^ { 4 }$ is $O \left( 5 x ^ { 4 } + 2 x ^ { 3 } - 1 \right.$ ) since $\left| x ^ { 4 } \right| \leq \left| 5 x ^ { 4 } + x ^ { 3 } \right| \leq \left| 5 x ^ { 4 } + 2 x ^ { 3 } - 1 \right|$ (if $x \geq 1$ ).

Question 2

assume that the number of multiplications of entries used to multiply a p × q and a q × r matrix is pqr.
-What is the best order to form the product ABC if A, B and C are matrices with dimensions 2 × 5, 5 × 7 and 7 × 3, respectively?

Accepted Answer

(AB)C uses 2 · 5 · 7 + 2 · 7 · 3 = 112 multiplications, fewer than A(BC), which uses 5 · 7 · 3 + 2 · 5 · 3 = 135.

Question 3

ﬁnd the "best" big-O notation to describe the complexity of the algorithm. Choose your answers from the following: $$1 , \log _ { 2 } n , n , n \log _ { 2 } n , n ^ { 2 } , n ^ { 3 } , \ldots , 2 ^ { n } , n !$$
-The worst-case analysis of a linear search of a list of size n (counting the number of comparisons)

Accepted Answer

n

Question 4

ﬁnd the best big-O function for the function. Choose your answer from among the following: $\text { 1, } \log _ { 2 } n , n , n \log _ { 2 } n , n ^ { 2 } , n ^ { 3 } , \ldots , 2 ^ { n } , n !$
-$$\frac { 3 - 2 n ^ { 4 } - 4 n } { 2 n ^ { 3 } - 3 n }$$

Accepted Answer

The answer of ﬁnd the best big-O function for the...

Question 5

Describe an algorithm that takes a list of n integers a1, a2, . . . , an and ﬁnds the number of integers each greater than ﬁve in the list.

Accepted Answer

procedure greatertha

Question 6

ﬁnd the "best" big-O notation to describe the complexity of the algorithm. Choose your answers from the following: $$1 , \log _ { 2 } n , n , n \log _ { 2 } n , n ^ { 2 } , n ^ { 3 } , \ldots , 2 ^ { n } , n !$$
-The number of print statements in the following:

Accepted Answer

The answer of ﬁnd the "best" big-O notation to describe...

Question 7

List all the steps that the naive string matcher uses to match the pattern xy in the text yxyxxy.

Accepted Answer

The steps are listed, delimited by semic

Question 8

Use the deﬁnition of big-O to prove that $$1 ^ { 3 } + 2 ^ { 3 } + \cdots + n ^ { 3 } \text { is } O \left( n ^ { 4 } \right) \text {. }$$

Accepted Answer

The answer of Use the deﬁnition of big-O to prove...

Question 9

ﬁnd the "best" big-O notation to describe the complexity of the algorithm. Choose your answers from the following: $$1 , \log _ { 2 } n , n , n \log _ { 2 } n , n ^ { 2 } , n ^ { 3 } , \ldots , 2 ^ { n } , n !$$
-An iterative algorithm to compute n!, (counting the number of multiplications)

Accepted Answer

The answer of ﬁnd the "best" big-O notation to describe...

Question 10

Use the definition of big-O  to prove that $\frac { 3 n - 8 - 4 n ^ { 3 } } { 2 n - 1 }$ is $O \left( n ^ { 2 } \right)$

Accepted Answer

The answer of Use the definition of big-O  to...

Question 11

Suppose you have two diﬀerent algorithms for solving a problem. To solve a problem of size n, the ﬁrst algorithm uses exactly $n \sqrt { n }$ operations and the second algorithm uses exactly $n ^ { 2 } \log n$ operations. As n grows, which algorithm uses fewer operations?

Accepted Answer

The ﬁrst algorithm u

Question 12

List all the steps the binary search algorithm uses to search for 27 in the following list: 5, 6, 8, 12, 15, 21, 25, 31.

Accepted Answer

The consecutive choices of sub

Question 13

Use the deﬁnition of big-O to prove that $$1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \cdots + ( n - 1 ) \cdot n \text { is } O \left( n ^ { 3 } \right)$$

Accepted Answer

The answer of Use the deﬁnition of big-O to prove...

Question 14

Describe an algorithm that takes a list of n integers $( n \geq 1 )$and ﬁnds the average of the largest and smallest

Accepted Answer

procedure avgmaxmin

Question 15

ﬁnd the "best" big-O notation to describe the complexity of the algorithm. Choose your answers from the following: $$1 , \log _ { 2 } n , n , n \log _ { 2 } n , n ^ { 2 } , n ^ { 3 } , \ldots , 2 ^ { n } , n !$$
-An algorithm that ﬁnds the average of n numbers by adding them and dividing by n

Accepted Answer

The answer of ﬁnd the "best" big-O notation to describe...

Question 16

assume that the number of multiplications of entries used to multiply a p × q and a q × r matrix is pqr.
-What is the most eﬃcient way to multiply the matrices A1, A2, A3 of sizes 10 × 50, 50 × 10, 10 × 40?

Accepted Answer

(A₁A₂)A₃, 900

Question 17

Give a big-O estimate for the number of operations (where an operation is an addition or a multiplication) used in this segment of an algorithm:

Accepted Answer

The answer of Give a big-O estimate for the number...

