text stringlengths 16 3.88k | source stringlengths 60 201 |
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two nonzero
subrepresentations. Obviously, irreducible implies indecomposable, but not vice versa.
Typical problems of representation theory are as follows:
1. Classify irreducible representations of a given algebra A.
2. Classify indecomposable representations of A.
3. Do 1 and 2 restricting to finite dimensional ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
δ(h) = x
�
�x
y
−
�
�y
, δ(e) = x
�
�y
, δ(f ) = y
�
.
�x
(ii) Any indecomposable finite dimensional representation of U is irreducible. That is, any finite
dimensional representation of U is a direct sum of irreducible representations.
As another example consider the representation theory of quivers.
A quiv... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
−−�−
−�−−�
|
�
�−−�−−�−−�−−�−−�−−
|
�
The graphs listed in the theorem are called (simply laced) Dynkin diagrams. These graphs
arise in a multitude of classification problems in mathematics, such as classification of simple Lie
algebras, singularities, platonic solids, reflection groups, etc. In fact, if we needed to m... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
are used only in its proof; a purely group-theoretical proof of this
theorem (not using representations) exists but is much more difficult!
�
1.2 Algebras
Let us now begin a systematic discussion of representation theory.
Let k be a field. Unless stated otherwise, we will always assume that k is algebra... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
. A = EndV – the algebra of endomorphisms of a vector space V over k (i.e., linear maps, or
operators, from V to itself). The multiplication is given by composition of operators.
4. The free algebra A = k
x1, ..., xn�
x1, ..., xn, and multiplication in this basis is simply concatenation of words.
. A basis of this a... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
fies δ(ab) = δ(b)δ(a) and δ(1) = 1.
EndV ;
⊃
The usual abbreviated notation for δ(a)v is av for a left module and va for the right module.
Then the property that δ is an (anti)homomorphism can be written as a kind of associativity law:
(ab)v = a(bv) for left modules, and (va)b = v(ab) for right modules.
Here are s... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
, V2 be two representations of an algebra A. A homomorphism (or in
V2 is a linear operator which commutes with the action of A, i.e.,
tertwining operator) θ : V1
V1. A homomorphism θ is said to be an isomorphism of representations
θ(av) = aθ(v) for any v
if it is an isomorphism of vector spaces. The set (space) ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
will
see below that the converse statement is false in general.
One of the main problems of representation theory is to classify irreducible and indecomposable
representations of a given algebra up to isomorphism. This problem is usually hard and often can
be solved only partially (say, for finite dimensional repres... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
V2 we have I = V2.
Corollary 1.17. (Schur’s lemma for algebraically closed fields) Let V be a finite dimensional
irreducible representation of an algebra A over an algebraically closed field k, and θ : V
V is an
Id for some ∂
intertwining operator. Then θ = ∂
·
k (a scalar operator).
⊃
�
Remark. Note that this C... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
commutative). Thus, by Schur’s lemma, δ(a) is
A. Hence every subspace of V is a subrepresentation. But V is
a scalar operator for any a
irreducible, so 0 and V are the only subspaces of V . This means that dim V = 1 (since V = 0).
�
Example 1.19. 1. A = k. Since representations of A are simply vector spaces, V = A... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
in this basis
for any linear operator B : V
is a direct sum of Jordan blocks. This implies that all the indecomposable representations of A are
k, with δ(x) = J�,n. The fact that these representations are indecomposable and
V
pairwise non-isomorphic follows from the Jordan normal form theorem (which in particular ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
νV (z). Show that νV : Z(A)
z
homomorphism. It is called the central character of V .
⊃
�
(b) Show that if V is an indecomposable finite dimensional representation of A then for any
Z(A), the operator δ(z) by which z acts in V has only one eigenvalue νV (z), equal to the
k is a
z
scalar by which z acts on some ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
that D is at most countably dimensional. Suppose θ is not a scalar, and consider the subfield
C(θ)
D. Show that C(θ) is a transcendental extension of C. Derive from this that C(θ) is
uncountably dimensional and obtain a contradiction.
V is a scalar operator.
→
1.4
Ideals
A left ideal of an algebra A is a subspace... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
S
S
•
◦
�
�φ = span
{
◦
}
�r = span
{
◦
}
1.5 Quotients
Let A be an algebra and I a two-sided ideal in A. Then A/I is the set of (additive) cosets of I.
Let β : A
β(b) := β(ab).
This is well defined because if β(a) = β(a�) then
A/I be the quotient map. We can define multiplication in A/I by β(a)
·
⊃
β(a�
b) = β(a... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
be any ideal in A containing all homogeneous
polynomials of degree
N . Show that A/I is an indecomposable representation of A.
⊂
Problem 1.25. Let V = 0 be a representation of A. We say that a vector v
V is cyclic if it
generates V , i.e., Av = V . A representation admitting a cyclic vector is said to be cyclic.... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
generated by x1, . . . , xn with defining relations f1 = 0, . . . , fm =
, we say that the algebra
1.7 Examples of algebras
1. The Weyl algebra, k
x, y
/
◦
�
◦
yx
−
xy
−
2. The q-Weyl algebra, generated by x, x−
x−
1x = yy−
1 = y−
1y = 1.
