Algorithms 1 - M4

None of these answers

Select the best answer. At each call, Merge divides the problem in ______ subproblems. 5 3 None of these answers 4 2

2

Select the best answer. At each call, Merge-Sort divides the problem in ______ subproblems. 8 2 16 4 None of these answers

log x - log 2

Select the best answer. log (x/2) = _________________. Unless specified otherwise, log(n) is log base 2. log x - log 2 None of these answers log x / log 2 (log x ) / 2 log x + log 2

log x - log 4

Select the best answer. log (x/4) = _________________. Unless specified otherwise, log(n) is log base 2. log x + log 4 None of these answers log x - log 4 (log x ) / 4 log x / log 4

(log_b⁡(x))/(log_b(a))

Select the best answer. log_a(⁡x)= __________ where log_a is the logarithm base a. Recall than ln(x) is the natural logarithm and lg or log are logarithm base 2. [] ln⁡(x) [] (ln⁡(a))/(ln⁡(x)) [] None of these answers [] ln⁡(x/a) [] (log_b⁡(x))/(log_b(a))

(ln⁡(x))/(ln⁡(a))

Select the best answer. log_a(⁡x)= __________ where log_a is the logarithm base a. Recall that ln(x) is the natural logarithm and lg or log are logarithm base 2. [] (ln⁡(x))/(ln⁡(a)) [] ln⁡(x/a) [] None of these answers [] ln⁡(x) [] ln⁡(x)/lg⁡(x)

3

The divide-and-conquer involves _______ step(s) 4 None of these answers 3 1 2

16, 4, n, 1

We plan to use the Master Theorem to solve this recurrence relation: T(n) = 16T(n/4) + n. Using the Master theorem below, identify/match the parameters a, b, f(n), and the case that applies. Based on the Master Theorem, a = ____, b= ______, f(n) = _____, and the case that applies is _______(Provide only the case number 1, 2, or 3. If no case applies, provide the number 7). If a number is a fraction x/y, enter "x/y". Do not try to compute x/y.

e. None of these answers

We plan to use the Master Theorem to solve this recurrence relation: T(n) = 2T(n/2) + n log n. Using the Master theorem below, select the solution for the above recurrence relation. a. Theta(n log_2 n) b. Theta(n^2 log n) c. Theta(n^2) d. Theta(n) e. None of these answers

e. None of these answers

We plan to use the Master Theorem to solve this recurrence relation: T(n) = 2^nT(n/2) + n^n. Using the Master theorem below, select the solution for the above recurrence relation. a. Theta(n^2 log n) b. Theta(2^n) c. Theta(n^2) d. Theta(n) e. None of these answers

4, 2, cn, 1

We plan to use the Master Theorem to solve this recurrence relation: T(n) = 4T(n/2) + cn. Using the Master theorem below, identify/match the parameters a, b, f(n), and the case that applies. Based on the Master Theorem, a = ____, b= ______, f(n) = _____, and the case that applies is _______(Provide only the case number 1, 2, or 3. If no case applies, provide the number 7). If a number is a fraction x/y, enter "x/y". Do not try to compute x/y.

b. Theta(n^2)

We plan to use the Master Theorem to solve this recurrence relation: T(n) = 4T(n/2) + n / log n. Using the Master theorem below, select the solution for the above recurrence relation. a. Theta(log_2 n) b. Theta(n^2) c. Theta(n log_2 n) d. Theta(n^2 log_2 n) e. None of these answers

7, 2, n^2, 1

We plan to use the Master Theorem to solve this recurrence relation: T(n) = 7T(n/2) + n^2. Using the Master theorem below, identify/match the parameters a, b, f(n), and the case that applies. Based on the Master Theorem, a = _____, b= _____, f(n) = _____, and the case that applies is _______(Provide only the case number 1, 2, or 3. If no case applies, provide the number 7). If a number is a fraction x/y, enter "x/y". Do not try to compute x/y.

b. Theta(n^2)

We plan to use the Master Theorem to solve this recurrence relation: T(n) = 7T(n/3) + n^2 . Using the Master theorem below, select the solution for the above recurrence relation. a. Theta(log_2 (n)) b. Theta(n^2) c. Theta(n log_2 (n)) d. Theta(n^2 log_2 n) e. None of these answers

4

log_10(10^4) = ____ 10,000 9.21 3 None of these answers 4

5

log_2(32) = ____

5

log_b(b^5) = 5 None of these answers 5.b 5^b b

b

log_m(m^b) = ____

2

At each call, Merge-Sort divides the problem in ______ subproblems.

