Question 1

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

Accepted Answer

procedure $\operatorname { avgmaxmin } \left( a _ { 1 } , \ldots , a _ { n } \text { : integers } \right)$
$\text { max } : = a _ { 1 }$
$\min : = a _ { 1 }$
for $i : = 2 \text { to } n$
 if $a _ { i } > \max \text { then } \max : = a _ { i }$
if $a _ { i } < \min \text { then } \min : = a _ { i }$
return $( \max + \min ) / 2$

Question 2

In questions 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 lists all ways to put the numbers 1, 2, 3, . . . , n in a row.

Accepted Answer

$$n !$$

Question 3

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: $$\begin{array} { l } 
t : = 0 \
\text { for } i = 1 \text { to } n \
\quad \text { for } j = 1 \text { to } n \
\quad t : = ( i t + j t + 1 ) ^ { 2 }
\end{array}$$

Accepted Answer

$$O \left( n ^ { 2 } \right)$$

Question 4

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: $$\begin{array} { l } 
t : = 1 \
\text { for } i = n \text { to } n ^ { 2 } \
t : = t + 2 i t
\end{array}$$

Accepted Answer

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

Question 5

Prove that $x ^ { 3 } + 7 x + 2 \text { is } \Omega \left( x ^ { 3 } \right)$

Accepted Answer

The answer of Prove that \(x ^ { 3 }...

Question 6

Prove or disprove that the greedy algorithm for making change always uses the fewest coins possible when the
denominations available are 1-cent coins, 8-cent coins, and 20-cent coins.

Accepted Answer

A) True 
 B)False

Question 7

Arrange the functions $n ^ { 3 / 2 } , \log \left( n ^ { n } \right) , \left( n ^ { 100 } \right) ^ { n }$ 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 8

Prove or disprove that the greedy algorithm for making change always uses the fewest coins possible when the
denominations available are pennies (1-cent coins), nickels (5-cent coins), and quarters (25-cent coins).

Accepted Answer

A) True 
 B)False

Question 9

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

Accepted Answer

The answer of assume that the number of multiplications of...

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 prove...

Question 11

assume that the number of multiplications of entries used to multiply a p × q and a q × r matrix is
pqr.
-Whatand 6 ×is 12,therespbestectivorderely?to form the product ABC if A, B and C are matrices with dimensions 8 × 3, 3 × 6

Accepted Answer

The answer of assume that the number of multiplications of...

Question 12

In questions 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: $$\begin{array} { l } 
i : = 1 , j : = 1 \
\text { while } i \leq n \
\text { while } j \leq i \
\text { print "hello"; } \
j : = j + 1 \
i : = i + 1
\end{array}$$

Accepted Answer

The answer of In questions ﬁnd the "best" big-O notation...

Question 13

Describe an algorithm that takes a list of  n  integers (n ≥ 1) and finds the location of the last even integer in the list. and returns 0 if there aer no even integers in the list .

Accepted Answer

procedure 11ec7c1b_0541_9ce4_bf14_9bb232

Question 14

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

Accepted Answer

The answer of In questions ﬁnd the "best" big-O notation...

Question 15

Express a brute-force algorithm that finds the largest product of two numbers in a list a1, a2, . . . , an (n ≥ 2) that is less than a threshold  N .

Accepted Answer

procedure largestproduct 11ec7c19_e34f_d

Question 16

Describe an algorithm that takes a list of integers  a1, a2, . . . , an (n ≥ 2) and finds the second-largest integer in the sequence by going through the list and keeping track of the largest and second-largest integer encountered.

Accepted Answer

procedure 11ec7c1b_9310_b6ae_bf14_e75e91

Question 17

In questions ﬁnd the "best" big-O notation to describe the complexity of the algorithm. Choose your
answers from the following:
1, $$\log _ { 2 } n , n , n \log _ { 2 } n , n ^ { 2 } , n ^ { 3 } , \ldots , 2 ^ { n } , n !$$
-A linear search to ﬁnd the smallest number in a list of n numbers.

Accepted Answer

The answer of In questions ﬁnd the "best" big-O notation...

Question 18

In questions ﬁnd the best big-O function for the function. Choose your answer from among the following:
1, $$\log _ { 2 } n , n , n \log _ { 2 } n , n ^ { 2 } , n ^ { 3 } , \ldots , 2 ^ { n } , n !$$
-$$f ( n ) = 1 + 4 + 7 + \cdots + ( 3 n + 1 )$$

Accepted Answer

The answer of In questions ﬁnd the best big-O function...

Question 19

Use the definition of big_$O$ to prove that $\frac { 6 n + 4 n ^ { 5 } - 4 } { 7 n ^ { 2 } - 3 }$ is $O \left( n ^ { 3 } \right)$.

Accepted Answer

The answer of Use the definition of big_$O$ to prove...