Question 18

ﬁnd the "best" big-O notation to describe the complexity of the algorithm. Choose your answers from the following: $$1 , \log _ { 2 } n , n , n \log _ { 2 } n , n ^ { 2 } , n ^ { 3 } , \ldots , 2 ^ { n } , n !$$
-An algorithm that prints all bit strings of length n.

Accepted Answer

The answer of ﬁnd the "best" big-O notation to describe...

Question 19

Arrange the functions $n ^ { 3 / 2 } , \log \left( n ^ { n } \right) , \left( n ^ { 100 } \right) ^ { n } \text { and } \log ( n ! )$ in a list so that each function is big-O of the next function.

Accepted Answer

The answer of Arrange the functions \(n ^ { 3...

Question 20

List all the steps that insertion sort uses to sort 8, 20, 13, 16, 9

Accepted Answer

The lists after each insertion

Exam 3: A: Algorithms

Prove that 5x4+2x3−1 is Θ(x4)5 x ^ { 4 } + 2 x ^ { 3 } - 1 \text { is } \Theta \left( x ^ { 4 } \right)5x4+2x3−1 is Θ(x4)

assume that the number of multiplications of entries used to multiply a p × q and a q × r matrix is pqr. -What is the best order to form the product ABC if A, B and C are matrices with dimensions 2 × 5, 5 × 7 and 7 × 3, respectively?

Describe an algorithm that takes a list of n integers a1, a2, . . . , an and ﬁnds the number of integers each greater than ﬁve in the list.

List all the steps that the naive string matcher uses to match the pattern xy in the text yxyxxy.

Use the deﬁnition of big-O to prove that 13+23+⋯+n3 is O(n4). 1 ^ { 3 } + 2 ^ { 3 } + \cdots + n ^ { 3 } \text { is } O \left( n ^ { 4 } \right) \text {. }13+23+⋯+n3 is O(n4).

Use the definition of big-O to prove that 3n−8−4n32n−1\frac { 3 n - 8 - 4 n ^ { 3 } } { 2 n - 1 }2n−13n−8−4n3​ is O(n2)O \left( n ^ { 2 } \right)O(n2)

Suppose you have two diﬀerent algorithms for solving a problem. To solve a problem of size n, the ﬁrst algorithm uses exactly nnn \sqrt { n }nn​ operations and the second algorithm uses exactly n2log⁡nn ^ { 2 } \log nn2logn operations. As n grows, which algorithm uses fewer operations?

List all the steps the binary search algorithm uses to search for 27 in the following list: 5, 6, 8, 12, 15, 21, 25, 31.

Use the deﬁnition of big-O to prove that 1⋅2+2⋅3+3⋅4+⋯+(n−1)⋅n is O(n3)1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \cdots + ( n - 1 ) \cdot n \text { is } O \left( n ^ { 3 } \right)1⋅2+2⋅3+3⋅4+⋯+(n−1)⋅n is O(n3)

Describe an algorithm that takes a list of n integers (n≥1)( n \geq 1 )(n≥1) and ﬁnds the average of the largest and smallest

assume that the number of multiplications of entries used to multiply a p × q and a q × r matrix is pqr. -What is the most eﬃcient way to multiply the matrices A1, A2, A3 of sizes 10 × 50, 50 × 10, 10 × 40?

Give a big-O estimate for the number of operations (where an operation is an addition or a multiplication) used in this segment of an algorithm:

Arrange the functions n3/2,log⁡(nn),(n100)n and log⁡(n!)n ^ { 3 / 2 } , \log \left( n ^ { n } \right) , \left( n ^ { 100 } \right) ^ { n } \text { and } \log ( n ! )n3/2,log(nn),(n100)n and log(n!) in a list so that each function is big-O of the next function.

List all the steps that insertion sort uses to sort 8, 20, 13, 16, 9

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

Prove that $5 x ^ { 4 } + 2 x ^ { 3 } - 1 \text { is } \Theta \left( x ^ { 4 } \right)$

Describe an algorithm that takes a list of n integers a₁, a₂, . . . , a_n and ﬁnds the number of integers each greater than ﬁve in the list.

Use the deﬁnition of big-O to prove that $1 ^ { 3 } + 2 ^ { 3 } + \cdots + n ^ { 3 } \text { is } O \left( n ^ { 4 } \right) \text {. }$

Use the definition of big-O to prove that $\frac { 3 n - 8 - 4 n ^ { 3 } } { 2 n - 1 }$ is $O \left( n ^ { 2 } \right)$

Suppose you have two diﬀerent algorithms for solving a problem. To solve a problem of size n, the ﬁrst algorithm uses exactly $n \sqrt { n }$ operations and the second algorithm uses exactly $n ^ { 2 } \log n$ operations. As n grows, which algorithm uses fewer operations?

Use the deﬁnition of big-O to prove that $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \cdots + ( n - 1 ) \cdot n \text { is } O \left( n ^ { 3 } \right)$

Describe an algorithm that takes a list of n integers $( n \geq 1 )$ and ﬁnds the average of the largest and smallest

assume that the number of multiplications of entries used to multiply a p × q and a q × r matrix is pqr. -What is the most eﬃcient way to multiply the matrices A₁, A₂, A₃ of sizes 10 × 50, 50 × 10, 10 × 40?

Arrange the functions $n ^ { 3 / 2 } , \log \left( n ^ { n } \right) , \left( n ^ { 100 } \right) ^ { n } \text { and } \log ( n ! )$ in a list so that each function is big-O of the next function.