Download Matrices - Computer Science

DISCRETE COMPUTATIONAL
STRUCTURES
CS 23022
Fall 2005
CS 23022 OUTLINE
1. Sets
9.
Matrices & Closures
2. Logic
10.
Counting Principles
3. Proof Techniques
11.
Discrete Probability
4. Algorithms
12.
Congruences
5. Integers & Induction
13.
Recurrence Relations
6. Relations & Posets
14.
Algorithm Complexity
7. Functions
15.
Graph Theory
8. Boolean Algebra &
Combinatorial
Circuits
16.
Trees & Networks
17.
Grammars & Languages
Learning Objectives
Learn about Boolean expressions
Become aware of the basic properties of
Boolean algebra
Explore the application of Boolean algebra in the
design of electronic circuits
Learn the application of Boolean algebra in
switching circuits
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Two-Element Boolean Algebra
Let B = {0, 1}.
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Two-Element Boolean Algebra
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Two-Element Boolean Algebra
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Two-Element Boolean Algebra
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Minterm
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Maxterm
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Boolean Algebra
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Boolean Algebra
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
 The Karnaugh map, or K-map for short, can be used to
minimize a sum-of-product Boolean expression.
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CS 23022 OUTLINE
1. Sets
9.
Matrices & Closures
2. Logic
10.
Congruences
3. Proof Techniques
11.
Counting Principles
4. Algorithms
12.
Discrete Probability
5. Integers & Induction
13.
Recurrence Relations
6. Relations & Posets
14.
Algorithm Complexity
7. Functions
15.
Graph Theory
8. Boolean Algebra &
Combinatorial Circuits
16.
Trees & Networks
17.
Grammars & Languages
Learning Objectives
Learn about matrices and their relationship with
relations
Become familiar with Boolean matrices
Learn the relationship between Boolean matrices
and different closures of a relation
Explore how to find the transitive closure using
Warshall’s algorithm
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Matrices
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Matrices
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Matrices – terms : equal , square
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Matrices- terms: zero matrix, diagonal elements
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Matrices- terms: diagonal matrix, identity matrix
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Matrices – Matrix Sum
 Two matrices are added only if they have the same number
of rows and the same number of columns
 To determine the sum of two matrices, their corresponding
elements are added
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Matrices – Matrix Addition Example
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Matrices- Multiply a Constant x Matrix
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Matrices – Matrix Difference
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Matrices - Properties
Commutative and Associative properties of Matrix addition
Distributive property of multiplication over addition ( subtraction )
-only holds for a constant time a matrix sum (difference)
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Matrices
The multiplication AB of matrices A and B is defined
only if the number of columns of A is the same as the
number of rows of B
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Matrices
Figure 4.1
 Let A = [aij]m×n be an m ×
n matrix and B = [bjk ]n×p
be an n × p matrix. Then
AB is defined
 To determine the (i, k)th
element of AB, take the ith
row of A and the kth column
of B, multiply the
corresponding elements,
and add the result
 Multiply corresponding
elements as in Figure 4.1
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Matrices
Note that the dimensions of AB are m x p.
Then (AB) x C is defined and has dimensions m x q
Convince yourself that A x (BC) is defined and also has dimensions m x q
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Matrices – Matrix transpose
 The rows of A are the columns of AT and the columns of
A are the rows of AT
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Matrices - Symmetric
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Matrices
Boolean (Zero-One) Matrices
Matrices whose entries are 0 or 1
Allows for representation of matrices in a convenient
way in computer memory and for the design and
implementation of algorithms to determine the
transitive closure of a relation
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Matrices
 Boolean (Zero-One) Matrices
 The set {0, 1} is a lattice under the usual “less than or equal
to” relation, where for all a, b ∈ {0, 1}, a ∨ b = max{a, b} and
a ∧ b = min{a, b}
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Matrices – Logical Operations
Note: join is the OR operation; meet is the AND operation
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Matrices
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Matrices – Boolean Product
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The Matrix of a Relation and Closure
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The Matrix of a Relation and Closure
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The Matrix of a Relation and Closure
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The Matrix of a Relation and Closure
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 ALGORITHM 4.3: Compute the transitive closure
 Input: M —Boolean matrices of the relation R
n—positive integers such that n × n specifies the size of M
 Output: T —an n × n Boolean matrix such that T is the
transitive closure of M
 1. procedure transitiveClosure(M,T,n)
 2. begin
 3.
A := M;
 4.
T := M;
 5.
for i := 2 to n do
 6.
begin
 7.
A :=
//A = Mi
 8.
∨ Mi
T := T ∨ A;
//T= M ∨ M2 ∨ · · ·
 9.
end
 10. end
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Warshall’s Algorithm for Determining the
Transitive Closure
Previously, the transitive closure of a relation R was
M R then
found by computing the matrices
and
taking the Boolean join
n
This method is expensive in terms of computer time
Warshall’s algorithm: an efficient algorithm to
determine the transitive closure
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Warshall’s Algorithm for Determining the
Transitive Closure
Let A = {a1, a2, . . . , an} be a finite set, n ≥ 1, and
let R be a relation on A.