1
.
�
1, y, y−
1 with defining relations yx = qxy and xx−
1 =
Proposition. ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
just a formal symbol, so really E = k[a][t, t−
variable, and E = tak[a][t, t−
a
s a representation of A with action given by xf
i
uppose now that we have a nontrivial
S
= and yf =
i
ij x y = 0. Then the operator
linear relation
d(ta n)
+
dt
(where
df
dt
j
tf
c
let a be
Then E
1).
ta+n
−
cts by zero in E. ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
is faithful if δ is injective.
⊃
For example, k[t] is a faithful representation of the Weyl algebra, if k has characteris
check it!), but not in characteristic p, where (d/dt)pQ = 0 for any polynomial Q. Howe
(
epresentation E = tak[a][t, t−
r
1], as we’ve seen, is faithful in any characteristic.
tic zero
ver, the
P
... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
y in V . Show that
1v
{
−
}
Problem 1.27. Let q be a nonzero complex number, and A be the q-Weyl algebra over C generated
by x 1 and y±
1 with defining relations xx = x−
1y = 1, and xy = qyx.
1
1, yy− = y−
1x =
−
±
1
(a) What is the center of A for different q? If q is not a root of unity, what are the two-sided
id... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
in the sense that for each quiver Q, there exists a certain algebra PQ, called the path
algebra of Q, such that a representation of the quiver Q is “the same” as a representation of the
algebra PQ. We shall first define the path algebra of a quiver and then justify our claim that
representations of these two objects a... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
��⊗ ah = ah, piah = 0 for i = h��
We now justify our statement that a representation of a quiver is the same thing as a represen
tation of the path algebra of a quiver.
Let V be a representation of the path algebra PQ. From this representation, we can construct a
ph⊗⊗ V
representation of Q as follows: let Vi = piV... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
Definition 1.35. A subrepresentation of a representation (Vi, xh) of a quiver Q is a representation
(Wi, x�h) where Wi
Wh⊗⊗ for
E.
all h
Wh⊗⊗ and x�h = xh|W ⊗h : Wh⊗
I and where xh(Wh⊗ )
Vi for all i
−⊃
∧
∧
�
�
Definition 1.36. The direct sum of two representations (Vi, xh) and (Wi, yh) is the representation... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
function of dimensions of A[n]). Often this series converges to a rational⎨ function, and
the answer is written in the form of such function. For example, if A = k[x] and deg(xn) = n then
A[m]
E.
−⊃
→
∞
∧
h
2
A(t) = 1 + t + t + ... + t + ... =
n
1
t
1
−
Find the Hilbert series of:
(a) A = k[x1, ..., xm] (w... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
nition 1.39. (g, [ , ]) is a Lie algebra if [ , ] satisfies the Jacobi identity
Example 1.40. Some examples of Lie algebras are:
[a, b] , c
[b, c] , a
+
�
�
+
�
[c, a] , b
= 0.
�
(2)
�
�
1. Any space g with [ , ] = 0 (abelian Lie algebra).
2. Any associative algebra A with [a, b] = ab
−
ba .
3. Any subspace U ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
bras
arise as spaces of infinitesimal automorphisms (=derivations) of associative algebras. In fact, they
similarly arise as spaces of derivations of any kind of linear algebraic structures, such as Lie algebras,
Hopf algebras, etc., and for this reason play a very important role in algebra.
Here are a few more conc... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
] = Y .
5. so(n), the space of skew-symmetric n
×
n matrices, with [a, b] = ab
−
ba.
Exercise. Show that Example 1 is a special case of Example 5 (for n = 3).
Definition 1.42. Let g1, g2 be Lie algebras. A homomorphism � : g1
linear map such that �([a, b]) = [�(a), �(b)].
g2 of Lie algebras is a
−⊃
Definition 1... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
the associative algebra generated by the ⎨xi’s with the
U
k
k cij xk.
xjxi
=
−
⎨
Remark. This is not a very good definition since it depends on the choice of a basis. Later we
will give an equivalent definition which will be basis-independent.
Exercise. Explain why a representation of a Lie algebra is the same thin... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
w
w
v2
−
�
W, a
�
k.
�
�
where v
�
w, v
�
(w1 + w2)
−
v
�
w1
v
�
−
w2, av
�
w
−
a(v
�
w), v
�
aw
−
a(v
�
w),
Exercise. Show that V
W generated by v
group V
W can be equivalently defined as the quotient of the free abelian
w, v
W by the subgroup generated by
V, w
�
�
w
�
w,... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
(2, 0) - bilinear forms, of type (2, 1)
- algebra structures, etc.
(V ⊕)�
�
�
n
n = V
If V
is finite dimensional with basis e , i = 1, ..., N , and ei is the dual basis of V ⊕, then a basis
i
of E is the set of vectors
and a typical element of E is
e
i1 �
...
�
ein �
j1
e
...
