Unformatted text preview: f ac u lty of sci ence
d ep ar tment of m athemat ics MATH 8954 Fall 2010
Course Schedule Lecture outline: Optimization algorithms (2)
Week Date Sections Part/ References Topic/Sections MACM 201 Discrete Mathematics II Notes/Speaker FS2009
We continuefrom
our
exploration of optimization algorithms, by looking at algorithms that applies
again
on 7graphs
with weighted
and aim
at computing a spanning tree of minimum weight.
1
Sept
I.1, I.2, I.3
methods
Combinatorial edges,Symbolic
2 14 I.4, I.5, I.6 4 28 II.4, II.5, II.6 Structures Unlabelled structures FS: Part A.1, A.2
Applications of such questions
are numerous, from computer networks to computational biology,
Comtet74
3
21
II.1, II.2, II.3
Labelled structures I
Handout #1
through algorithmic theory.
(self study)
Labelled structures II We will study two algorithms:
PrimCombinatorial
and Kruskal algorithms. Both are typical examples of
Combinatorial
5
Oct 5
III.1, III.2
Asst #1 Due
Parameters
greedy algorithms: atparameters
any
given
time
they
make
a
decision
(here to add an edge to an evolving
FS A.III
6
12
IV.1, IV.2
Multivariable GFs
(selfstudy)
set of edges that, ultimately, will form a spanning tree) that is the best one at the time it is
7
19
IV.3, IV.4
Complex
Analysis
Analytic
Methodsthat it
made
without
trying to
ensure
is optimal
over the course of the whole algorithm. But
FS: Part B: IV, V, VI
8
26
Analysis
then,
we
canIV.5prove
that
thisB4 greedy Singularity
decision
is actually a good one to make (i.e. it does not
Appendix
V.1
Stanley 99: Ch. 6
9 us
Novaway
2
Due
take
from finding
spanning
tree).AsstIn#2 terms
of proofs, proving that a greedy
Asymptotic
methods
Handout a
#1 minimum
(selfstudy)
9
VI.1
algorithm
actually
computes an optimal solution to Sophie
the problems it aims to solve is not easy,
10
Introduction to Prob.
Mariolys
but we12 will A.3/
seeC two examples.
18 IX.1 20 IX.2 23 IX.3 25 IX.4 13 30 IX.5 14 Dec 10 11 12 Random Structures
and Limit Laws
FS: Part C
(rotating
presentations) Limit Laws and Comb Marni Discrete Limit Laws Sophie Combinatorial
instances of discrete Mariolys Continuous Limit Laws Marni QuasiPowers and
Gaussian limit laws Sophie Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11Dec09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 1 f ac u lty of sci ence
d ep ar tment of m athemat ics MATH 8954 Fall 2010
Course Schedule MACM 201 Discrete Mathematics II
Week first
Date start
Sections
Part/ References
We
by defining
what isTopic/Sections
a minimumNotes/Speaker
spanning tree.
from FS2009
1
Sept 7 I.1, I.2,weight
I.3
Symbolic methods
Combinatorial
Definition:
of a spanning
tree. Let G = (V , E) be a graph, w an
Structures
2
14
I.4, I.5, I.6
Unlabelled structures
Part A.1,
A.2 T = (V , E 0 ) be spanning tree of G.
edgeweighting
ofFS:
G,
and
Comtet74
3 21 II.1, II.2, II.3 Handout #1
(self study) Labelled structures I The
weight
ofII.6T is the sum of Labelled
the weights
of the edges in P:
4
28
II.4, II.5,
structures II
Combinatorial
Combinatorial X
5
Oct 5
III.1, III.2
Asst #1 Due
parameters
Parameters
w(P)
=
w(e)
FS A.III
6 12 IV.1, IV.2 (selfstudy) Multivariable GFs 7 19 IV.3, IV.4 Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(selfstudy) Complex Analysis e∈E 0 26
Analysis tree of G whose weight is minimum
A8 minimum
spanning tree is aSingularity
spanning
IV.5 V.1
9
Nov 2
Asst #2 Due
Asymptotic methods
among
all spanning trees of G.
