1	CS 332: Algorithms Introduction Proof By Induction Asymptotic notation
2	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
3	The Course Instructor: David Luebke luebke@cs.virginia.edu Office: Olsson 219 Office hours: 2-3 Monday, 10-11 Thursday TA: Pavel Sorokin Office hours and location TBA
4	The Course Grading policy: Homework: 30% Exam 1: 15% Exam 2: 15% Final: 35% Participation: 5%
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 Who has not taken CS 302?
6	The Course Format Three lectures/week Homework most weeks Problem sets Maybe occasional programming assignments Two tests + final exam
7	Review: Induction Suppose S(k) is true for fixed constant k Often k = 0 S(n) à S(n+1) for all n >= k Then S(n) is true for all n >= k
8	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+1
9	Induction Example: Gaussian Closed Form Prove 1 + 2 + 3 + … + n = n(n+1) / 2 Basis: If n = 0, then 0 = 0(0+1) / 2 Inductive hypothesis: Assume 1 + 2 + 3 + … + n = n(n+1) / 2 Step (show true for n+1): 1 + 2 + … + n + n+1 = (1 + 2 + …+ n) + (n+1) = n(n+1)/2 + n+1 = [n(n+1) + 2(n+1)]/2 = (n+1)(n+2)/2 = (n+1)(n+1 + 1) / 2
10	Induction Example: Geometric Closed Form Prove a⁰ + a¹ + … + aⁿ = (aⁿ⁺¹ - 1)/(a - 1) for all a ¹ 1 Basis: show that a⁰ = (a⁰⁺¹ - 1)/(a - 1) a⁰ = 1 = (a¹ - 1)/(a - 1) Inductive hypothesis: Assume a⁰ + a¹ + … + aⁿ = (aⁿ⁺¹ - 1)/(a - 1) Step (show true for n+1): a⁰ + a¹ + … + aⁿ⁺¹ = a⁰ + a¹ + … + aⁿ + aⁿ⁺¹ = (aⁿ⁺¹ - 1)/(a - 1) + aⁿ⁺¹ = (aⁿ⁺¹⁺¹ - 1)/(a - 1)
11	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
12	Asymptotic Performance In this course, we care most about asymptotic performance How does the algorithm behave as the problem size gets very large? Running time Memory/storage requirements Bandwidth/power requirements/logic gates/etc.
13	Asymptotic Notation By now you should have an intuitive feel for asymptotic (big-O) notation: What does O(n) running time mean? O(n²)? 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
14	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 Except, of course, function calls Constant word size Unless we are explicitly manipulating bits
15	Input Size Time and space complexity This is generally a function of the input size E.g., sorting, multiplication How we characterize input size depends: Sorting: number of input items Multiplication: total number of bits Graph algorithms: number of nodes & edges Etc
16	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 y = m * x + b c = 5 / 9 * (t - 32 ) z = f(x) + g(y) We can be more exact if need be
17	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”? Random (equally likely) inputs Real-life inputs
18	The End