2

At each call, Merge-Sort divides the sequence in _______ sequences. 5 4 2 3 None of these answers

[] It calls Merge [] It divides the sequence in two subsequences at each step. [] It calls Merge-Sort [] It is recursive

Check all that apply. Check all that apply to the Merge-Sort algorithm. [] It calls Merge [] It divides the sequence in two subsequences at each step. [] It runs in O(n) [] It calls Merge-Sort [] It is recursive

Merge Sort Strassen's

Check all that apply. These algorithms use a divide-and-conquer strategy: [] Merge Sort [] Naïve sorting algorithm presented in previous modules [] Strassen's [] Insertion Sort

f1(n), f3(n), f4(n)

Consider the Merge procedure below. The running time of Merge belongs to f1(n), f2(n), f3(n), and/or f4(n) as defined below: f1(n) = theta(n) f2(n) = big O(log(n)) f3(n) = omega(n) f4(n) = big O(n^2)

f1(n), f2(n), f3(n)

Consider the Merge procedure below. The running time of Merge belongs to f1(n), f2(n), f3(n), and/or f4(n) as defined below: f1(n) = theta(n) f2(n) = omega(log log(n)) f3(n) = omega(log (n)) f4(n) = big O(log(n))

16

Consider the Merge procedure. MERGE(A, p, q, r) 1 n1 = q - p + 1 2 n2 = r - q 3 let L[1...n1 + 1] and R[1...n2 + 1] be new arrays 4 for i = 1 to n1 5 L[i] = A[p + i - 1] 6 for j = 1 to n2 7 R[j] = A[q + j] 8 L[n1 + 1] = infinity 9 R[n2 + 1] = infinity 10 i = 1 11 j = 1 12 for k = p to r 13 if L[i] R[j] 14 A[k] = L[i] 15 i = i + 1 16 else A[k] = R[j] 17 j = j + 1 If we call Merge(A, 10, 19,25), the body of the for-loop at Line 12 will run ____ times.

10

Consider the Merge procedure. MERGE(A, p, q, r) 1 n1 = q - p + 1 2 n2 = r - q 3 let L[1...n1 + 1] and R[1...n2 + 1] be new arrays 4 for i = 1 to n1 5 L[i] = A[p + i - 1] 6 for j = 1 to n2 7 R[j] = A[q + j] 8 L[n1 + 1] = infinity 9 R[n2 + 1] = infinity 10 i = 1 11 j = 1 12 for k = p to r 13 if L[i] R[j] 14 A[k] = L[i] 15 i = i + 1 16 else A[k] = R[j] 17 j = j + 1 If we call Merge(A, 10, 19,25), the body of the for-loop at Line 4 will run ____ times.

6

Consider the Merge procedure. MERGE(A, p, q, r) 1 n1 = q - p + 1 2 n2 = r - q 3 let L[1...n1 + 1] and R[1...n2 + 1] be new arrays 4 for i = 1 to n1 5 L[i] = A[p + i - 1] 6 for j = 1 to n2 7 R[j] = A[q + j] 8 L[n1 + 1] = infinity 9 R[n2 + 1] = infinity 10 i = 1 11 j = 1 12 for k = p to r 13 if L[i] R[j] 14 A[k] = L[i] 15 i = i + 1 16 else A[k] = R[j] 17 j = j + 1 If we call Merge(A, 10, 19,25), the body of the for-loop at Line 6 will run ____ times.

