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CS 332: Algorithms
Introduction
Proof By Induction
David Luebke
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7/27/2016
The Course
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Purpose: a rigorous introduction to the design
and analysis of algorithms
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Not a lab or programming course
Not a math course, either
Textbook: Introduction to Algorithms, Cormen,
Leiserson, and Rivest (CLR)
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The “Big White Book”
An excellent reference you should own
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7/27/2016
The Course
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Instructor: David Luebke
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[email protected]
Office: Olsson 219
TA: Emily Evans
 Grader: Wei Lin Zhong
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David Luebke
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7/27/2016
The Course
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Grading policy (overrides web page):
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Homework: 40%
Midterm: 20%
Final: 35%
Participation: 5%
David Luebke
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7/27/2016
The Course
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Prerequisites:
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CS 302 w/ grade of C- or better
CS 216 w/ grade of C- or better
These are strictly enforced!
Who’s not registered? Write down:
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Your name, year, and major (inc. Echols/Rodman)
Any special reason why you should be let into the
course instead of others
David Luebke
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7/27/2016
The Course
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Format
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Two lectures/week
Homework every week
 Problem
sets
 Very occasional programming assignments
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Two tests (tentative)
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Review: Induction
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Suppose
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S(k) is true for fixed constant k
 Often
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k=0
S(n)  S(n+1) for all n >= k
Then S(n) is true for all n >= k
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Proof By Induction
Claim:S(n) is true for all n >= k
 Basis:
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Inductive hypothesis:
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Show formula is true when n = k
Assume formula is true for an arbitrary n
Step:
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Show that formula is then true for n+1
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7/27/2016
Induction Example:
Gaussian Closed Form
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Prove 1 + 2 + 3 + … + n = n(n+1) / 2
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Basis:
 If
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n = 0, then 0 = 0(0+1) / 2
Inductive hypothesis:
 Assume
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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
David Luebke
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7/27/2016
Induction Example:
Geometric Closed Form
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Prove a0 + a1 + … + an = (an+1 - 1)/(a - 1) for
all a != 1
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Basis: show that a0 = (a0+1 - 1)/(a - 1)
a0 = 1 = (a1 - 1)/(a - 1)
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Inductive hypothesis:
 Assume
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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)
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7/27/2016
Induction
We’ve been using weak induction
 Strong induction also holds
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Basis: show S(0)
Hypothesis: assume S(k) holds for arbitrary k <= n
Step: Show S(n+1) follows
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7/27/2016
Induction
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Another variation:
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Basis: show S(0), S(1)
Hypothesis: assume S(n) and S(n+1) are true
Step: show S(n+2) follows
Convenient to prove that the ith Fibonacci
number Fi is given by:
i  ˆ i
Fi 
5
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7/27/2016
Inductive Proof?
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Claim:
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Basis:
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All pens are black
Suppose the number of pens is 0, trivially all the
pens are black
Hypothesis:
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Assume if there are n pens, they are all black
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Inductive Proof?
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Step:
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Given n+1 pens, remove one pen, the remaining n
pens are black by the inductive hypothesis
 Call
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Replace the first pen and remove another, the
remaining n pens are black by inductive hypothesis
 Call
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this set of n pens T1
this set of n pens T2
The union of T1 and T2 is the original set of n+1
pens. Both subsets are black, so entire set is black.
What is wrong here?
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7/27/2016
Up Next
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In this course, we care most about asymptotic
performance
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How does the algorithm behave as the problem
size gets very large?
 Running
time
 Memory/storage requirements
 Bandwidth/power requirements/logic gates/etc.
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Coming up:
 Asymptotic
performance of two search algorithms,
 A formal introduction to asymptotic notation
 Solving recurrences
David Luebke
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7/27/2016