CS6515 - Algorithms- Exam 1 Questions and Answers Already Passed Latest 2025
Steps to solve a Dynamic Programming Problem - Answers 1. Define the Input and Output.
2. Define entries in table, i.e. T(i) or T(i, j) is...
3. Define a Recurrence relationship - Based on a subproblem to the main problem. (hint: use a prefix of
the original input 1 < i < n).
4. Define the Pseudocode.
5. Define the Runtime of the algorithm. Use Time Function notation here => T(n) = T(n/2) + 1...
DP: Types of Subproblems (4) - Answers Input = x1, x2, ..., xn
1) Subproblem = x1, x2, ..., xi ; O(n)
2) Subproblem = xi, xi+1, ..., xj ; O(n^2)
Input = x1, x2, ..., xn; y1, y2, ..., ym
1) Subproblem = x1, x2, ..., xi; y1, y2, ..., yj ; O(mn)
Input = Rooted Binary Tree
1) Subproblem = Smaller rooted binary tree inside the Input.
DC: Geometric Series - Answers Given r = common ratio and a = first term in series
=> a + ar + ar^2 + ar^3 + ... + ar^(n-1)
=> a * [(1 - r^n) / (1-r)]
, DC: Arithmetic Series - Answers Given d = common difference and a = first term in series => a + (a + d) +
(a + 2d) + ... + (a + (n-1)d
Sum = n/2 * [2*a + (n-1)d]
DC: Solving Recurrences - Master Theorem - Answers If T(n) = aT([n/b]) + O(n^d) for constants a>0, b>1,
d>=0:
T(n) = {
O(n^d) if d > logb(a)
O((n^d)logn) if d = logb(a)
O(n^(logb(a))) if d < logb(a)
}
Nth roots of Unity - Answers (1, 2*PI*j/n) for j = 0, 1, ..., n-1
*Around the Unit Circle!
Steps to solve for FFT - Answers 1) Write out Matrix Coefficient Form based on n (size of input) Mn(w) = [
1 1 ... 1
1 w ... w^n-1
...
1 w^n-1 ... w^((n-1)*(n-1)) ]
2) Find value for w = e^(2*PI*i)/n, Substitute in Mn(w).
3) For the input coefficients into nx1 matrix. I.E. [4 0 1 1], let known as B.
4) Evaluate FFT:
a) FFT of Input = Mn(w) x B
b) Inverse FFT of Input = 1/n * Mn(w^-1) x B
Steps to solve a Dynamic Programming Problem - Answers 1. Define the Input and Output.
2. Define entries in table, i.e. T(i) or T(i, j) is...
3. Define a Recurrence relationship - Based on a subproblem to the main problem. (hint: use a prefix of
the original input 1 < i < n).
4. Define the Pseudocode.
5. Define the Runtime of the algorithm. Use Time Function notation here => T(n) = T(n/2) + 1...
DP: Types of Subproblems (4) - Answers Input = x1, x2, ..., xn
1) Subproblem = x1, x2, ..., xi ; O(n)
2) Subproblem = xi, xi+1, ..., xj ; O(n^2)
Input = x1, x2, ..., xn; y1, y2, ..., ym
1) Subproblem = x1, x2, ..., xi; y1, y2, ..., yj ; O(mn)
Input = Rooted Binary Tree
1) Subproblem = Smaller rooted binary tree inside the Input.
DC: Geometric Series - Answers Given r = common ratio and a = first term in series
=> a + ar + ar^2 + ar^3 + ... + ar^(n-1)
=> a * [(1 - r^n) / (1-r)]
, DC: Arithmetic Series - Answers Given d = common difference and a = first term in series => a + (a + d) +
(a + 2d) + ... + (a + (n-1)d
Sum = n/2 * [2*a + (n-1)d]
DC: Solving Recurrences - Master Theorem - Answers If T(n) = aT([n/b]) + O(n^d) for constants a>0, b>1,
d>=0:
T(n) = {
O(n^d) if d > logb(a)
O((n^d)logn) if d = logb(a)
O(n^(logb(a))) if d < logb(a)
}
Nth roots of Unity - Answers (1, 2*PI*j/n) for j = 0, 1, ..., n-1
*Around the Unit Circle!
Steps to solve for FFT - Answers 1) Write out Matrix Coefficient Form based on n (size of input) Mn(w) = [
1 1 ... 1
1 w ... w^n-1
...
1 w^n-1 ... w^((n-1)*(n-1)) ]
2) Find value for w = e^(2*PI*i)/n, Substitute in Mn(w).
3) For the input coefficients into nx1 matrix. I.E. [4 0 1 1], let known as B.
4) Evaluate FFT:
a) FFT of Input = Mn(w) x B
b) Inverse FFT of Input = 1/n * Mn(w^-1) x B
