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Draw the diagram for the process of algorithm analysis and design. Explain the process.
Explain with example the Asymptotic Notations used in the analysis of algorithms.
List the Basic Efficiency classes and briefly describe them.
Write the general plan to analyze non recursive algorithms.
Write the general plan to analyze recursive algorithms.
For each of the following algorithms, indicate
(i) a natural size metric for its inputs,
(ii) its basic operation, and
(iii) whether the basic operation count can be different for inputs of the same size:
a. computing the average of n numbers
b. computing n/n!
c. finding the smallest element in a list of n numbers
d. reverse display a list of n numbers
e. reverse a list of n numbers
f. pen-and-pencil algorithm for adding two n-digit decimal integers
7. Consider the definition-based algorithm for finding the difference between two n × n
matrices. What is its basic operation? How many times is it performed as a function of the
matrix order n? As a function of the total number of elements in the input matrices?
8. Consider the definition-based algorithm for finding the for the inverse of an n × n matrices.
What is its basic operation? How many times is it performed as a function of the matrix
order n? As a function of the total number of elements in the input matrices?
9. Consider a classic sequential search that scans a list to search the occurrences of a given
search key in the list. Discuss the worst case, average case, and best case efficiency of
classic sequential search.
10. Glove selection:
There are 22 gloves in a drawer: 5 pairs of red gloves, 4 pairs of yellow, and 2 pairs of green.
You select the gloves in the dark and can check them only after a selection has been made.
What is the smallest number of gloves you need to select to have at least one matching pair
in the best case? In the worst case?
11. Missing socks
Imagine that after washing 5 distinct pairs of socks, you discover that two socks are missing.
Of course, you would like to have the largest number of complete pairs remaining. Thus, you
are left with 4 complete pairs in the best-case scenario and with 3 complete pairs in the
worst case. Assuming that the probability of disappearance for each of the 10 socks is the
same, find the probability of the best-case scenario; the probability of the worst-case
scenario; the number of pairs you should expect in the average case.
12. For each of the following functions, indicate how much the function’s value will change if its
argument is increased threefold.
a. n log 2n
b. n
c. n3
d. n2
e. n!
f. 2n
13. For each of the following pairs of functions, indicate whether the first function of each of the
following pairs has a lower, same, or higher order of growth (to within a constant multiple)
than the second function.
a. n(n + 1) and 2000n2
b. 100n2 and 0.01n3
c. log2 n and ln n
2
2
n−1
n
d. log2 n and log2 n
e. 2 and 2
f. (n − 1)! and n!
14. Use the most appropriate notation among O, Ω , and Θ to indicate the time efficiency of the
expression 50x + 7
a. in the worst case.
b. in the best case.
c. in the average case.
15. Use the informal definitions of O, Ω, and Θ to determine whether the following assertions
are true or false.
a. 2n(n − 1)/2 ∈ O(n3)
b. 2n(n − 1)/2 ∈ O(n2)
c. 2n(n − 1)/2 ∈ Θ (n3)
d. 2n(n − 1)/2 ∈ Ω(n)
16. Prove that the following functions are listed in increasing order of their order of growth.
log n,
n,
n log2 n,
n2 ,
n3 ,
2n ,
n!
17. List the following functions according to their order of growth from the lowest to the highest:
(n2 + 3)!, 2 lg(n + 50)5, 33n,
0.05n10 + 3n3 + 1,
ln3n, √n,
32n
18. Compute the following sums.
a. 1 + 2 + 3 + 4 + ⋯ + 100.
b. 10 + 20 + 30 + 40 + ⋯ + 1000
𝑖
c. ∑𝑛𝑖=2 ∑𝑛𝑗=1
𝑗
19. Find the order of growth of the following sums. Use the Θ(g(n)) notation with the simplest
function g(n) possible.
a. ∑𝑛𝑖=1 𝑖 5 + 2
b. ∑𝑛𝑖=1 𝑖 2
20. Find the order of growth of the following algorithm:
maxval ←A[0]
for i ←1 to n − 1 do
if A[i]>maxval
maxval←A[i]
return maxval
21. Find the order of growth of the following algorithm:
for i ← 0 to n − 2 do
for j ← i to n − 1 do
if A[i]= A[j] return false
return true
22. Find the order of growth of the following algorithm:
for i ← 0 to n − 1 do
for j ← 0 to n − 1 do
C[i, j] ← 0.0
for k ← 0 to n − 1 do
C[i, j] ← C[i, j] + A[i, k] * B[k, j]
return C
23. Find the order of growth of the following algorithm:
count ← 0
while n > 1 do
count ← count+1
𝑛 ← ⌊𝑛/2⌋
return count
24. Find the order of growth of the following algorithm:
if n = 0 return 1
else return F(n - 1) * n
25. Consider the following algorithm and answer the following:
ALGORITHM Foo(n)
//Input: A nonnegative integer n
Sum←0
for i←1 to n do
Sum←Sum + i/i!
return Sum
a. What does this algorithm compute?
b. What is its basic operation?
c. How many times is the basic operation executed?
d. What is the efficiency class of this algorithm?
26. Consider the following algorithm and answer the following:
ALGORITHM Enigma(A[0..n − 1, 0..n − 1])
//Input: A matrix A[0..n − 1, 0..n − 1] of real numbers
for i ←0 to n − 2 do
for j ←i + 1 to n − 1 do
if A[i, j ] ≠ A[j, i]
return false
return true
a. What does this algorithm compute?
b. What is its basic operation?
c. How many times is the basic operation executed?
d. What is the efficiency class of this algorithm?
27. Find the time efficiency class of the following algorithm.
ALGORITHM GE(A[0..n − 1, 0..n])
//Input: An n × (n + 1) matrix A[0..n − 1, 0..n] of real numbers
for i ←0 to n − 2 do
for j ←i + 1 to n − 1 do
for k←i to n do
A[j, k]←A[j, k]− A[i, k] ∗ A[j, i]/ A[i, i].
28. Consider the following recursive algorithm for computing the sum of the following series:
S(n) = 1/1!+ 2/2!+ . . . + n/n! .
ALGORITHM S(n)
//Input: A positive integer n
// Procedure: fact(n) returns the factorial of the number passed as parameter
//Output: The sum of the series: S(n) = 1/1!+ 2/2!+ . . . + n/n!
if n = 1 return 1
else return S(n − 1) + n/fact(n)
29. What is a brute-force algorithm? Explain with an example.
30. What is the time efficiency of the brute-force algorithm for computing an as a function of n?
31. What are the advantages of the brute-force algorithm over other efficient algorithms?
32. Using the Brute force approach with selection sorting to sort an array of n integers using.
Give the pseudocode of it and find its time efficiency class.
33. Using the Brute force approach with bubble sorting to sort an array of n integers using. Give
the pseudocode of it and find its time efficiency class.
34. Using the Brute force approach with sequential search in an array of n integers using. Give
the pseudocode of it and find its time efficiency class.
35. Using the Brute force approach for finding a pattern in an array of size n. Give the
pseudocode of it and find its time efficiency class.