10 9 VI.1 Sophie 12
A.3/ C
to Prob.
Mariolys
Problem
statement: minimum Introduction
spanning
tree.
18
IX.1
Limit Laws and Comb
Marni
11
Input.
A
connected
graph
G
=
(V
,
E),
V
=
{v1, . . . , vn}, E = {e1, . . . , em},
Random Structures
20
IX.2
Discrete Limit Laws
Sophie
and Limit Laws
and 23a nonnegative
edgeweighting
of E. Mariolys
FS: Part
C
Combinatorial
IX.3
12 (rotating
presentations) instances of discrete 25
Continuous
Marnithat T = (V , E ) is a minimum
Output
A IX.4
subset ET of n − 1 edges
ofLimitELaws
such
T
QuasiPowers and
13
30
IX.5
Sophie
spanning
tree of G.
Gaussian limit laws
14
Dec 10
Example. Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11Dec09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 2 MATH 8954 Fall 2010
Course Schedule f ac u lty of sci ence
d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II
Week
Date
Sections
Part/ References
Notes/Speaker
Application
1: approximation
forTopic/Sections
the Travelling
Salesman Problem.
from FS2009
The
Travelling
Problem
asks to find a Hamilton path of minimum
1
Sept 7 I.1, I.2, I.3 Salesman
Symbolic methods
Combinatorial
Structures
2
14
I.5, I.6
Unlabelled structures
weight.
ToI.4,ensure
such
FS:
Part A.1,aA.2Hamiltonian path exists, we assume that the graph G
Comtet74
II.1, II.2, II.3
Labelled structures I
Handout #1
is 3the21 complete
graph
Kn. If there
are edges we want to exclude for sure from
(self
study)
4
28
II.4, II.5, II.6
Labelled structures II
any Hamilton path, we can just give them a very high weight.
5 Oct 5 III.1, III.2 Example.
6
12
IV.1, IV.2
7 19 8 26 9 Nov 2 IV.3, IV.4
IV.5 V.1 Combinatorial
parameters
FS A.III
(selfstudy) Combinatorial
Parameters Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(selfstudy) Complex Analysis Asst #1 Due Multivariable GFs Singularity Analysis
Asymptotic methods Asst #2 Due 9 VI.1 12 A.3/ C Introduction to Prob. Mariolys 18 IX.1 Limit Laws and Comb Marni 20 IX.2 Discrete Limit Laws Sophie 23 IX.3 Combinatorial
instances of discrete Mariolys 25 IX.4 Continuous Limit Laws Marni 13 30 IX.5 QuasiPowers and
Gaussian limit laws Sophie 14 Dec 10 10 11 12 Random Structures
and Limit Laws
FS: Part C
(rotating
presentations) Sophie Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11Dec09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 3 f ac u lty of sci ence
d ep ar tment of m athemat ics MATH 8954 Fall 2010
Course Schedule MACM 201 Discrete Mathematics II
Week
Date
Sections
Part/ References
Topic/Sections
Here
again,
the problem
is difficult:
finding aNotes/Speaker
Hamiltoni path of minimum weight
from FS2009
in1 a Sept
nonnegatively
weighted Symbolic
complete
graph is NPcomplete. There is no
7 I.1, I.2, I.3
methods
Combinatorial
Structures
algorithm
that
2
14
I.4,
I.5, I.6 will work in a reasonable
Unlabelled structurestime for all graphs.
