4833/6833 Info

CSE 4833/6833 Introduction to Analysis of Algorithms (Spring 2018)

Asymptotic analysis

Asymptotic notation: Big-O, Big-Theta, Big-Omega, little-O, little-omega
Master Theorem for solving recurrences of the form $T(n) = aT\left(\frac{n}{b}\right) + \Theta\left(n^d\right)$ , where the base case $T(1)$ is equal to some constant value:

Case 1: if

a > b^d

, then

T(n) = \Theta(n^{log_{b}a})

Case 2: if

a = b^d

, then

T(n) = \Theta(n^{d} \log n)

Case 3: if

a < b^d

, then

T(n) = \Theta(n^d)

^{}

a < b^{d},

Numerical algorithms

Euclid's algorithm for greatest common denominator
Bisection algorithm (binary search) for square root
Divide-and-conquer (Karatsuba) multiplication

Search for an item in an array

Array sorting algorithms

Algorithm	Time Complexity			Space Complexity
	Best	Average	Worst	Worst
Insertion Sort	`Ω(n)`	`Θ(n^2)`	`O(n^2)`	`O(1)`	Stable
Selection Sort	`Ω(n^2)`	`Θ(n^2)`	`O(n^2)`	`O(1)`
Quicksort	`Ω(n log(n))`	`Θ(n log(n))`	`O(n^2)`	`O(log(n))`
Mergesort	`Ω(n log(n))`	`Θ(n log(n))`	`O(n log(n))`	`O(n)`	Stable
Heapsort	`Ω(n log(n))`	`Θ(n log(n))`	`O(n log(n))`	`O(1)`
Bucket Sort	`Ω(n+k)`	`Θ(n+k)`	`O(n^2)`	`O(n)`
Radix Sort	`Ω(nk)`	`Θ(nk)`	`O(nk)`	`O(n+k)`	Stable
Counting Sort	`Ω(n+k)`	`Θ(n+k)`	`O(n+k)`	`O(k)`	Stable

Selection problem

Quickselect algorithm

Convex hull problem

Quickhull algorithm