�
�
e
jm
,
N
i1,...,in,j... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
, if A : V
V � �
B : V
Bw (check that this is well defined!)
are linear maps, then
w) = Av
(A
one can define the linear map A
B)(v
W
⊃
�
�
V � and B : W
W �
⊃
W � given by the formula
⊃
�
�
�
The most important properties of tensor products are summarized i... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
n-th symmetric, respectively exterior, power
nV ? If dimV = m, what are their
of V . If
vi}
dimensions?
is a basis of V , can you construct a basis of S nV,
s(T ) where T
−
√
�
{
(e) If k has characteristic zero, find
such that T = sT for all transpositions s, and
for all transpositions s.
a natural identifica... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
is a left A-module. Namely, V
W freely generated by formal symbols
w, v
V
is the abelian group which is the quotient of the group V
v
W , modulo the relations
V , w
•
�
�
�
w
(v1 + v2)
�
v1
w
v2
−
�
�
w, v
�
(w1 + w2)
−
v
w1
v
−
�
�
w2, va
�
w
v
−
�
aw, a
A.
�
−
Exercise. Through... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
.)
bw
W, b
V, w
W ) onto the space V
B
�
�B W by v
v
w
�
�
−
V and w
�
�
�
�
|
�B W the
. We denote the projection of
�B w. (Note that this
If, additionally, A is another k-algebra, and if the right B-module structure on V is part of an
�B w for
�B W becomes a left A-module by a (v
�B w) = av
(A... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
. Prove that (V
The isomorphism (from left to right) is given by (v
w
�C X ∪= V
�B w)
�B W )
W and x
�C x
X.
�
�
�
(b) If A, B, C are three algebras, and if V is an (A, B)-bimodule and W an (A, C)-bimodule,
then the vector space HomA (V, W ) (the space of all left A-linear homomorphisms from V to W )
HomA... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
to give more conceptual (i.e., coordinate free) definitions
of the free algebra, polynomial algebra, exterior algebra, and universal enveloping algebra of a Lie
algebra.
Namely, given a vector space V , define
b := a
its tensor algebra T V over a field k to be T V =
b, a
n
n 0V � ,
∧
m. Observe that a choice of a... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
∧
0SnV ,
V =
√
0
n
∧
�
√
n
V .
1.12 Hilbert’s third problem
Problem 1.51. It is known that if A and B are two polygons of the same area then A can be cut
by finitely many straight cuts into pieces from which one can make B. David Hilbert asked in 1900
whether it is true for polyhedra in 3 dimensions. In particula... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
Conclude
that xk + x−
1 = 2/3
k has denominator 3k and get a contradiction.
roots of the equation x+x−
(c) Using (a) and (b), show that the answer to Hilbert’s question is negative. (Compute the
Dehn invariant of the regular tetrahedron and the cube).
1.13 Tensor products and duals of representations of Lie algebr... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
55. According to the above, a representation of sl(2) is just a vector space V with a
triple of operators E, F, H such that HE
F E = H (the
corresponding map δ is given by δ(e) = E, δ(f ) = F , δ(h) = H).
EH = 2E, HF
F H =
2F, EF
−
−
−
−
Let V be a finite dimensional representation of sl(2) (the ground field in... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
F N
¯
= 0 on V (∂), and
¯
V (∂), by (b). Use the fact that Pk(x) does not have multiple roots).
(e) Let Nv be the smallest N satisfying (c). Show that ∂ = Nv
1.
−
(f) Show that for each N > 0, there exists a unique up to isomorphism irreducible representation
of sl(2) of dimension N . ... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
) Show that
for some nonnegative integer ∂.
V ,
(use that the generalized eigenspace decomposition of C must be a decomposition of representations).
C has only one eigenvalue
namely
on
�
+2)
(�
2
(i) Show that V has a subrepresentation W = V� such that V /W = nV� for some n (use (h)
and the fact that V is the s... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
of the representations and the Jordan normal
form theorem)
(m) (Clebsch-Gordan decomposition) Find the decomposition into irreducibles of the represen
tation V�
�
Vµ of sl(2).
V (x) = T r(e
Hint. For a finite dimensional representation V of sl(2) it is useful to introduce the character
C. Show that νV W (x) = νV... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
algebra g over a field k is said to be solvable if there
exists n such that K n(g) = 0. Prove the Lie theorem: if k = C and V is a finite dimensional
irreducible representation of a solvable Lie algebra g then V is 1-dimensional.
�
Hint. Prove the result by induction in dimension. By the induction assumptio... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
element x of a Lie algebra g let ad(x) denote the operator g
g, y
ad(x)2(y) = ad(y)n+1(x) = 0.
⊃
[x, y]. Consider the Lie algebra gn generated by two elements x, y with the defining relations
�⊃
(a) Show that the Lie algebras g1, g2, g3 are finite dimensional and find their dimensions.
(b) (harder!) Show that the Li... | https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/6b7e3b7277dc86249ec40bb134186639_MIT18_712F10_ch1.pdf |
Lecture 2
6.006 Fall 2011
Lecture 2: Models of Computation
Lecture Overview
• What is an algorithm? What is time?
