Chapter 3 Algorithms

Question Mode Multiple Select Question Select all that apply Which of the following are true statements?

(log2 n)^7 is O(n^8). n^5 is O(n^7). n^8 is O(3^n).

Question Mode Multiple Choice Question Sort into increasing order the elements in the list 8, -3, 5, 1, 3, 7, 6, 4.

-3, 1, 3, 4, 5, 6, 7, 8

Multiple Choice Question What change is made by the cashier's algorithm for 40 cents using quarters, dimes, and pennies (but no nickels)?

1 quarter, 1 dime, and 5 pennies

Question Mode Multiple Select Question Select all that apply For which sets of coin values does the greedy change-making algorithm always make change using the least number of coins? (Select all that apply.)

1, 5, 10, 25 1, 5, 25 1, 5, 10, 25, 50, 100

Question Mode Ordering Question Click and drag on elements in order List these functions so that each function in the list is big-O of the function below it.

1. 1,000,000 2. (log x)^2 3. 107x +108 4. x^3 +x^2 +17 5. 2^x 6. 3^x

Question Mode Ordering Question Click and drag on elements in order Order the following steps of a trace of the binary search algorithm used to search for 13 in the list 10, 11, 12, 13, 14.

1. Compare 13 to 12 and set I to 4. 2. Compare 4 to 5 and set m to 4. 3. Compare 13 to 13 and set j to 4. 4. Compare 4 to 4. Then compare 13 to 13, set location to 4, and report that 13 is at location 4.

Order the following steps of a trace of the linear search algorithm used to search for 13 in the list 14, 13, 12.

1. Set i to 1 2. Compare 1 with 3 and 13 to 14 3. Set i to 2. 4. Compare 2 with 3 and 13 to 13 5. Compare 2 with 3 and set location to 2. 6. Report that 13 is at location 2.

Question Mode Ordering Question Click and drag on elements in order Rank the following functions so that each function is big-O of the one above it, that is rank them in descending order.

1. n! 2. 2^n 3. n^2 4. n log n 5. n 6. log n

The number of character comparisons used by the naive string matcher to look for the pattern is in the text Mississippi is

13

Question Mode Multiple Choice Question What change is made by the cashier's algorithm for 66 cents using quarters, dimes, nickels, and pennies?

2 quarters, 1 dime, 1 nickel, and 1 penny

What is the list at the end of the first pass of the bubble sort algorithm when given the list 4, 2, 6, 7, 5 as input.

2, 4, 6, 5, 7

Which of the following is the sequence at the end of the second pass when the insertion sort algorithm is used to sort the list 4, 2, 6, 7, 5?

2, 4, 6, 7, 5

The number of character comparisons used by the naive string matcher to look for the pattern of in the text love is

4

Question Mode Multiple Select Question Select all that apply Which of the following shifts s are found by the naive string matcher in searching for the pattern is in the text Mississippi?

4 1

Multiple Choice Question Using the greedy algorithm for scheduling talks, find the greatest number of talks that can be scheduled if the starting and ending times of the possible talks are 8:00-8:50, 8:30-9:00, 8:55-9:45, 9:15-9:55, 10:00-10:30, 10:15-10:25, 10:30-10:55, 11:00-11:35, 10:55-11:25, 10:45-11:30.

5

Multiple Select Question Select all that apply Which of the following statements apply to greedy algorithms?

A proof is required to show a particular greedy algorithm always produces an optimal solution. They solve optimization problems. To show that a greedy algorithm does not always produce an optimal solution, a counterexample is sufficient. They select the best choice at each step according to some criteria.

Question Mode Multiple Choice Question Use bubble sort to sort 4, 3, 5, 2, 6, showing the list obtained at each step.

After the first pass 3, 4, 2, 5, 6 After the second pass 3, 2, 4, 5, 6 After the third pass 2, 3, 4, 5, 6 After the fourth pass 2, 3, 4, 5, 6

Question Mode Multiple Select Question Select all that apply Which of the statements below are true about the halting problem? Recall that the halting problem asks whether there can be a procedure that takes as input any computer program and any input to this program and determines whether when given this input the program will stop. (Select all that apply.)

Alan Turing was able to resolve the halting problem. You can determine that no such procedure exists using a proof by contradiction.

Multiple Select Question Select all that apply Which pair of witnesses can be used to show that 10x is O(x2)?

C = 1 and k = 10 C = 10 and k = 1 C = 15 and k = 10

Question Mode Multiple Select Question Select all that apply Which pairs of numbers C, k are witnesses to the fact that x2 + 3x is O(x2)?

C = 4, k = 1 C = 4, k = 2 C = 5, k = 1

Matching Question Match the properties of an algorithm in the left column with their corresponding descriptions on the right.