FS: Part A.1, A.2
3 21 II.1, II.2, II.3 Comtet74
Handout #1
(self study) Labelled structures I To4 make
the problem easier, Labelled
we assume
that our weighted complete graph
28
II.4, II.5, II.6
structures II
satisfies
the
triangle
inequality:
for every x,Assty,#1zDue∈ V , w({x, y}) ≤ w({x, z}) +
Combinatorial
Combinatorial
5
Oct 5
III.1, III.2
parameters
Parameters
w({y, z}).
FS A.III
6 12 IV.1, IV.2 (selfstudy) Multivariable GFs 7
19
IV.3, IV.4
Now
we introduce
an Methods
algorithmComplex
thatAnalysis
uses a spanning tree.
Analytic
8 26 IV.5 V.1 FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(selfstudy) Singularity Analysis 1.
a Minimum Spanning
Tree T of
G,Duerooted at an arbitrary vertex;
9 Compute
Nov 2
Asst #2
Asymptotic methods
10 9 VI.1 18 IX.1 Sophie 2. Order
the vertices of V according
to a preorder
traversal of T .
12
A.3/ C
Introduction to Prob.
Mariolys 11 Limit Laws and Comb Marni Claim.
LetIX.2 P = xRandom
, xn be Discrete
the path
in GSophie
defined by the preorder traversal
20
Limit Laws
1 , . . .Structures
and Limit Laws
Part C
Combinatorial
of G 23givenIX.3by theFS:
above
algorithm.
Then w(P)
Mariolys is at most twice the cost of a
(rotating
instances of discrete
12
Hamilton
path of presentations)
minimum weight
in G. In other words, our algorithm is an
25
IX.4
Continuous Limit Laws Marni
approximation
algorithm for the
TSP problem
with approximation factor equal
QuasiPowers
and
13
30
IX.5
Sophie
Gaussian limit laws
to142: Dec
it 10runs quickly (we
will see finding a minimum
spanning tree can be done
Presentations
Asst #3 Due
quickly) and the solution it outputs has cost at most twice the cost of a best
Hamilton path. Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11Dec09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 4 MATH 8954 Fall 2010
Course Schedule f ac u lty of sci ence
d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II
Week
Date
Sections
Example.
from FS2009
1 Sept 7 I.1, I.2, I.3 2 14 I.4, I.5, I.6 3 21 II.1, II.2, II.3 4 28 II.4, II.5, II.6 5 Oct 5 III.1, III.2 6 12 IV.1, IV.2 7 19 IV.3, IV.4 8 26 9 Nov 2 IV.5 V.1 Part/ References Topic/Sections Combinatorial
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study) Symbolic methods Combinatorial
parameters
FS A.III
(selfstudy) Combinatorial
Parameters Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(selfstudy) Complex Analysis Notes/Speaker Unlabelled structures
Labelled structures I
Labelled structures II
Asst #1 Due Multivariable GFs Singularity Analysis
Asymptotic methods Asst #2 Due 9 VI.1 12 A.3/ C Introduction to Prob. Mariolys 18 IX.1 Limit Laws and Comb Marni 20 IX.2 Discrete Limit Laws Sophie 23 IX.3 Combinatorial
instances of discrete Mariolys 25 IX.4 Continuous Limit Laws Marni 13 30 IX.5 QuasiPowers and
Gaussian limit laws Sophie 14 Dec 10 10 11 12 Random Structures
and Limit Laws
FS: Part C
(rotating
presentations) Sophie Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11Dec09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 5 MATH 8954 Fall 2010
Course Schedule f ac u lty of sci ence
d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II
Week
Date
Sections
Proof
of the
claim. Part/ References
from FS2009
1 Sept 7 I.1, I.2, I.3 2 14 I.4, I.5, I.6 3 21 II.1, II.2, II.3 4 28 II.4, II.5, II.6 5 Oct 5 III.1, III.2 6 12 IV.1, IV.2 7 19 IV.3, IV.4 8 26 9 Nov 2 IV.5 V.1 Topic/Sections Combinatorial
Structures
FS: Part A.1, A.2
Comtet74
Handout #1
(self study) Symbolic methods Combinatorial
parameters
FS A.III
(selfstudy) Combinatorial
Parameters Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(selfstudy) Complex Analysis Notes/Speaker Unlabelled structures
Labelled structures I
Labelled structures II
Asst #1 Due Multivariable GFs Singularity Analysis
Asymptotic methods Asst #2 Due 9 VI.1 12 A.3/ C Introduction to Prob. Mariolys 18 IX.1 Limit Laws and Comb Marni 20 IX.2 Discrete Limit Laws Sophie 23 IX.3 Combinatorial
instances of discrete Mariolys 25 IX.4 Continuous Limit Laws Marni 13 30 IX.5 QuasiPowers and
Gaussian limit laws Sophie 14 Dec 10 10 11 12 Random Structures
and Limit Laws
FS: Part C
(rotating
presentations) Sophie Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11Dec09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 6 f ac u lty of sci ence
d ep ar tment of m athemat ics MATH 8954 Fall 2010
Course Schedule MACM 201 Discrete Mathematics II
Week
Date
Sections
Part/ References
Topic/Sectionsoutbreak.