• Random access machine
• Pointer machine
• Python model
• Document distance: problem & algorithms
History
Al-Khw¯arizm¯ı “al-kha-raz-mi” (c. 780-850)
• “father of algebra” with his book “The Com... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-fall-2011/6b9b20992d8c6a0f3f10a34ff7878aa9_MIT6_006F11_lec02.pdf |
/null (a.k.a. reference)
• weaker than (can be implemented on) RAM
2
012...345...word} Lecture 2
6.006 Fall 2011
Python Model
Python lets you use either mode of thinking
1. “list” is actually an array → RAM
L[i] = L[j] + 5 → Θ(1) time
2. object with O(1) attributes (including references) → pointer machine
x =... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-fall-2011/6b9b20992d8c6a0f3f10a34ff7878aa9_MIT6_006F11_lec02.pdf |
θ(1)
(e)
b = x in L ≡
& L.index(x)
& L.find(x)
for y in L:
if x == y:
b = T rue;
break
else
b = F alse
(cid:41)
(cid:41)
⎫
⎪⎬
⎪⎭
θ(1 + |L2|) time
θ(j − i + 1) = O(|L|)
θ(1)
θ(index of x) = θ(|L|)
⎫
⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎭
(f) len(L) → θ(1) time - list stores its length in a field
(g) L.sort() → θ(|L| log ... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-fall-2011/6b9b20992d8c6a0f3f10a34ff7878aa9_MIT6_006F11_lec02.pdf |
documents, detecting
duplicates (Wikipedia mirrors and Google) and plagiarism, and also in web search (D2 =
query).
Some Definitions:
• Word = sequence of alphanumeric characters
• Document = sequence of words (ignore space, punctuation, etc.)
The idea is to define distance in terms of shared words. Think of docume... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-fall-2011/6b9b20992d8c6a0f3f10a34ff7878aa9_MIT6_006F11_lec02.pdf |
re.findall (r“ w+”, doc) → what cost?
in general re can be exponential time
→ for char in doc:
Θ(|doc|)
if not alphanumeric
add previous word
(if any) to list
start new word
⎫
⎪⎪⎪⎪⎪⎬
⎫
⎪⎬ Θ(1)
⎪⎪⎪⎪⎪⎭
⎪⎭
(2) sort word list
for word in list:
← O(k log k · |word|) where k is #words
⎫
O( |
word )
|
= O(|do... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-fall-2011/6b9b20992d8c6a0f3f10a34ff7878aa9_MIT6_006F11_lec02.pdf |
rst word of each list
if words equal: ← O(|word|)
total += count1 * count2
if word1 ≤ word2: ← O(|word|)
advance list1
else:
advance list2
repeat either until list done
Dictionary Approach
(2)’
count = {}
for word in doc:
if word in count: ← Θ(|word|) + Θ(1) w.h.p
⎫
⎪⎬
count[word] += 1
else
Θ(1)
⎪⎭
co... | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-fall-2011/6b9b20992d8c6a0f3f10a34ff7878aa9_MIT6_006F11_lec02.pdf |
: 1.8 — (3) (full dictionary)
• docdist8: 0.2 — whole doc, not line by line
7
MIT OpenCourseWare
http://ocw.mit.edu
6.006 Introduction to Algorithms
Fall 2011
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-006-introduction-to-algorithms-fall-2011/6b9b20992d8c6a0f3f10a34ff7878aa9_MIT6_006F11_lec02.pdf |
DECOMPOSITION,
ABSTRACTION,
FUNCTIONS
(download slides and .py files (cid:258)(cid:374)(cid:282)(cid:3)follow along!)
6.0001 LECTURE 4
6.0001 LECTURE 4
1
LAST TIME
while loops vs for loops
should know how to write both kinds
should know when to use them
guess-and-check and approximation methods
bisection m... | https://ocw.mit.edu/courses/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/6ba59859535f1566dd57a7279aeba5d1_MIT6_0001F16_Lec4.pdf |
� DECOMPOSITION IDEA: different devices work
together to achieve an end goal
6.0001 LECTURE 4
7
APPLY THESE CONCEPTS
TO PROGRAMMING!
