Operating System
dxf@bupt.edu.cn
Exam
40% unannounced quiz
60% final exam
Teaching method
Reference book
1.2
Reference book
System Concepts, Seventh Edition, Avi Silberschatz, Peter Baer
Galvin and Greg Gagne, John Wiley & Sons, Inc., 2004. ISBN 0471
8.6 Partial orderings
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science & Technology
Beijing University of Posts &
Telecommunications
Content
Partial Order
14/ 10/ 10
Linear order ( ?)
Quasiorder ( ?)
Product Partial Order ( ?)
Lexicographic Orde
Discrete Mathematical Structures
Trees ( ?)
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science & Technology
Beijing University of Posts &
Telecommunications
Application
?
?
?
?
?
14/ 9/ 4
College of Computer Science & Technology, BUPT
2
14/ 9/ 4
C
6.3 Bayes Theorem
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science & Technology
Beijing University of Posts &
Telecommunications
Example 1
14/ 9/ 3
We have two boxes. The first contains two
green balls and seven red balls; The second
contains fo
Discrete Mathematical Structures
Graph Coloring
?
Yang Juan
jyang@tseg.org
YangJuan
College of Computer Science & Technology
yangjuan@bupt.edu.cn
Beijing University of Posts &
CollegeofComputerScience&Technology
Telecommunications
BeijingUniversityofPosts
Discrete Mathematical Structures
Minimum Spanning Trees
( ?)
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science & Technology
Beijing University of Posts &
Telecommunications
Weighted graph ( ?)
14/ 9/ 3
Definition:a weighted graph is a graph for
w
Discrete Mathematical Structures
Shortest Path Problem
( ?)
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science & Technology
Beijing University of Posts &
Telecommunications
Many problems in the real life are related to
the shortest path problem.
Discrete Mathematical Structures
Euler Paths and Circuits ( ?
?)
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science & Technology
Beijing University of Posts &
Telecommunications
Problem
CanIdrawthefigureinonecontinuous
tracewithnolinebeingdrawntw
Discrete Mathematical Structures
Planar Graphs
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science & Technology
Beijing University of Posts &
Telecommunications
Planar Graphs ?
A graph is called planar if it can be drawn
in the plane in such a way
What are Graphs?
Not
Generalmeaningineverydaymath:
Aplotorchartofnumericaldatausinga
coordinatesystem.
Technicalmeaningindiscretemathematics:
Aparticularclassofdiscretestructures(tobe
defined)thatisusefulforrepresenting
relationsandhasaconvenientwebbylook
9.4: Connectivity
In an undirected graph, a path of length n
from u to v is a sequence of adjacent edges
going from vertex u to vertex v.
A path is a circuit if u=v.
A path pass through the vertices or traverses
the edges.
A path is simple if it contains
8.6 Partial orderings
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science &
Technology
Beijing University of Posts &
Telecommunications
Content
Partial Order
2013/11/6
Linear order
Quasiorder
Product Partial Order
Lexicographic Order
Hasse Diagram
8.5 Equivalence
Relations
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science &
Technology
Beijing University of Posts &
Telecommunications
Definition
A relation R on a set A is an equivalence
relation () iff R is
2013/11/6
reflexive
symmetric
tran
Relations
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science &
Technology
Beijing University of Posts &
Telecommunications
8 Relations
2013/11/6
8.1 Relations and Their Properties
8.2 n-ary Relations and Their Applications
n
8.3 Representing Relat
8.4 Closures of Relations
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science &
Technology
Beijing University of Posts &
Telecommunications
Closures of Relations
Definition
3 elements:
2013/11/6
The closure() of a relation R with respect to
propert
8.5 Equivalence Relations
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science & Technology
Beijing University of Posts &
Telecommunications
Definition
A relation R on a set A is an equivalence
relation ( ? ) iff R is
14/ 9/ 3
reflexive
symmetric
tr
8.4 Closures of Relations
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science & Technology
Beijing University of Posts &
Telecommunications
Closures of Relations
Definition
3 elements:
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The closure( ? ) of a relation R with respect to
prope
Relations
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science & Technology
Beijing University of Posts &
Telecommunications
8 Relations
14/ 9/ 3
8.1 Relations and Their Properties
?
8.2 n-ary Relations and Their Applications
n ?
8.3 Representing R
7.3: Divide & Conquer
R.R.s
Main points so far:
Many types of problems are solvable by
reducing a problem of size n into some
number a of independent subproblems, each
of size n/b, where a1 and b>1.
The time complexity to solve such problems
is given by
7.4: Generating Functions
Definition: The generating function for the
sequence a0,a1.ak of real numbers is the
infinite series
G ( x) =a0 +a1 x +L ak x k +L = ak x k
k =0
14/ 9/ 3
College of Computer Science & Technology, BUPT
Example 1
The generating fun
7.4: Generating
Functions
Definition: The generating function for the
sequence a0,a1.ak of real numbers is the
infinite series
G( x) a0 a1 x
2013/11/6
ak x k
College of Computer Science & Technology, BUPT
ak x k
k 0
Example 1
The generating functions f
7.3: Divide & Conquer
R.R.s
Main points so far:
Many types of problems are solvable by
reducing a problem of size n into some
number a of independent subproblems, each
of size n/b, where a1 and b>1.
The time complexity to solve such problems
is given by
Advanced Counting
Techniques
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science &
Technology
Beijing University of Posts &
Telecommunications
7 Advanced Counting
Techniques
7.1 Recurrence Relations
7.2 Solving Recurrence Relations
2013/11/6
Linear
6.3 Bayes Theorem
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science &
Technology
Beijing University of Posts &
Telecommunications
Example 1
2013/11/6
We have two boxes. The first contains two
green balls and seven red balls; The second
contains f
Advanced Counting
Techniques
Yang Juan
yangjuan@bupt.edu.cn
College of Computer Science & Technology
Beijing University of Posts &
Telecommunications
7 Advanced Counting
Techniques
7.1 Recurrence Relations
7.2 Solving Recurrence Relations
14/ 9/ 3
Linear