Introduction - Introduction to Algorithms - Lecture Slides, Slides of Computer Science

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Algorithms

Introduction

The Course

• Purpose: a rigorous introduction to the design

and analysis of algorithms

 Not a lab or programming course  Not a math course, either

• Textbook: Introduction to Algorithms ,

Cormen, Leiserson, Rivest, Stein

 The “Big White Book”  Second edition: now “Smaller Green Book”  An excellent reference you should own

The Course

• Grading policy:

 Homework: 30%  Exam 1: 15%  Exam 2: 15%  Final: 35%  Participation: 5%

The Course

• Prerequisites:

 CS 202 w/ grade of C- or better  CS 216 w/ grade of C- or better  CS 302 recommended but not required o Who has not taken CS 302?

Review: Induction

• Suppose

 S(k) is true for fixed constant k o Often k = 0  S(n)  S(n+1) for all n >= k

• Then S(n) is true for all n >= k

Proof By Induction

• Claim:S(n) is true for all n >= k
• Basis:

 Show formula is true when n = k

• Inductive hypothesis:

 Assume formula is true for an arbitrary n

• Step:

 Show that formula is then true for n+

Induction Example: Geometric Closed Form

• Prove a 0 + a^1 + … + a n^ = (an+1^ - 1)/(a - 1) for

all a ≠ 1

 Basis: show that a 0 = (a 0+1^ - 1)/(a - 1) a 0 = 1 = (a 1 - 1)/(a - 1)  Inductive hypothesis: o Assume a 0 + a 1 + … + a n^ = (a n+1^ - 1)/(a - 1)  Step (show true for n+1): a 0 + a 1 + … + a n+1^ = a 0 + a 1 + … + a n^ + a n+ = (a n+1^ - 1)/(a - 1) + a n+1^ = (a n+1+1^ - 1)/(a - 1)

Induction

• We’ve been using weak induction
• Strong induction also holds  Basis: show S(0)  Hypothesis: assume S(k) holds for arbitrary k <= n  Step: Show S(n+1) follows
• Another variation:  Basis: show S(0), S(1)  Hypothesis: assume S(n) and S(n+1) are true  Step: show S(n+2) follows

Asymptotic Notation

• By now you should have an intuitive feel for

asymptotic (big-O) notation:

 What does O(n) running time mean? O(n 2 )? O(n lg n)?  How does asymptotic running time relate to asymptotic memory usage?

• Our first task is to define this notation more

formally and completely

Analysis of Algorithms

• Analysis is performed with respect to a

computational model

• We will usually use a generic uniprocessor

random-access machine (RAM)

 All memory equally expensive to access  No concurrent operations  All reasonable instructions take unit time o Except, of course, function calls  Constant word size o Unless we are explicitly manipulating bits

Running Time

• Number of primitive steps that are executed

 Except for time of executing a function call most statements roughly require the same amount of time o y = m * x + b o c = 5 / 9 * (t - 32 ) o z = f(x) + g(y)

• We can be more exact if need be

Analysis

• Worst case

 Provides an upper bound on running time  An absolute guarantee

• Average case

 Provides the expected running time  Very useful, but treat with care: what is “average”? o Random (equally likely) inputs o Real-life inputs