1.4 Analysis of Algorithms

• introduction

• observations 

• mathematical models 

• order-of-growth classifications

• theory of algorithms

• memory

 

1.4.1 Introduction:

1. the cast of characters:

 

2.Focus on the running time:

Babbage machine - how many times do you have to turn the crank?

 

3. Algorithms performance

 

4. better performance (success algorithms)

minimize the running time

 

In the 1970s, Kanoof, use the scientific method to understand the performance of algorithms in operation.

 

5. Scientific method applied to analysis algorithms 

 

1.4.2 Observations

1. running time of the program 

Examples: 3-sum problem 

 

2. measuring the running time 

Plot running time T(N), input size N

3. log-log scale

Fit the line: 

 

4. prediction and validation

 

5. ratios 

 

6. Key factors determine the running time 

 

1.4.3 Mathematical models

Cost of basic operations

Example: 1-Sum

 

Simplifying the calculation

Tilde notation

Example: 2-Sum

Example: 3-Sum                                                                                                                       

Estimating a discrete sum(calculus)

Mathematical models for running time 

 

1.4.4 Order-of-growth classifications

 

Common order-of-growth classifications

Binary search demo

Mathematical analysis: Binary search

An  algorithm for 3-Sum

Compare running time 

 

1.4.5 Theory of Algorithms

notations

example 1: 1-sum

example 2: 3-sum

The GAP between lower and upper bound

 

 

1.4.6 Memory

 

Typical memory usage summary