6.0001 LECTURE 4
8
CREATE STRUCTURE with
DECOMPOSITION
in projector example, separate devices
in programming, divide code into modules
• are self-contained
• used to break up cod... | https://ocw.mit.edu/courses/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/6ba59859535f1566dd57a7279aeba5d1_MIT6_0001F16_Lec4.pdf |
bound to the value of
actual parameter when function is called
new scope/frame/environment created when enter a function
scope is mapping of names to objects
def f( x ):
x = x + 1
print('in f(x): x =', x)
return x
x = 3
z = f( x )
6.0001 LECTURE 4
14
VARIABLE SCOPE
def f( x ):
Global scope
f scope
x = x + 1
prin... | https://ocw.mit.edu/courses/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/6ba59859535f1566dd57a7279aeba5d1_MIT6_0001F16_Lec4.pdf |
func_b
func_c
return z()
print func_a()
print 5 + func_b(2)
print func_c(func_a)
Some
code
Some
code
Some
code
None
7
None
6.0001 LECTURE 4
z
func_a
func_a scope
returns None
returns None
24
SCOPE EXAMPLE
inside a function, can access a variable defined outside
inside a function, cannot modify a variable define... | https://ocw.mit.edu/courses/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/6ba59859535f1566dd57a7279aeba5d1_MIT6_0001F16_Lec4.pdf |
Global scope
g scope
Some
code
3
x
h
3
Some
code
g
x
z
x = 3
z = g(x)
6.0001 LECTURE 4
29
SCOPE DETAILS
def g(x):
def h():
x = 'abc'
x = x + 1
print('g: x =', x)
h()
return x
Global scope
g scope
Some
code
3
x
h
34
Some
code
g
x
z
x = 3
z = g(x)
6.0001 LECTURE 4
30
SCOPE DETAILS
def g(x):
def h():
x = 'abc'
x =... | https://ocw.mit.edu/courses/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/6ba59859535f1566dd57a7279aeba5d1_MIT6_0001F16_Lec4.pdf |
Python
Fall 2016
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-0001-introduction-to-computer-science-and-programming-in-python-fall-2016/6ba59859535f1566dd57a7279aeba5d1_MIT6_0001F16_Lec4.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.080 / 6.089 Great Ideas in Theoretical Computer Science
Spring 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
6.080/6.089 GITCS
March 4, 2008
Lecturer: Scott Aaronson
Scribe: Hristo Paskov
Lecture 8
1 Administriv... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
time? No. There is
so much that we don’t know about the limits of feasible computation, so we must savor what we
do know. There are more problems that we can solve in n3 than in n2 steps. Similarly, there are
more problems that we can solve in 3n than in 2n steps. The reason this is true is that we can
consider a p... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
. . . b2n
. . .
. . .
. . . bnn
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
c12
c11
c22
c21
. . .
. . .
cn1 cn2
. . . c1n
. . . c2n
. . .
. . .
. . . cnn
⎞
⎟
⎟
⎟
⎠
2.1.5 Straightforward way
The straightforward way takes n3 steps because of the way we multiply columns and rows. However,
there do exist better algorithms.
2.... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
have to look
at all entries of the matrices. Some people conjecture that for all � > 0, there exists an algorithm
that takes O(n2+�) time.
2.1.8 Practical considerations
If matrices are reasonably small, then you’re better off multiplying them with the na¨ıve O(n3)
algorithm. It’s an empirical fact that, as you go ... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
a best algorithm. We don’t know how often this
weird phenomenon arises, but we do know that it can in principle occur.
Incidentally, there’s a lesson to be learned from the story of matrix multiplication. It was
intuitively obvious to people that n3 was the best we could do, and then we came up with algorithms
that... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
Who invents O(n10,000) algorithms?
2. Subtle response - any criterion for efficiency has to meet the needs of practitioners and the
orists. It has to be convenient. Imagine a subroutine that takes polynomial time and an
algorithm that makes polynomial calls to the subroutine. Then the runtime is still polyno
mial sin... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
For example:
3, 8, 2, 4, 9, 1, 5, 7, 6
has several longest subsequences of length 4: (2,4,5,7), (2,4,5,6), (3,4,5,6), (3,4,5,7).
Solving this is manageable when we have 9 numbers, but what about when n = 1000? How do
we program a computer to do this?
4.2 A Polynomial Time Algorithm
One could try all possibilities... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
their actual spouses. A matching without any such instabilities is called a
stable marriage. Our goal is to give an efficient algorithm to find stable marriages, but the first
question is: does a stable marriage even always exist?
8-4
5.2 Victorian Romance Novel Algorithm
It turns out that the easiest way to show tha... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
.2.2 Termination
Does this algorithm terminate? Yes: in the worst case, every man would propose once to every
woman on his list. (Note that a man never reconsiders a woman who’s been crossed off.)
Next question: when the algorithm terminates is everyone matched up? Yes. Suppose for the
sake of contradiction there’s ... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
to loop through all n! possible matchings and
output a stable one when found. The above is a much more efficient algorithm, and it also provides
a proof that a solution exists.
6 Other Examples
After forty years these are some of the problems we know are solvable in polynomial time: Given
N men and N woman again, bu... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
that alternates between edges we haven’t used and edges we have. If we find
such a path, then we simply add all the odd-numbered edges to our matching and remove all the
even-numbered edges. In this approach, we keep searching in our graph till no more such paths
can be found. This approach leads to an O(n3) algorith... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
. . a2n
. . .
. . .
. . . ann
⎞⎛⎞
⎟
⎟
⎟
⎠
⎜
⎜
⎜
⎝
⎟
⎟
⎟
⎠
=
x1
x2
.
.
.
xn
⎞⎛
y1
y2
.
.
.
yn
⎟
⎟
⎟
⎠
⎜
⎜
⎜
⎝
Another example is solving linear system of equations. Instead of trying every possible vector, we
can use Gaussian Elimination. This takes O(n2) time to zero out rows below, and doing this n
8-6
t... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
�
⎟
⎟
⎠
≤
x1
x2
.