Notes/Speaker
Application
2: analysis
of a pathogen
from FS2009
In1 a work
I am
doing with
BC Center for Disease Control, we have
Sept 7 I.1,
I.2, I.3 currently
Symbolic the
methods
Combinatorial
Structures
2
14
I.4,the
I.5, I.6 genomes of the bacteria
Unlabelled structures
sequenced
responsible for the disease tuberculosis
FS: Part A.1, A.2
Comtet74
3
21
II.1, II.2, II.3
Labelled structures I
Handout #1
(Mycobacterium
tuberculosis)
taken
from a few hundred of patients of an out(self
study)
4
28
II.4, II.5, II.6
Labelled structures II
break of tuberculosis (TB) in BC. We want to see if we can cluster the patients
Combinatorial
Combinatorial
5
Oct 5
III.1, III.2
Asst #1 Due
Parameters
into
groups,
whereparameters
in
each
group,
all
patients
are likely to have been infected
FS A.III
6
12
IV.1, IV.2
Multivariable GFs
(selfstudy)
with
the same
strain
of the pathogen
(i.e. the same source).
7 19 IV.3, IV.4 Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(selfstudy) Complex Analysis This
has important applicationsSingularity
as not
all strains of TB are treated the same
8
26
Analysis
IV.5 V.1
9
Nov
2
Asst #2 Due and need to be treated with
way:
some
are resistant to firstline
antibiotics
Asymptotic methods
9
VI.1
Sophie
stronger
antiobtics,
but then you want to be
careful that they do not acquire
10
12
A.3/ C
Introduction to Prob.
Mariolys
resistance to these stronger antibiotics and then infect other patients other
18
IX.1
Limit Laws and Comb
Marni
11
resistance
spreads
in
the
population
and
this
is a very serious problem.
Random Structures
20
IX.2
Discrete Limit Laws
Sophie
and Limit Laws
FS: Part C
(rotating
presentations) Combinatorial
23
IX.3
Mariolys
The
techniques
we
use
rely
on
advanced
technology
(to sequence the genomes of
instances of discrete
12
25
IX.4
Continuous
Limit Laws Marni
the pathogen)
and advanced tools
in computational
biology (to detect mutations
QuasiPowers
and
in13some
– IX.5afew hundred – selected
genes
ofSophie
Mycobacterium tuberculosis in the
30
Gaussian limit laws
patients).
