Great Theoretical Ideas In Computer Science
A. Gupta
V. Guruswami
Lecture 7
CS 15-251
September 20, 2011
Fall 2011
Counting II:
Pigeons, Pirates, and Binomials
(
+
+
Plan
Carnegie Mellon University
)(
+
)=?
Multinomial coefficients
Pirates and Gold
Pigeon
15-251
Great Theoretical Ideas
in Computer Science
15-251
Great Theoretical Ideas
in Computer Science
www.cs.cmu.edu/~15251
Grading
Course Staff
Instructors
TAs
Anupam Gupta
Venkat Guruswami
Dmitriy Chernyak
Mark Wong Siang Kai
Ankur Parikh
Tim Wilson
Fin
15-251: Great Theoretical Ideas In Computer Science
Final : Spring 2011 Practice Test 2
Name:
Andrew ID:
Section:
INSTRUCTIONS:
Write your NAME, ANDREW ID, and SECTION above.
This is a closed book test. You may not use notes. You may not use a calculato
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1
1. Notation & Elementary Tools
summation symbol:
n
X
ak = am + am+1 + . . . + an1 + an
(n, m ZZ, n m)
ak = am am+1 . . . an1 an
(n, m ZZ, n m)
k=m
product symbol:
n
Y
k=m
binomial coefficients:
n
k
!
=
n!
k!(n k)!
Proof by induction:
Phrase the theore
15-251: Great Theoretical Ideas In Computer Science
Recitation 1
Announcements
Conceptual office hours are today, 6-8 PM in Gates 5 Carrel 1 (the double carrel). Use this time
to clarify concepts from lecture.
Check the website for times and locations f
: Trie
24 2016 .
24 2016 .
1 / 10
24 2016 .
2 / 10
cat
a small animal that is related
to lions and tigers and that is
often kept by people as a pet
dog
canid; especially : a highly variable
domestic mammal (Canis familiaris) closely
related to t
10/11/2011
15-251: Fall 2011
Algebraic Structures:
Groups Theory
Il est peu de notions en mathematiques qui
soient plus primitives que celle de loi de
composition.
- Nicolas Bourbaki
Lecture 13 (October 11, 2011)
Number Theory
Integers
Naturals
Number The
Number theory and
Cryptography
Lecture 12 (October 6, 2011)
p -1
p
1
How do you compute
58
using few multiplications?
First idea:
m me mod (pq)
5 52 53 54 55 56 57 58
=
= 5*5 52*5
m ( gr, m * hr)
How do you compute
Repeated squaring calculates
k
a2
in k m
Number Theory and
Modular Arithmetic
God made the integers; all else is
the work of man.
Lecture 11 (October 4, 2011)
0
7
p-1
p 1
- Leopold Kronecker, 1823-91.
1
6
2
5
f(pq) = (p-1)(q-1)
3
4
Divisibility:
An integer a divides b (written a|b)
if and only i
15-251
Great Theoretical Ideas
in Computer Science
15-251
Proof Techniques for
Computer Scientists
Induction
Inductive Reasoning
Lecture 2 (September 1, 2011)
Dominoes
Domino Principle:
Line up any number of
dominos in a row; knock
the first one over and
15-251
Great Theoretical Ideas
in Computer Science
Proofs and Logic
Lecture 3 (September 6, 2011)
P, P Q
Q
In mathematics, sometimes your intuition
can be dead wrong.
We know that 1 2
1+1 = 2
do we need a proof?
But heres a theorem of Banach & Tarski:
A s
15-251
Great Theoretical Ideas
in Computer Science
Bits of Wisdom on Solving
Problems, Writing Proofs, and
Enjoying the Pain: How to
Succeed in This Class
Lecture 4 (September 8, 2011)
What did our brains
evolve to do?
What were our brains
designed to do?
15-251
Great Theoretical Ideas
in Computer Science
15-251
Game Playing for
Computer Scientists
Combinatorial
Games
Lecture 5 (September 13, 2011)
A Take-Away Game
Two Players: I and II
A move consists of removing one,
two, or three chips from the pile
Pla
9/15/2011
15-251
15-251
Great Theoretical Ideas
in Computer Science
Counting for
Computer Scientists
Counting I: One-To-One
Correspondence
and Choice Trees
Lecture 6 (September 14, 2011)
In the next three lectures we will learn
some fundamental counting m
9/22/2011
Great Theoretical Ideas In Computer Science
A. Gupta
V. Guruswami
Lecture 8
CS 15-251
September 22, 2011
Fall 2011
Carnegie Mellon University
Counting III:
Catalan numbers,
Generating functions
Let us recap some key results
from last lecture
x1
Probability Theory I
(Flipping Coins for Computer Scientists)
Lecture 9 (September 27, 2011)
Some Puzzles
6 and 7 Are Equally Likely
Teams A and B are equally good
In any one game, each is equally likely to win
To reach either one, after 5 games, it
must
Probability Theory II
(Setting Expectations for
Computer Scientists)
Lecture 10 (September 29, 2011)
Some useful
sample spaces
E[X+Y] = E[X] + E[Y]
1) A fair coin
3) Two independent bias-p coin tosses
sample space S = cfw_H, T
Pr(H) = , Pr(T) = .
sample s
part I of a short course on
The replica method and its applications in
biomedical modelling and data analysis
ACC Coolen
Institute for Mathematical and Molecular Biomedicine, Kings College London
January 2014
ACC Coolen ([email protected])
1 / 67
replica method
A