11

Consider the Merge procedure. MERGE(A, p, q, r) 1 n1 = q - p + 1 2 n2 = r - q 3 let L[1...n1 + 1] and R[1...n2 + 1] be new arrays 4 for i = 1 to n1 5 L[i] = A[p + i - 1] 6 for j = 1 to n2 7 R[j] = A[q + j] 8 L[n1 + 1] = infinity 9 R[n2 + 1] = infinity 10 i = 1 11 j = 1 12 for k = p to r 13 if L[i] R[j] 14 A[k] = L[i] 15 i = i + 1 16 else A[k] = R[j] 17 j = j + 1 If we call Merge(A, 10, 19,25), the left subsequence L has __________ elements.

7

Consider the Merge procedure. MERGE(A, p, q, r) 1 n1 = q - p + 1 2 n2 = r - q 3 let L[1...n1 + 1] and R[1...n2 + 1] be new arrays 4 for i = 1 to n1 5 L[i] = A[p + i - 1] 6 for j = 1 to n2 7 R[j] = A[q + j] 8 L[n1 + 1] = infinity 9 R[n2 + 1] = infinity 10 i = 1 11 j = 1 12 for k = p to r 13 if L[i] R[j] 14 A[k] = L[i] 15 i = i + 1 16 else A[k] = R[j] 17 j = j + 1 If we call Merge(A, 10, 19,25), the right subsequence R has __________ elements.

L, A, R

Consider the Merge procedure. MERGE(A, p, q, r) 1 n1 = q - p + 1 2 n2 = r - q 3 let L[1...n1 + 1] and R[1...n2 + 1] be new arrays 4 for i = 1 to n1 5 L[i] = A[p + i - 1] 6 for j = 1 to n2 7 R[j] = A[q + j] 8 L[n1 + 1] = infinity 9 R[n2 + 1] = infinity 10 i = 1 11 j = 1 12 for k = p to r 13 if L[i] R[j] 14 A[k] = L[i] 15 i = i + 1 16 else A[k] = R[j] 17 j = j + 1 Merge will modify/initialize these sequences. n1 n2 L A R

T(n) = 2 T(n/2) + 5.n

Consider the code of Merge-Sort Merge-Sort(A, p, r) if (p < r) q = floor((p+q)/2) Merge-Sort(A, p, q) Merge-Sort(A, q+1, r) Merge(A, p, q, r) If T(n) is the running time of Merge-Sort for an n size input, then the relation recurrence is for n > 1 is _________________ if the running time for Merge is 5.n. None of these answers T(n) = T(n/2) + c.n T(n) = 2 T(n/2) + 5.n T(n) = 2 T(n/2) + c.n T(n) = T(n/2) + 5.n

[] T(n) = 2 T(n/2) + 5.n

Consider the code of Merge-Sort Merge-Sort(A, p, r) if (p < r) q = floor((p+q)/2) Merge-Sort(A, p, q) Merge-Sort(A, q+1, r) Merge(A, p, q, r) If T(n) is the running time of Merge-Sort for an n size input, then the relation recurrence is for n > 1 is _________________ if the running time for Merge is 5.n. [] T(n) = T(n/2) + c.n [] T(n) = T(n/2) + 5.n [] None of these answers [] T(n) = 2 T(n/2) + 5.n [] T(n) = 2 T(n/2) + c.n

T(n) = 2 T(n/2) + f(n)

Consider the code of Merge-Sort: Merge-Sort(A, p, r) if (p < r) q = floor((p+q)/2) Merge-Sort(A, p, q) Merge-Sort(A, q+1, r) Merge(A, p, q, r) If T(n) is the running time of Merge-Sort for an n size input, then the relation recurrence is for n > 1 is _________________ if the running time for Merge is f(n). T(n) = 2 T(n/2) + f(n) T(n) = 2 T(n/2) + c.n T(n) = 2 T(n/2) + 5.n None of these answers T(n) = T(n/2) + f(n)

2T(16) + 96

Consider the following recurrence relation: T(n) = c. if n=1 2T(n/2) + cn. if n>1 Let c be 3, then T(32) = _____ 2T(16) + 32 2T(16) + 96 None of these answers T(16) + 96 T(16) + 64