Warshall’s algorithm determines the transitive
closure by constructing a sequence of n Boolean
matrices
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Warshall’s Algorithm for Determining the
Transitive Closure
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Warshall’s Algorithm for Determining the
Transitive Closure
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Warshall’s Algorithm for Determining the
Transitive Closure
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Warshall’s Algorithm for Determining the
Transitive Closure
 ALGORITHM 4.4: Warshall’s Algorithm
 Input: M —Boolean matrices of the relation R
 n—positive integers such that n × n specifies the size of M
 Output: W —an n × n Boolean matrix such that
W is the transitive
closure of M
 1. procedure WarshallAlgorithm(M,W,n)
 2. begin
 3.
W := M;
 4.
for k := 1 to n do
 5.
 6.
 7.
 8.
 9.
for i := 1 to n do
for j := 1 to n do
if W[i,j] = 1 then
if W[i,k] = 1 and W[k,j] = 1 then
W[i,j] := 1;
 10. end
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Learning Objectives
Learn the basic counting principles—
multiplication and addition
Explore the pigeonhole principle
Learn about permutations
Learn about combinations
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Basic Counting Principles
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Basic Counting Principles
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Basic Counting Principles
 There are three boxes containing books. The first box
contains 15 mathematics books by different authors, the
second box contains 12 chemistry books by different authors,
and the third box contains 10 computer science books by
different authors.
 A student wants to take a book from one of the three boxes.
In how many ways can the student do this?
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Basic Counting Principles
Suppose tasks T1, T2, and T3 are as follows:
T1 : Choose a mathematics book.
T2 : Choose a chemistry book.
T3 : Choose a computer science book.
Then tasks T1, T2, and T3 can be done in 15, 12, and
10 ways, respectively.
All of these tasks are independent of each other.
Hence, the number of ways to do one of these tasks
is 15 + 12 + 10 = 37.
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Basic Counting Principles
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Basic Counting Principles
 Morgan is a lead actor in a new movie. She needs to shoot a
scene in the morning in studio A and an afternoon scene in
studio C. She looks at the map and finds that there is no direct
route from studio A to studio C. Studio B is located between
studios A and C. Morgan’s friends Brad and Jennifer are
shooting a movie in studio B. There are three roads, say A1, A2,
and A3, from studio A to studio B and four roads, say B1, B2, B3,
and B4, from studio B to studio C. In how many ways can
Morgan go from studio A to studio C and have lunch with Brad
and Jennifer at Studio B?
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Basic Counting Principles
 There are 3 ways to go from studio A to studio B and 4
ways to go from studio B to studio C.
 The number of ways to go from studio A to studio C via
studio B is 3 * 4 = 12.
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Basic Counting Principles
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Basic Counting Principles
 Consider two finite sets, X1 and X2. Then
 This is called the inclusion-exclusion principle for two finite sets.
 Consider three finite sets, A, B, and C. Then
 This is called the inclusion-exclusion principle for three finite sets.
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Pigeonhole Principle
The pigeonhole principle is also known as the Dirichlet
drawer principle, or the shoebox principle.
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Pigeonhole Principle
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Pigeonhole Principle
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Permutations
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Permutations
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Combinations
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Combinations
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Generalized Permutations and Combinations
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Generalized Permutations and Combinations
Consider 10 chips of 3 types ( R, W, B) with 5 R, 3 W and 2 B
Then, n = 10, k = 3, and n1=5, n2=3 and n3 =2.
The number of different arrangements of these 10 chips is:
C(10,5) * C(10-5,3) * C(10-5-3,2) = C(10,5) * C(5,3) * C(2,2)
= 10! / 5!(5!) * 5!/ 3!(2!) * 2! / 2!(1!) = 10! / 5!3!2! = n!/ n1!n2!n3!
= 10 * 9 * 8 * 7 * 6 / (3 * 2 * 1 * 2 * 1) = 5 * 9 * 8 * 7 = 2520
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Generalized Permutations and Combinations
Consider an 8-bit string. How many 8-bit strings contain exactly three 1s ?
Using the formula above, with n = 8 and k = 3, the answer is C(8,3).
C(8,3) = 8! / 3!(8-3)! = 8!/ 3!5! = 8 * 7 * 6 / 3 * 2 * 1 = 56
Examples: 11100000 00000111 00011100 01010100 etc
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Generalized Permutations and Combinations
Then n = 3, k=2 and the number of
integer solutions is:
C(3+2-1,2-1) = C(4,1) = 4
(0,3) , (1,2), (2,1) , (3,0)
y
Suppose we have x + y = 3, with x≥ 0,
y ≥0.
y=3-x
3.5
3
2.5
2
1.5
1
0.5
0
0, 3
1, 2
2, 1
3, 0
0
1
2
3
4
x
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Generalized Permutations and Combinations
Let objects = {1,2,3,4,5}, n = 5, r=3. Then the number of 3-combinations of
these objects ( with repetitions allowed) is C(5-1+3,3) = C(7,3) = 7! / 3!(4!) = 35
111, 222, 333, 444, 555
112, 113,114, 115
221, 223, 224 , 225
331, 332, 334, 335
441, 442, 443, 445
551, 552, 553, 554
123, 124, 125 , 134, 135,145
234, 235, 245
345
Repeat all 3 = 5 combinations
Repeat two of three = 20 combinations
No repeats = 10 combinations
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Permutations and Combinations
Permutations and Combinations - Rosen
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