Lecture Notes #11 for Ma/CS 6a October 28 and 30, 2013
Consequences of Knigs Theorem
o
Corollary 1 of Knigs Theoem. A bipartite graph G that is regular of degree d > 0
o
has a perfect matching.
Proof
Lecture Notes #13 for Ma/CS 6a November 6, 2013
Remarks on Maxow-Mincut
Prosition 1. An unaugmentable feasible ow f (i.e. one for which there are no
f -augmenting paths from s to t) is a maxow (a maxi
Lecture Notes for Ma/CS 6a October 16, 2013
Some easy and some hard problems about graphs
Suppose G is a large graph. For example, you may imagine that G has 10000 or so
vertices and average degree 10
Lecture Notes #12 for Ma/CS 6a November 4, 2013
Augmenting paths and the Maxow-Mincut Theorem
Theorem 1. (Maxow-Mincut) Given a network with digraph D, source s, sink t, and
capacity function c there
Lecture Notes for Ma/CS 6a October 30, 2013
The integers
It is necessary to know the axioms (rules or laws of algebra) for working with the
integers . . . , 3, 2, 1, 0, 1, 2, 3, . . . Axioms for the n
Lecture Notes #10 for Ma/CS 6a October 25, 2013
Matchings in bipartite graphs, continued
A matching is a graph M all of whose vertices have degree 1. By a matching in a graph
G we mean a subgraph M th
Lecture Notes #9 for Ma/CS 6a October 21 & 23, 2013
Notation: We will start writing V (G) and E (G) for the vertex and edge sets of a
graph G.
Cheapest (or minimal) spanning trees
Suppose each edge of
Lecture Notes #8 for Ma/CS 6a October 18, 2013
The greedy algorithm, cont.
Theorem 1. Let k be a positive integer. If G is a nite graph with all vertices of degree
k and at least one vertex of degree
Lecture Notes for Ma/CS 6a October 9, 2013
Binomial numbers
We use the term n-set to mean a set of n elements. Sometimes we say a set of
cardinality n, or of size n, to mean the same thing. Also, we w
Lecture Notes for Ma/CS 6a October 14, 2013
Introduction to graphs
Graph theory has many technical terms. Unfortunately, perhaps, terminology diers
from one author or school of graph theory to another
Lecture Notes for Ma/CS 6a October 7, 2013
Primality testing
For the RSA cryptosystem, we need a large (perhaps 300-digit) integer m and the
value of (m). We propose to take m = pq where p and q are d
Lecture Notes for Ma/CS 6a October 4, 2013
Fermats little theorem and the Euler phi-function
Every one knows that the square of an odd number is odd and the square of an even
number is even. This is t
Lecture Notes for Ma/CS 6a September 2, 2013 Revised Sept. 3
Congruences
For integers a, b, m, the statement a b (mod m) (read a is congruent to b modulo
m) means m | (b a).
It may seem silly to intro
1
+
+
+
+
+
+
+
+
+
+
x2
1
x1
1
x
1 l L
(l)
xj
=
d
s
l = L
(s)
xd
+
+
h (x )
(l1) (l)
j
(l )
wij = xi
(l1)
(l)
wij
i=0
(s) = tanh(s)
(l1)
xi
d( l )
(l1)
i
(l)
(l1) 2
= (1 (xi
)
(l)
w
d(H)
10
10
5
10
H
0
10
5
10
up
20
UNKNOWN TARGET FUNCTION
f: X Y
80
100
120
140
160
180
200
N d
DISTRIBUTION
on
X
TRAINING EXAMPLES
( x1 , y1 ), . , ( xN , y )
N
ALGORITHM
60
PROBABIL
y = f (x)
h(x), f (x)
8
>+1
<
>
:
1
1
E (h) =
N
target function f:
x)
XY
plus noise
TRAINING EXAMPLES
( x1 , y1 ), . , ( xN , y )
N
UNKNOWN
INPUT
DISTRIBUTION
P (x)
N
h(xn), f (xn)
n=1
(