.
.
xn
⎞⎛
y1
y2
.
.
.
yn
⎟
⎟
⎟
⎠
⎜
⎜
⎜
⎝
What about solving a system of linear inequalities? This is a problem called linear programming,
whose basic theory was developed by George Dantzig shortly after World War II. As an operations
researcher during the war, Dantzig had faced problems that i... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
War II, computer science (like most
other sciences) would probably not have made so many advances in such a short period of time. If
you’re interested in learning more about the history of wartime research (besides the Manhattan
Project, which everyone knows about), check out Alan Turing: The Enigma by Andrew Hodges... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
O(n) time.
6.6 Can we solve these in polynomial time?
On the other hand, what if we want to know whether our map has a three-coloring? This seems
harder since we are no longer always forced to color each country a specific color—in many cases
we’ll have two choices. It’s not obvious how to solve the problem efficientl... | https://ocw.mit.edu/courses/6-080-great-ideas-in-theoretical-computer-science-spring-2008/6bb9d9288a25008309519b3bc454f79b_lec8.pdf |
Guidelines on Formulating a Management Problem as
a Linear Programming Model – Prof. Stephen Graves
General Rules of Thumb
• There is usually more than one correct formulation of a problem.
•
•
Sometimes it is unclear whether to use inequality or equality constraints.
Sometimes our intuition tells us that a constr... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/6bd2fc985c7646001c20d2c0bf63b44c_lec5_formulating_mgmt_problem.pdf |
Horizon
Bulk Road Salt
Month
12 Months: April ---- March
Production Locations:
4 Mines
Stockage Locations:
4 Mines
50 Stockpiles
100's of Customer Sites
Transportation:
Mines to Stockpiles by Rail ( Barges)
Stockpiles to Customers by Truck
Issues:
Production Plan for Seasonal Demand
Transportation and In... | https://ocw.mit.edu/courses/15-066j-system-optimization-and-analysis-for-manufacturing-summer-2003/6bd2fc985c7646001c20d2c0bf63b44c_lec5_formulating_mgmt_problem.pdf |
2.997 Decision-Making in Large-Scale Systems
MIT, Spring 2004
March 3
Handout #12
Lecture Note 9
1 Explicit Explore or Exploit (E3) Algorithm
Last lecture, we studied the Q-learning algorithm:
Qt+1(xt, at) = Qt(xt, at) + βt g (xt) + π min Qt(xt+1, a ≤) − Qt(xt, at) .
�
at
�
a
�
An important characteristic of Q... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/6be403afd3e9dabd97dc6d869cc11d52_lec_9_v1.pdf |
replaced with their true values.
ˆ
ˆ
We now introduce the algorithm.
1.1 Algorithm
We will first consider a version of E3 which assumes knowledge of J �; the assumption will be lifted later.
The E3 algorithm proceeds as follows.
1. Let N = ≥. Pick arbitrary state x0. Let k = 0.
2. If xk → N , perform “balanced wan... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/6be403afd3e9dabd97dc6d869cc11d52_lec_9_v1.pdf |
,MN
ˆ
∅� � β with high probability
2
(iv) If exploitation is not possible, then there is an exploration policy that reaches an unknown state after
T transitions with high probability.
To show the first main point, we consider the following lemma.
Lemma 1 Suppose a state x has been visited at least m times with eac... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/6be403afd3e9dabd97dc6d869cc11d52_lec_9_v1.pdf |
Proof: Trivial for x /
→ N since Ju,MN (x) = gmax ∀ Ju(x). If x → N , take T = inf{t : xt → N }. Then
/
1−�
Ju(x) = E
�
T −1
�
πt g
u(xt) +
πt g
u(xt)
�
t=T
�
πt gu(xt) + πT gmax
1 − π
�
t=0
�
T −1
� E
�
= Ju,MN (x)
t=0
�
To prove the main point iii(b), we first introduce the following definition.
�
Definiti... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/6be403afd3e9dabd97dc6d869cc11d52_lec_9_v1.pdf |
(p) = Pu,M (x0, x1)Pu,M (x1, x2) . . . Pu,M (xT −1, xT )
is the probability of observing path p and
T
g
u(p) =
t
g
π u(xt)
t=0
�
is the discounted cost associated with path p.
By selecting T properly, we can have
�
�
�
�
�
a(x, y) − Pˆa(x, y) � β.
�
πt gu(xt)
�
πT gmax
1 − π
� δ
E
�
t=T +1
�
��
�
�
�
�
R... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/6be403afd3e9dabd97dc6d869cc11d52_lec_9_v1.pdf |
∗R
�
p∗R
�
�
�
�
�
�
�
Pˆ
u(p)ˆgu(p)
�
�
�
�
�
�
�
(β + 2�) |S| T gmax
1 − π
(1 − �)Pa(xt, xt+1) � Pˆa(xt, xt+1) � (1 + �)Pa(xt, xt+1)
where � = � . Therefore,
�
(1 − �)T Pu(p) � Pˆu(p) � (1 + �)T Pu(p).
