CS6515 - Algorithms- Exam 1 2025/2026
Exam Questions and Answers | A+
Score Assured
Steps to solve a Dynamic Programming Problem - 🧠 ANSWER ✔✔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) - 🧠 ANSWER ✔✔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 - 🧠 ANSWER ✔✔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 - 🧠 ANSWER ✔✔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 - 🧠 ANSWER ✔✔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 - 🧠 ANSWER ✔✔(1, 2*PI*j/n) for j = 0, 1, ..., n-1
*Around the Unit Circle!
Steps to solve for FFT - 🧠 ANSWER ✔✔1) Write out Matrix Coefficient
Form based on n (size of input) Mn(w) = [ 1 1 ... 1
COPYRIGHT©NINJANERD 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER: 619652435. TERMS OF USE. PRIVACY
STATEMENT. ALL RIGHTS RESERVED
3
Exam Questions and Answers | A+
Score Assured
Steps to solve a Dynamic Programming Problem - 🧠 ANSWER ✔✔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) - 🧠 ANSWER ✔✔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 - 🧠 ANSWER ✔✔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 - 🧠 ANSWER ✔✔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 - 🧠 ANSWER ✔✔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 - 🧠 ANSWER ✔✔(1, 2*PI*j/n) for j = 0, 1, ..., n-1
*Around the Unit Circle!
Steps to solve for FFT - 🧠 ANSWER ✔✔1) Write out Matrix Coefficient
Form based on n (size of input) Mn(w) = [ 1 1 ... 1
COPYRIGHT©NINJANERD 2025/2026. YEAR PUBLISHED 2025. COMPANY REGISTRATION NUMBER: 619652435. TERMS OF USE. PRIVACY
STATEMENT. ALL RIGHTS RESERVED
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