CS6515 - ALGORITHMS- EXAM 1 | 98 QUESTIONS AND
ANSWERS | 2026 UPDATE | WITH COMPLETE SOLUTION
How do you tell if a graph has negative edges? Answer - when fitting graph on
a table, if the number of moves decrease the w() from edge to edge, then there
is a negative edge;
check from 1 to n
Why is all pairs Dist(y,z) n^2? Answer - Because it builds a two dim table!
what is the run time of bellman ford algoirthm?
How about if you had to do it for all edges? Answer - O(nm)
O(n^2m)
FLoyd-Warshall run time? Answer - O(n^3)
what is the base case for the bellman ford algorithm? Answer - D(0,s,t)
how does bellman ford and floyd differ when it comes to detecting negative
weight cycles? Answer - bellman == can only find it if it can be access from the
"s" or start vertex
,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
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).
ANSWERS | 2026 UPDATE | WITH COMPLETE SOLUTION
How do you tell if a graph has negative edges? Answer - when fitting graph on
a table, if the number of moves decrease the w() from edge to edge, then there
is a negative edge;
check from 1 to n
Why is all pairs Dist(y,z) n^2? Answer - Because it builds a two dim table!
what is the run time of bellman ford algoirthm?
How about if you had to do it for all edges? Answer - O(nm)
O(n^2m)
FLoyd-Warshall run time? Answer - O(n^3)
what is the base case for the bellman ford algorithm? Answer - D(0,s,t)
how does bellman ford and floyd differ when it comes to detecting negative
weight cycles? Answer - bellman == can only find it if it can be access from the
"s" or start vertex
,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
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).