4
... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/6be403afd3e9dabd97dc6d869cc11d52_lec_9_v1.pdf |
Then we have
J N
u
� ,T (x) =
J N
u
� ,T (x) > J �
T (x) + β.
P N
u
� (q)g N
u
(q) +
P N
u
� (p)g N
u
(p)
and
Therefore
q∗N
�
r
�
path in N
path outside N
J �
T (x) =
�
⎢⎦
�
�
Pu� (q)gu(q) +
⎢⎦
Pu� (q)gu(q).
�
q
�
r
�
J N
u� ,T (x) − J �
u
� ,T (x) =
r
�
⎤
⎥
⎥
�
which implies
P N
u
� (p) g ... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/6be403afd3e9dabd97dc6d869cc11d52_lec_9_v1.pdf |
4, which shows that
each attempt to explore is successful with some non negligible probability. By applying the Chernoff bound,
it can be shown that, after a number of attempts that is polynomial in the quantities of interest, exploration
will occur with high probability.
5
... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/6be403afd3e9dabd97dc6d869cc11d52_lec_9_v1.pdf |
Probability Review
15.075 Cynthia Rudin
A probability space, defined by Kolmogorov (1903-1987) consists of:
• A set of outcomes S, e.g.,
for the roll of a die, S = {1, 2, 3, 4, 5, 6},
1
2
for the roll of two dice, S =
,
temperature on Monday, S = [ 50, 50].
1
1
−
,
2
1
,
1
3
, . . . ,
6
6
• A set of events, wh... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/6c4b98949fe3e0acae5f75245e4b09dc_MIT15_075JF11_chpt02.pdf |
Do not confuse independence with disjointness. Disjoint events A and B have P (A ∩ B) = 0.
For a partition B1, . . . , Bn, where Bi ∩ Bj = ∅ for i
= j and B1 ∪ B2 ·
· · Bn = S then
A = (A ∩ B1) ∪ (A ∩ B2), . . . , ∪(A ∩ Bn)
and thus
n
P (A) =
P (A ∩ Bi) =
P (A|Bi)P (Bi)
i=1
i
from the definition of conditional ... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/6c4b98949fe3e0acae5f75245e4b09dc_MIT15_075JF11_chpt02.pdf |
The expected value (mean) of an r.v. X is:
(cid:88)
E(X) = µ =
xf (x)
(discrete)
x
(cid:90)
E(X) = µ =
xf (x)dx
(continuous).
x
Expectation is linear, meaning
E(aX + bY ) = aE(X) + bE(Y ).
Roulette
The variance of an r.v. X is:
V ar(X) = σ2 = E(X − µ)2 .
Variance measures dispersion around the mean. Varia... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/6c4b98949fe3e0acae5f75245e4b09dc_MIT15_075JF11_chpt02.pdf |
Cov(X, Y ) = E[(X − µx)(Y − µy)] = E(XY ) − µxE(Y ) − µyE(X) + µxµy = 0.
Useful relationships:
1. Cov(X, X) = E[(X − µx)2] = V ar(X)
2. Cov(aX + c, bY + d) = ab Cov(X, Y )
3. V ar(X±Y ) = V ar(X)+V ar(Y )±2Cov(X, Y ) where Cov(X, Y ) is 0 if X and Y are indep.
The correlation coefficient is a normalized version of c... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/6c4b98949fe3e0acae5f75245e4b09dc_MIT15_075JF11_chpt02.pdf |
for that, it’s
¯
X = 1
n
• Do
¯es X have anything to do with the average pizza sales per day? In other words, does
measuring X tell us anything about the Xi’s? For instance, (on average) is X close to
the average sales per day, E(Xi)?
i Xi.
¯
¯
(cid:80)
5
• Does it matter what the distr... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/6c4b98949fe3e0acae5f75245e4b09dc_MIT15_075JF11_chpt02.pdf |
the distribution of the Xi’s is, the sample mean approaches the true
mean.
¯
Proof Weak LLN using Chebyshev
Section 2.7. Selected Discrete Distributions
Bernoulli X ∼ Bernoulli(p) “coin flipping”
f (x) = P (X = x) =
(cid:26) p
1 − p
if x = 1 “heads”
if x = 0 “tails”
Binomial X ∼ Bin(n, p) “n coins flipping,” “... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/6c4b98949fe3e0acae5f75245e4b09dc_MIT15_075JF11_chpt02.pdf |
to draw x balls ways to draw n − x balls
(cid:19)
(cid:18)
f (x) =
with attribute
without attribute
ways to draw n balls
=
M
x
N − M
n − x
(cid:19)
.
(cid:18)
N
n
Multinomial Distribution “generalization of binomial”
Think of customers choosing backpacks of different colors. A random group of n customers
... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/6c4b98949fe3e0acae5f75245e4b09dc_MIT15_075JF11_chpt02.pdf |
ar(X) = λ.