12

Consider the following recurrence relation: T(n) = c. if n=1 2T(n/2) + cn. if n>1 Let c be 3. If we expand T(2), then T(2) = _____

4

Consider the following recurrence relation: T(n) = c. if n=1 2T(n/2) + cn. if n>1 When developing the recurrence tree, the 3rd level ("3rd call") has __________ nodes

2^(i-1)

Consider the following recurrence relation: T(n) = c. if n=1 2T(n/2) + cn. if n>1 When developing the recurrence tree, the ith level ("ith call") has __________ nodes. None of these answers 2^(i-1) 2^i 2^(i+1) 2^(i-2)

5.n

Consider the following recurrence relation: T(n) = c. if n=1 2T(n/2) + cn. if n>1 When developing the recurrence tree, the sum of the costs of all nodes at the ith level ("ith call") is ____________ when c = 5. 2^(5.n) 5 n 5.n None of these answers

5.n

Consider the following recurrence relation: T(n) = c. if n=1 2T(n/2) + cn. if n>1 When developing the recurrence tree, the sum of the costs of all nodes at the third level ("third call") is ____________ when c = 5. None of these answers 2^(5.n) 5 5.n n

4

Consider the following recurrence relation: T(n) = { c if n = 1 2T(n/2)+cn if n>1 Let c be 4, then T(1) =____ n c.n 4 4.n None of these answers

1

Consider the following recurrence relation: T(n) = { c if n = 1 2T(n/2)+cn if n>1 When developing the recurrence tree, the 1st level ("1st call") has __________ nodes

200.n.log n

Consider the recurrence relation: T(n) = 2.T(n/2) + 200n. A good guess could be _________________. Unless specified otherwise, log(n) is log base 2. 100.n.log n 100log n n.log n None of these answers 200.n.log n

4.n.(log n - 1)+4.n

Consider the recurrence relation: T(n) = 2.T(n/2) + 4.n. John guesses that T(n) = 4.n.log(n). Substituting T(n/2) by the guess in the recurrence relation will lead to T(n) = ______________. Unless specified otherwise, log(n) is log base 2. 2.n.(log n - log 2)+2.n None of these answers 2.n.(log n - log 2)+4.n 4.n.(log n - log 2)+2.n 4.n.(log n - 1)+4.n

None of these answers

Consider the recurrence relation: T(n) = 2.T(n/2) + 4.n. John guesses that T(n) = 4.n.log(n). Substituting T(n/2) by the guess in the recurrence relation will lead to T(n) = ______________. Unless specified otherwise, log(n) is log base 2. 2.n.log n - 4.n + 2.n 2.n.log n - 2.n + 2.n None of these answers 2.n.log n +2.n 4.n.log n - 2.n +4.n

4.n.log n - 4n + 4.n

Consider the recurrence relation: T(n) = 2.T(n/2) + 4.n. John guesses that T(n) = 4.n.log(n). Substituting T(n/2) by the guess in the recurrence relation will lead to T(n) = ______________. Unless specified otherwise, log(n) is log base 2. 4.n.(log n - 1)+2.n 2.n.(log n - 2)+2.n 2.n.log n +2.n 4.n.log n - 4n + 4.n None of these answers

4.n.log(n/2)+4.n

Consider the recurrence relation: T(n) = 2.T(n/2) + 4.n. John guesses that T(n) = 4.n.log(n). Substituting T(n/2) by the guess in the recurrence relation will yield T(n) = ______________. Unless specified otherwise, log(n) is log base 2. 4.n.log(n)-4.n 4.n.log(n/2)+4.n 4.n.log(n/2)-4.n None of these answers 2.n.log(n/2)

7.n.log n

Consider the recurrence relation: T(n) = 2.T(n/2) + 7.n. A good guess could be _________________. Unless specified otherwise, log(n) is log base 2. 7.log n 4.n.log n 7.n.log n None of these answers n.log n

recursive

Divide-and-conquer strategy uses _____________ algorithms. [] divisive [] iterative [] None of these answers [] recursive [] procedural