Chapter 2 The Fundamentals: Algorithms, the Integers, and Matrices

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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.

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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.

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

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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.

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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 !!!
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Asymptotic Notations O-notation
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Example 3 Prove 2n1O(n)
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2.2 The Integers and Division

• We discuss the properties of integers which
  belongs to
• the branch of Mathematics called number theory.

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• 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

1. There are s and t such that bas and cat.
   Therefore, bca(st).
2. There is s such that bas. Therefore, bca(sc)
3. There are s and t such that bas and cbt,
   therefore, ca(st)

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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).
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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.
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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).
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Solution
Hence, gcd(414,662)2, since 2 is the last
nonzero remainder.
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• Representation of integers

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• Algorithm for integer operations

Addition of a and b
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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
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2.6 Matrices
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• Algorithms for matrix multiplication

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• Transposes and powers of matrices

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• Zero-One Matrices

A matrix with entries that are either 0 or 1 is
called a zero-one matrix.
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