7
Exponential Distribution X ∼ Exp(λ) “waiting times for Poisson events”
f (x) = λe−λx for x ≥ 0
Gamma Distribution X ∼ Gamma(λ, r) “sums of r iid exponential r.v.’s,” “sums of waiting
times for Poisson events”
f (x) =
λrx
r−1 −λx
e
Γ(r)
for x ≥ 0,
where Γ(r) is the “Gamma” function, w... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/6c4b98949fe3e0acae5f75245e4b09dc_MIT15_075JF11_chpt02.pdf |
Table A.3 in your book has values for Φ(z) for many possible z’s. Read the table
left to right, and up to down. So if the entries in the table look like this:
z
0.03
−2.4
0.0075
This means that for z = −2.43, then Φ(z) = P (Z ≤ z) = 0.0075. So the table relates z to
Φ(z). You can either be given z and need Φ(z) ... | https://ocw.mit.edu/courses/15-075j-statistical-thinking-and-data-analysis-fall-2011/6c4b98949fe3e0acae5f75245e4b09dc_MIT15_075JF11_chpt02.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
6.854J / 18.415J Advanced Algorithms
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
�
�
18.415/6.854 Advanced Algorithms
September 15, 2008
Goldberg-Tarjan Min-Cost Circulation Algorithm
Lecturer: Michel X. Goeman... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/6c53462d9af606e71fa8950a51dfccfa_lec4.pdf |
with cost −1.
2.1 Choice of cycle �
As in the Ford-Fulkerson algorithm, the question is which negative-cost cycle to choose.
1. (Weintraub 1972). One idea is to try choosing the maximum improvement cycle, where
the difference in cost is as large as possible. One can show that the number of iterations is
polynomial ... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/6c53462d9af606e71fa8950a51dfccfa_lec4.pdf |
with respect to costs c + �}.
For any �, we can decide if there is a negative cost cycle by using the Bellman-Ford algorithm.
Now, perform binary search to find the smallest � for which no such cycle exists. In the next
problem set we will show a result by Karp, which finds the cycle of minimum mean cost in
O(nm) tim... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/6c53462d9af606e71fa8950a51dfccfa_lec4.pdf |
that � is monotonically non-increasing in general. First, we need the following strong relationship
between �(f ) and µ(f ), and this really justifies the choice of cycle of Goldberg and Tarjan.
Theorem 1 For all circulations f , �(f ) = −µ(f ).
Proof: We first show that µ(f ) � −�(f ). From the definition of �(f ) the... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/6c53462d9af606e71fa8950a51dfccfa_lec4.pdf |
every vertex can be reached (by the direct path). Note that the shortest paths are well-defined
since there are no negative cost cycles with respect to c� . By the optimality property of shortest
c(�)
|�|
=
lect-2
c’(v,w)
w
s
0
0
v
0
Figure 1: p(v) is the length of the shortest path from s to v.
paths, p(... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/6c53462d9af606e71fa8950a51dfccfa_lec4.pdf |
On the other hand, new edges may be created with a reduced cost of +�(f ). More
formally, Ef � ≤ Ef → {(w, v) : (v, w) ≥ �}. So for all (v, w) ≥ Ef � it holds that cp(v, w) � −�(f ).
�
Thus we have that �(f �) � �(f ).
2.3 Analysis for Integer-valued Costs
We now prove a polynomial bound on the number of iteration... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/6c53462d9af606e71fa8950a51dfccfa_lec4.pdf |
respectively. Let
A be the set of edges in Efi such that cp(v, w) < 0 (we should emphasize that this is for the p
corresponding to the circulation f we started from). We now show that as long as �i ≤ A, then
|A| strictly decreases. This is because cancelling a cycle removes at least one arc with a negative
reduced ... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/6c53462d9af606e71fa8950a51dfccfa_lec4.pdf |
(m2n2 log(nC)).
2.4 Strongly Polynomial Analysis
In this section we will remove the dependence on the costs. We will obtain a strongly polynomial
bound for the algorithm for solving the minimum cost circulation problem. In fact we will show
that this bound will hold even for irrational capacities. The first strongly... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/6c53462d9af606e71fa8950a51dfccfa_lec4.pdf |
(v, w). Let E< = {(x, y) : f �(x, y) < f (x, y)}.
We can see that E< ≤ Ef � by definition of Ef � . Furthermore, from flow conservation, we know that
there exists a cycle � ≥ Ef � containing the edge (v, w). Indeed, by flow decomposition, we know
that the circulation f − f � can be decomposed into (positive net) flows a... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/6c53462d9af606e71fa8950a51dfccfa_lec4.pdf |
v, w) < −2n�(f �) which means that it
�
was not �(f )-fixed. Thus (v, w) becomes �(f �)-fixed and the claim is proven.
Notice that if e is fixed, it will remain fixed as we iterate the algorithm. An immediate con
sequence of the above lemma then is a bound on the number of iterations in the Goldberg-Tarjan
algorithm. ... | https://ocw.mit.edu/courses/6-854j-advanced-algorithms-fall-2008/6c53462d9af606e71fa8950a51dfccfa_lec4.pdf |
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