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PPT – Chapter 2 The Fundamentals: Algorithms, the Integers, and Matrices PowerPoint presentation | free to download - id: 77ba4a-NjUyM

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Chapter 2 The Fundamentals Algorithms, the

Integers, and Matrices

- Algorithm specifying a sequence of steps used

to solve a problem. - Computational complexity of an algorithm what

are the computer resources needed to use this

algorithm to solve a problem of a specified size?

- Integers properties of integers, division of

integers, algorithms involving integers. - Matrices basic material about matrices, matrix

arithmetic.

2.1 Algorithms

Definition 1. An algorithm is a finite sequence

of precise instructions for performing a

computation or for solving a problem.

- Solution
- Set the temporary maximum equal to the first

integer in the sequence. - Compare the next integer in the sequence to the

temporary maximum, and set the larger one to be

temporary maximum. - Repeat the previous step if there are more

integers in the sequence. - Stop when there are no integers left in the

sequence. The temporary maximum at this point is

the maximum in the sequence.

- The properties of algorithms
- Input
- Output
- Definiteness
- Correctness
- Finiteness
- Effectiveness
- Generality

Example 2 Describe an algorithm for finding an

element x in a list of distinct elements

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2.2 Complexity of Algorithms

Assume that both algorithms A and B solve the

problem P. Which one is better?

- Time complexity the time required to solve a

problem of a specified size. - Space complexity the computer memory required

to solve a problem of a specified size.

The time complexity is expressed in terms of the

number of operations used by the algorithm.

- Worst case analysis the largest number of

operations needed to solve the given problem

using this algorithm. - Average case analysis the average number of

operations used to solve the problem over all

inputs.

Number of operations

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Orders of Growth

Running Time necessary operations

Operation Per second

instant 1 second 11.5 days Never end days

instant Instant 1 second Never end days

Using silicon computer, no matter how fast CPU

will be you can never solve the problem whose

running time is exponential !!!

Asymptotic Notations O-notation

Example 3 Prove 2n1O(n)

2.2 The Integers and Division

- We discuss the properties of integers which

belongs to - the branch of Mathematics called number theory.

- Basic properties of divisibility of integers

- Theorem 1. Let a, b, and c be integers. Then
- If ab and ac, then a(bc).
- If ab, then abc for all integers c.
- If ab and bc, then ac.

Proof

- There are s and t such that bas and cat.

Therefore, bca(st). - There is s such that bas. Therefore, bca(sc)
- There are s and t such that bas and cbt,

therefore, ca(st)

Definition 2. A positive integer p greater than

1 is called prime if the only positive factors of

p are 1 and p. A positive integer that is greater

than 1 and is not prime is called composite.

- The primes less than 100 are 2, 3, 5, 7, 11, 13,

17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,

67, 71, 73, 79, 83, 89 and 97.

Theorem 2. Every positive integer can be written

uniquely as the uniquely as the product of primes

in order of increasing size.

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Definition 3. Let a and b be integers. The

largest integer d such that da and db is called

the greatest common divisor of a and b, denoted

by gcd(a,b).

Definition 4. The integers a and b are relatively

prime if gcd(a,b)1.

Definition 5. The least common multiple of the

positive integers a and b is the smallest

positive integer that is divisible by both a and

b, denoted by lcm(a,b).

Definition 6. Let a be an integer and m be a

positive integer. We denoted by a mod m the

remainder when a is divided by m.

2.4 Integers and Algorithms

- The Euclidean Algorithm Find the greatest

common divisor of two positive integers

Lemma 1 Let abqr, where a,b,q, and r are

integers. Then gcd(a,b)gcd(b,r).

Solution

Hence, gcd(414,662)2, since 2 is the last

nonzero remainder.

- Representation of integers

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- Algorithm for integer operations

Addition of a and b

Multiplication of a and b

a

1 1 0 1 0 1 1 1

1 0 1 1 0 0 0 0 1 1 0 1 0 0 0

0 1 0

b

a shifts 0 place

a shifts 1 place

a shifts 3 places

2.6 Matrices

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- Algorithms for matrix multiplication

- Transposes and powers of matrices

- Zero-One Matrices

A matrix with entries that are either 0 or 1 is

called a zero-one matrix.

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