Correctness : Produces the right output values for each set of input values Definiteness: Steps are precisely defined Finiteness: Always produces the output after a finite number of steps Effectiveness: Each step can be performed exactly and in a finite amount of time Generality: Is applicable for all problems of the desired form, not just for a particular set of input values Input: Operates on values from a specified set Output: A solution of the proper form is produced for each set of input values

Question Mode Multiple Choice Question Which of these rules for selecting a talk from those compatible with talks already scheduled defines a greedy algorithm that always yields the most talks from a given list of possible talks?

Earliest finish time

Which of these statements about the binary search algorithm are true?

For all search terms and all sequences, the binary search algorithm continues splitting the sequence until it only contains one term. The number of comparisons performed by the algorithm when given a search term that is in the sequence is independent of the search term. Given a sequence, each integer from 1 to the size of the sequence can be the output of the binary search algorithm for some search term.

Question Mode Multiple Select Question Select all that apply Which of these statements are true about the bubble sort algorithm as specified in the text.

For some input, the algorithm performs exactly one interchange. For some input, the algorithm does not perform any interchanges. The bubble sort algorithm's first pass always makes the same number of comparisons for lists of the same size.

Question Mode Multiple Select Question Select all that apply Which of these statements are true? (Select all that apply.)

If f1(x) is O(g1(x)) and f2(x) is O(g2(x)) then, (f1 + f2)(x) is O(max(|g1(x)|, |g2(x)|)). If f1(x) is O(g(x)) and f2(x) is O(g(x)) then, (f1 f2)(x) is O(g(x)2). If f1(x) is O(g(x)) and f2(x) is O(g(x)) then, (f1 + f2)(x) is O(g(x)).

Question Mode Multiple Select Question Select all that apply Which of these statements are true about the insertion sort algorithm as specified in the book.

Linear search is used to insert the next element on each pass. For all sequences of the same size, the algorithm always uses the same number of passes. After inserting the jth element, the first j elements are in order.

Question Mode Multiple Choice Question Using the greedy change-making algorithm, determine the coins required in making change for 46 cents, using quarters, dimes, nickels, and pennies.

One quarter, two dimes, and a penny

Which of these statements about the binary search algorithm are true?

The number of comparisons performed by the algorithm when given a search term that is in the sequence is independent of the search term. Given a sequence, each integer from 1 to the size of the sequence can be the output of the binary search algorithm for some search term. For all search terms and all sequences, the binary search algorithm continues splitting the sequence until it only contains one term.

Question Mode Multiple Choice Question Using the greedy change-making algorithm, determine the minimum number of coins required to make change for 36 cents, using quarters, dimes, nickels, and pennies.

Three coins

Multiple Choice Question What steps need to be done to show that the algorithm for finding the maximum element in a finite sequence is correct?

We must show that when the algorithm terminates, the value of the variable max equals the maximum term of the sequence.

Question Mode Multiple Choice Question Sort the elements in the list k, m, h, g, e, a, b, d, so that they are in increasing alphabetical order.

a, b, d, e, g, h, k, m

Question Mode Multiple Select Question Select all that apply Which of the following are correct statements?

f(n) = (n + 1) log(n2 + 1) + 3n2 is O(n2). f(n) = 3n log(n!) + (n2 + 3) log n is O(n2 log n).

Question Mode Multiple Select Question Select all that apply Which of the following are correct statements?

f(n) = 2n2 + 8 log n is Θ(n2) f(n) = ∑ni=1i is Θ(n2)

Multiple Select Question Select all that apply Which of these statements are true?

f(x) = 17x^2 is O(16x^2). f(x) = 1 is O(x). f(x) = 40x^3 is O(x^4).

Question Mode Multiple Select Question Select all that apply Which of these statements are true?

f(x) = 40x^3 is O(x^4). f(x) = 17x^2 is O(16x^2). f(x) = 1 is O(x).

Question Mode Multiple Select Question Select all that apply Define f(x) = x2 + 2x. Select all of the functions g(x) below for which f(x) is Ω(g(x))?

g(x) = x log x g(x) = x3 g(x) = 2x g(x) = x2

Question Mode Multiple Select Question Select all that apply Which of these statements are true?

n3^ is not O(n^2). 3x^4 + 1000x is O(x^4).

Multiple Choice Question Let f and g be functions from the set of real numbers to the set of real numbers, then f(x) is O(g(x)) if there are constants C and k such that

|f(x)| ≤ C|g(x)| whenever x > k

Question Mode Multiple Choice Question Which of these inequalities correctly completes this definition? If f and g are functions from the set of integers or the set of real numbers to the set of real numbers, then f(x) is Ω(g(x)) if there are positive constants C and k such that _____ whenever x > k.

|f(x)| ≥ C|g(x)|