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David Luebke 1 04/19/23
CS 332: Algorithms
Introduction
Proof By Induction
David Luebke 2 04/19/23
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, and Rivest (CLR) The “Big White Book” An excellent reference you should own
David Luebke 3 04/19/23
The Course
Instructor: David Luebke [email protected] Office: Olsson 219
TA: Emily Evans Grader: Wei Lin Zhong
David Luebke 4 04/19/23
The Course
Grading policy (overrides web page): Homework: 40% Midterm: 20% Final: 35% Participation: 5%
David Luebke 5 04/19/23
The Course
Prerequisites: 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: Your name, year, and major (inc. Echols/Rodman) Any special reason why you should be let into the
course instead of others
David Luebke 6 04/19/23
The Course
Format Two lectures/week Homework every week
Problem sets Very occasional programming assignments
Two tests (tentative)
David Luebke 7 04/19/23
Review: Induction
Suppose S(k) is true for fixed constant k
Often k = 0 S(n) S(n+1) for all n >= k
Then S(n) is true for all n >= k
David Luebke 8 04/19/23
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
David Luebke 9 04/19/23
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
David Luebke 10 04/19/23
Induction Example:Geometric Closed Form
Prove a0 + a1 + … + an = (an+1 - 1)/(a - 1) for all a != 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)
David Luebke 11 04/19/23
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
David Luebke 12 04/19/23
Induction
Another variation: 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:
5
ˆ ii
iF
David Luebke 13 04/19/23
Inductive Proof?
Claim: All pens are black
Basis: Suppose the number of pens is 0, trivially all the
pens are black Hypothesis:
Assume if there are n pens, they are all black
David Luebke 14 04/19/23
Inductive Proof?
Step: Given n+1 pens, remove one pen, the remaining n pens
are black by the inductive hypothesis Call this set of n pens T1
Replace the first pen and remove another, the remaining n pens are black by inductive hypothesis
Call 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?
David Luebke 15 04/19/23
Up Next
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.
Coming up: Asymptotic performance of two search algorithms, A formal introduction to asymptotic notation Solving recurrences