Question 20

Use the definition of big-$O$ to prove that  1.2+2.3+3.4+...+  $( n - 1 )$ .  $n$  is $O \left( n ^ { 3 } \right)$

Accepted Answer

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

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

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: t:=0 for i=1 to n for j=1 to n t:=(it+jt+1

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: t:=1 for i=n to t:=t+2it

Prove that $x ^ { 3 } + 7 x + 2 \text { is } \Omega \left( x ^ { 3 } \right)$

Prove or disprove that the greedy algorithm for making change always uses the fewest coins possible when the denominations available are 1-cent coins, 8-cent coins, and 20-cent coins.

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

Prove or disprove that the greedy algorithm for making change always uses the fewest coins possible when the denominations available are pennies (1-cent coins), nickels (5-cent coins), and quarters (25-cent coins).

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

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

assume that the number of multiplications of entries used to multiply a p × q and a q × r matrix is pqr. -Whatand 6 ×is 12,therespbestectivorderely?to form the product ABC if A, B and C are matrices with dimensions 8 × 3, 3 × 6

Describe an algorithm that takes a list of n integers (n ≥ 1) and finds the location of the last even integer in the list. and returns 0 if there aer no even integers in the list .

Express a brute-force algorithm that finds the largest product of two numbers in a list a1, a2, . . . , an (n ≥ 2) that is less than a threshold N .

Describe an algorithm that takes a list of integers a1, a2, . . . , an (n ≥ 2) and finds the second-largest integer in the sequence by going through the list and keeping track of the largest and second-largest integer encountered.

In questions ﬁnd the "best" big-O notation to describe the complexity of the algorithm. Choose your answers from the following: 1, $\log _ { 2 } n , n , n \log _ { 2 } n , n ^ { 2 } , n ^ { 3 } , \ldots , 2 ^ { n } , n !$ -A linear search to ﬁnd the smallest number in a list of n numbers.

In questions ﬁnd the best big-O function for the function. Choose your answer from among the following: 1, $\log _ { 2 } n , n , n \log _ { 2 } n , n ^ { 2 } , n ^ { 3 } , \ldots , 2 ^ { n } , n !$ - $f ( n ) = 1 + 4 + 7 + \cdots + ( 3 n + 1 )$

Use the definition of big_ $O$ to prove that $\frac { 6 n + 4 n ^ { 5 } - 4 } { 7 n ^ { 2 } - 3 }$ is $O \left( n ^ { 3 } \right)$ .

Use the definition of big- $O$ to prove that 1.2+2.3+3.4+...+ $( n - 1 )$ . $n$ is $O \left( n ^ { 3 } \right)$

The Foundations: Logic and Proofs

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

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Advanced Counting Techniques

Relations

Graphs

Trees

Boolean Algebra

Modeling Computation

Filters

Exam 3: Algorithms

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

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: t:=0 for i=1 to n for j=1 to n t:=(it+jt+1

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: t:=1 for i=n to t:=t+2it

Prove that x3+7x+2 is Ω(x3)x ^ { 3 } + 7 x + 2 \text { is } \Omega \left( x ^ { 3 } \right)x3+7x+2 is Ω(x3)

Prove or disprove that the greedy algorithm for making change always uses the fewest coins possible when the denominations available are 1-cent coins, 8-cent coins, and 20-cent coins.

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

Prove or disprove that the greedy algorithm for making change always uses the fewest coins possible when the denominations available are pennies (1-cent coins), nickels (5-cent coins), and quarters (25-cent coins).

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

Use the definition of big- OOO 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)

assume that the number of multiplications of entries used to multiply a p × q and a q × r matrix is pqr. -Whatand 6 ×is 12,therespbestectivorderely?to form the product ABC if A, B and C are matrices with dimensions 8 × 3, 3 × 6

Describe an algorithm that takes a list of n integers (n ≥ 1) and finds the location of the last even integer in the list. and returns 0 if there aer no even integers in the list .

Express a brute-force algorithm that finds the largest product of two numbers in a list a1, a2, . . . , an (n ≥ 2) that is less than a threshold N .

Describe an algorithm that takes a list of integers a1, a2, . . . , an (n ≥ 2) and finds the second-largest integer in the sequence by going through the list and keeping track of the largest and second-largest integer encountered.

Use the definition of big_ OOO to prove that 6n+4n5−47n2−3\frac { 6 n + 4 n ^ { 5 } - 4 } { 7 n ^ { 2 } - 3 }7n2−36n+4n5−4​ is O(n3)O \left( n ^ { 3 } \right)O(n3) .

Use the definition of big- OOO to prove that 1.2+2.3+3.4+...+ (n−1)( n - 1 )(n−1) . nnn is O(n3)O \left( n ^ { 3 } \right)O(n3)

The Foundations: Logic and Proofs

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

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Advanced Counting Techniques

Relations

Graphs

Trees

Boolean Algebra

Modeling Computation

Filters

Prove that $x ^ { 3 } + 7 x + 2 \text { is } \Omega \left( x ^ { 3 } \right)$

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

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

Use the definition of big_ $O$ to prove that $\frac { 6 n + 4 n ^ { 5 } - 4 } { 7 n ^ { 2 } - 3 }$ is $O \left( n ^ { 3 } \right)$ .

Use the definition of big- $O$ to prove that 1.2+2.3+3.4+...+ $( n - 1 )$ . $n$ is $O \left( n ^ { 3 } \right)$