But once wePresentations
have this set of mutations
14
Dec 10
Asst #3 Due for all our patients, we use
two simple tools we did see in MACM201: the Hamming distance and minimum
spanning trees. Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11Dec09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 7 MATH 8954 Fall 2010
Course Schedule f ac u lty of sci ence
d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II
Week do
Date
Sections
Part/ References
We
thefrom
following:
FS2009
1 Sept 7 I.1, I.2, I.3 Topic/Sections Notes/Speaker Symbolic methods Combinatorial
• We build a complete
graph G = (V , E) where each vertex E represents
Structures
2
14
I.4, I.5, I.6
Unlabelled structures
FS: Part A.1, A.2
a
patient
and
the
edge
between
two vertices x and y is weighted by the
Comtet74
3
21
II.1, II.2, II.3
Labelled structures I
Handout #1
(self study) between
distance
the
genomes
of Mycobacterium tuberculosis
4 Hamming
28
II.4, II.5, II.6
Labelled
structures
II
sequenced
in Combinatorial
both patients.Combinatorial
5
Oct 5
III.1, III.2
Asst #1 Due
parameters
FS A.III
(selfstudy) 6 Parameters 12
IV.1, IV.2
Multivariable GFs
• We
compute
a minimum spanning
tree in this graph, and from this tree, we
7
19
IV.3, IV.4
Complex Analysis
Methods
can
easily
seeAnalytic
(visualize)
potential
clusters of patients infected by pathogens
FS: Part B: IV, V, VI
8
26
Singularity
Analysis
Appendix B4
IV.5genomes
V.1
whose
Stanleyhave
99: Ch. 6 very similar genomes, forming the clusters we are
9
Nov 2
Asst #2 Due
Asymptotic methods
Handout #1
looking
for. (selfstudy)
9
VI.1
Sophie 10 12 A.3/ C Introduction to Prob. Mariolys 18 IX.1 Limit Laws and Comb Marni 20 IX.2 Discrete Limit Laws Sophie 23 IX.3 Combinatorial
instances of discrete Mariolys 25 IX.4 Continuous Limit Laws Marni 13 30 IX.5 QuasiPowers and
Gaussian limit laws Sophie 14 Dec 10 11 12 Random Structures
and Limit Laws
FS: Part C
(rotating
presentations) Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11Dec09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 8 MATH 8954 Fall 2010
Course Schedule f ac u lty of sci ence
d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II
Week
Date
Sections
Part/ References
Topic/Sections
Notes/Speaker
Algorithmic
principles.
from FS2009
We
to look
at two algorithms:
1 are
Sept 7going
I.1, I.2, I.3
Symbolic methodsPrim’s algorithm and Kruskal algorithm.
Combinatorial
Structures
2
14
I.4, I.5, I.6
Unlabelled structures
FS: Part
A.1, A.2
Both
algorithms
are
greedy
algorithm:
Comtet74
3 21 II.1, II.2, II.3 Handout #1
(self study) Labelled structures I 4 they
28
II.4, II.5, II.6
structures IIE
1.
want
to compute a setLabelled
of edges
T that forms a spanning tree of G,
5 Oct 5 III.1, III.2 6 12 IV.1, IV.2 Combinatorial
parameters
FS A.IIIT
(selfstudy) 2. they start with E empty, Combinatorial
Parameters Asst #1 Due Multivariable GFs 7 they
19
IV.3, IV.4
Complex
3.
repeatedly
(n−1
if GAnalysis
has n vertices) add one edge to the current
Analytic
Methodstimes
FS: Part B: IV, V, VI
8
26
Singularity Analysis
Appendix B4
set
ETIV.5of
edges,
V.1
9 Stanley 99: Ch. 6
Handout #1
(selfstudy) Nov 2 Asymptotic methods Asst #2 Due 4. Kruskal’s
algorithm adds an edge that does
9
VI.1
Sophienot create a cycle, of min. weight
12
A.3/all
C
Introduction to Prob.
Mariolys
among
such possible edges, 10 11 18 IX.1 Limit Laws and Comb Marni 5. Prim’s
adds
an edge
connected
Random
Structures
20
IX.2algorithm
Discrete Limit
Laws
Sophieto ET , of min. weight among all
and Limit Laws
FS:
Part C
Combinatorial
such
possible
edges.
