CS-2852 Data Structures Week 10, Class 1 Lab 8 notes Big-O revisited CS-2852 Dr. Josiah Yoder Slide style: Dr. Hornick 1

CS-2852Data Structures

Week 10, Class 1 Lab 8 notes Big-O revisited

CS-2852Dr. Josiah Yoder

Slide style: Dr. Hornick1

Running time


Slide style: Dr. HornickContent: Dr. Hasker

2

f(n) f(50) f(100) f(100)/f(50)

1 1 1 1

log2 n 5.64 6.64 1.18

n 50 100 2

n2 2500 10,000 4

n3 12,500 100,000 8

2n 1e15 1e30 1e15

n! 3e64 9e157 3e93

Big-O Motivation

Want to ignore minor details Focus on what really makes the difference as

n grows Each function on the previous slide is in a

class of its own Want to find that class Multiplication by scalar doesn’t matter Addition of lower-order operations doesn’t

matterCS-2852

Dr. Josiah YoderSlide style: Dr. Hornick

3

Big-O Definition

T(n) = O(f(n)

if and only if

There exist n0 and c such that

T(n) ≤ cf(n) for all n > n0



Simplifying Big-O expressions Addition

O(1) < O(log n) < O(nk) < O(kn) < O(n!)

e.g. O(nk+n!) = O(n!) e.g. O(log n + n2) = O(n2) e.g. O(n log n + n2) = O(n2) e.g. O(n log n + n) = O(n log n



Simplifying Big-O expresionsMultiplication by scalar

O(kf(n)) = O(f(n)) for any fixed k

e.g. O(5) = O(1) e.g. O(2 log2 n + 5) = O(log n)



Strategies for determining Big-O

Analysis of Code Intuition based on

Structure of data Analysis of Code



Big-O based on analysis of Code

public void f() {

if(isG()) {

h();

x++;

} else {

for(int i=0;i<j();i++) {

k();

for(A a: list) {

m();

}

}

} // end of f

max – sequence of simple expressions

max – different if clauses

prod – number of iterations over loop * contents of loop

subs – method calls



Big-O based analysis of a Recursive algorithm

Each recursive call should be O(1) except for method calls No loops

So order is number of recursive calls needed Number of recursive calls can be exponential

e.g. simple implementation of Fibbonaci Often, number of recursive calls is O(n) or

even O(log n)



Example

Suppose binary search of an array is implemented recursively. What is the Big-O running time?



Example

Suppose binary search of a Red-Black tree is implement recursively. What is the Big-O running time?



Example

Suppose binary search of a simple BinarySearchTree is implemented recursively. What is the Big-O running time?



Example

Suppose binary search of a simple BinarySearchTree is implemented iteratively (with a loop). What is the Big-O running time?



Example

Suppose we insert n items into an ArrayList using add(E). What is the Big-O running time?



Example

Suppose we insert n items into an ArrayList using add(0, E). What is the Big-O running time?



Example

Suppose we insert n items into a LinkedList using add(0, E). What is the Big-O running time?



Example

Suppose we insert n items into a LinkedList using add(E). What is the Big-O running time?



Example

Suppose we insert just 1 item into a LinkedList using add(n/2, e). What is the Big-O running time?



Example

Suppose we insert one item into an empty hash-table.

What is the Big(O) running time?



Example

Suppose we insert n items into an empty hash-table, and then remove them


Assume: No collisions occur



Example

Suppose we insert n items into an empty hash-table, and then remove them


Assume: Collisions always occur



Example

Suppose we insert an item into a properly-implemented stack.

What is the Big-O running time?



Example

Suppose we remove an item from a properly-implemented circular queue (the wrap-around array implementation)

What is the Big-O running time?



Example

What is the Big-O running time of the word search lab?

(in terms of the size of the grid)



CS-2852 Dr. Josiah Yoder


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CS-2852 Data Structures Week 10, Class 1 Lab 8 notes Big-O revisited CS-2852 Dr. Josiah Yoder Slide style: Dr. Hornick 1