CS6515 - Algorithms Questions with Verified
Correct Answers
Steps to solve a Dynamic Programming Problem
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)
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
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
Given d = common difference and a = first term in series => a + (a + d) + (a + 2d) + ... + (a +
(n-1)d
Sum = n/2 [2a + (n-1)d]
DC: Solving Recurrences - Master Theorem
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
(1, 2PIj/n) for j = 0, 1, ..., n-1
*Around the Unit Circle!
Steps to solve for FFT
, 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^(2PIi)/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
Euler's Formula
e^ix = cosx + isinx
Imaginary Number Multiples
i = i, i^2 = -1, i^3 = -i, i^4 = 1
i = -i, i^2 = -1, i^3 = i, i^4 = 1
Omega(w)
w = (1, 2PI / n) = e^(2PI*i/n)
DC Algorithms and Runtimes (6)
MergeSort: O(nlogn) - Split the input array into two halves and recursively sort and merge at
the end.
Correct Answers
Steps to solve a Dynamic Programming Problem
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)
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
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
Given d = common difference and a = first term in series => a + (a + d) + (a + 2d) + ... + (a +
(n-1)d
Sum = n/2 [2a + (n-1)d]
DC: Solving Recurrences - Master Theorem
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
(1, 2PIj/n) for j = 0, 1, ..., n-1
*Around the Unit Circle!
Steps to solve for FFT
, 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^(2PIi)/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
Euler's Formula
e^ix = cosx + isinx
Imaginary Number Multiples
i = i, i^2 = -1, i^3 = -i, i^4 = 1
i = -i, i^2 = -1, i^3 = i, i^4 = 1
Omega(w)
w = (1, 2PI / n) = e^(2PI*i/n)
DC Algorithms and Runtimes (6)
MergeSort: O(nlogn) - Split the input array into two halves and recursively sort and merge at
the end.