23
IX.3
Mariolys 12 25 IX.4 (rotating
presentations) instances of discrete Continuous Limit Laws Marni QuasiPowers andof G with n vertices, n − 1 edges and
Kruskal’s
algorithm builds a subgraph
13
30
IX.5
Sophie
Gaussian limit laws
that
is
acyclic: a spanning
tree.
14
Dec 10
Presentations
Asst #3 Due Prim’s algorithm builds a subgraph of G with n vertices, n − 1 edges and that
is connected: a spanning tree.
The key question is: why this approach to take at any time the best available edge ensures the resulting spanning tree is of minimum weight among all
spanning trees?
Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11Dec09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 9 MATH 8954 Fall 2010
Course Schedule f ac u lty of sci ence
d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II
Week
Date
Sections
Part/ References
Topic/Sections
Notes/Speaker
Kruskal’s
algorithm.
from FS2009
Input:
= (V , E) withSymbolic
V methods
= n.
1
Septa
7 graph
I.1, I.2, I.3 G Combinatorial
2
14
I.5, I.6
Unlabelled structures
Output:
aI.4,subet
EStructures
ofA.1,EA.2that forms
a minimum weight spanning tree of G.
FS:
T Part
3 21 II.1, II.2, II.3 1. Initialization.
28
II.4, II.5, II.6 4
5
6
7 11 12 Combinatorial
Parameters 12 IV.1, IV.2 (selfstudy) Multivariable GFs 19 IV.3, IV.4 Analytic Methods
FS: Part B: IV, V, VI
Appendix B4
Stanley 99: Ch. 6
Handout #1
(selfstudy) Complex Analysis Nov 2 Asst #1 Due Singularity Analysis
Asymptotic methods Asst #2 Due QuasiPowers and
Gaussian limit laws Sophie (a)
For
9
VI.1 i from 1 to n − 2 Do /* Assume
Sophie ET = {e1 , . . . , ei } */
12
C
i.A.3/Let
e be an edge ofIntroduction
E − EtoTProb.thatMariolys
satisfies
18
IX.1
Limit Laws and Comb
Marni
A. (V , E
{e}) is Discrete
acyclic,
T ∪Structures
Random
20
IX.2
Limit Laws
Sophie
and Limit Laws
C
B. e isFS:ofPartmin.weight
among all edges
of E − ET that satisfies A.
Combinatorial
23
IX.3
Mariolys
(rotating
instances of discrete
ii.IX.4Add e presentations)
to ET and denote
by
.
25
Continuousit
Limit
Lawsei+1
Marni 13
30
IX.5
Illustration.
14 Labelled structures II (b) Add to ET an edge of E of minimum weight. Call it e1. 26
2. Main
loop.
IV.5 V.1 10 Labelled structures I Combinatorial Oct 5
III.1, III.2
(a)
Set
ET =parameters
∅.
FS A.III 8
9 Comtet74
Handout #1
(self study) Dec 10 Presentations Asst #3 Due Dr. Marni MISHNA, Department of Mathematics, SIMON FRASER UNIVERSITY
Version of: 11Dec09 Dr. Cedric Chauve, Spring 2017 Optimization 2: Minimum Spanning Trees 10 MATH 8954 Fall 2010
Course Schedule f ac u lty of sci ence
d ep ar tment of m athemat ics MACM 201 Discrete Mathematics II
Week
Date
Sections
Part/ References
Topic/Sections
Notes/Speaker
Prim’s
algorithm.
from FS2009
Input:
= (V , E) withSymbolic
V methods
= n.
1
Septa
7 graph
I.1, I.2, I.3 G Combinatorial
2
14
I.5, I.6
Unlabelled structures
Output:
aI.4,subet
EStructures
ofA.1,EA.2that forms
a minimum weight spanning tree of G.
FS...
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 Spring '09
 MarniMishna

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