Exam 3: Algorithms

Find the best $\text { big- } O \text { function for } n ^ { 3 } + \sin n ^ { 7 } \text {. }$

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: while $n > 1$ print "hello"; $n : = \lfloor n / 2 \rfloor$

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

In questions find the best big-O function for the function. Choose your answer from among the following: 1, $\log _ { 2 } n , n , n \log _ { 2 } n , n ^ { 2 } , n ^ { 3 } , \ldots , 2 ^ { n } , n !$ - $3 n ^ { 4 } + \log _ { 2 } n ^ { 8 }$

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 $\mathbf { A } _ { 1 } , \mathbf { A } _ { 2 } , \mathbf { A } _ { 3 } \text { of sizes } 10 \times 50,50 \times 10,10 \times 40 \text { ? }$

Prove that $\frac { x ^ { 3 } + 7 x ^ { 2 } + 3 } { 2 x + 1 }$ is $\Theta \left( x ^ { 2 } \right)$ .

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

Show how the binary search algorithm searches for 27 in the following list: 5 6 8 12 15 21 25 31.

Let $f ( n ) = 3 n ^ { 2 } + 8 n + 7$ . Show that f(n) is $O \left( n ^ { 2 } \right)$ .find C and k from the definition.

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 binary search of n elements.

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 finds the average of n numbers by adding them and dividing by n.

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 worst-case analysis of a linear search of a list of size n (counting the number of comparisons).

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?

Arrange the following functions in a list so each is big- $O$ of the next one in the list: $n ^ { 3 } + 88 n ^ { 2 } + 3 , \log n ^ { 4 } , 3 ^ { n }$ $n ^ { 2 } \log n , n \cdot 2 ^ { n } , 10000$

Show that $\sum _ { j = 1 } ^ { n } \left( j ^ { 3 } + j \right) \text { is } O \left( n ^ { 4 } \right)$

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.

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 prints all bit strings of length n.

In questions find the best big-O function for the function. Choose your answer from among the following: 1, $\log _ { 2 } n , n , n \log _ { 2 } n , n ^ { 2 } , n ^ { 3 } , \ldots , 2 ^ { n } , n !$ - $\lceil n + 2 \rceil \cdot \lceil n / 3 \rceil$

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

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