Chapter IV Algorithm for I3ab Ams at itc

CHAPTER IV
ALGORITHMS
Institute of Technology of Cambodia
ITC
November 18, 2023
Mr. PEN CHENTRA AMS November 18, 2023 1 / 31

Contents
1 Introduction
2 Examples of Algorithms
3 Analysis of Algorithsms
4 Recursive Algorithms

Contents
1 Introduction

Introduction
An algorithm is a step-by-step method of solving some problem.
The word “algorithm” derives from the name of the ninth-century
Persian mathematician al-Khowarizmi. Today, “algorithm” typically
refers to a solution that can be executed by a computer.

Character of algorithm
Character
Algorithms typically have the following characteristics:
Input The algorithm receives input.
Output The algorithm produces output.
Precision The steps are precisely stated.
Determinism The intermediate results of each step of execution are
unique and are determined only by the inputs and the
results of the preceding steps.
Finiteness The algorithm terminates; that is, it stops after finitely
many instructions have been executed.
Correctness The output produced by the algorithm is correct; that is,
the algorithm correctly solves the problem. Generality
The algorithm applies to a set of inputs.

Finding the Maximum of Three Numbers
Finding the Maximum of Three Numbers
Figure: Pseudocode

Finding the Maximum Value in a Sequence
Finding the Maximum Value in a Sequence
Figure: Psedocode

Contents
1 Introduction

Searching
Text search

Example
Example 1 (Example)
Shows a trace of Algorithm where we are searching for the pattern
“001” in the text “010001”

Sorting
Example 2
Suppose that i = 4 and s1 = 8, s2 = 13, s3 = 20, s4 = 27 is
8 13 20 27
If s5 is 16, after it is inserted,s1, · · · , s5 becomes
8 13 16 20 27

Sorting
Figure: Insertion Sort

Randomized Algorithm
A randomized algorithm does not require that the intermediate results
of each step of execution be uniquely defined and depend only on the
inputs and results of the preceding steps. By definition, when a
randomized algorithm executes, at some points it makes random
choices. In practice, a pseudorandom number generator is used

Example

Example
Figure

Contents
1 Introduction

Required time
Time to Execute an Algorithm if One Step Takes 1 Microsecond to
Execute. lg n denotes log2 n (the logarithm of n to base 2)
Figure: Table of required time to execute

Case time
We can ask for the minimum time needed to execute the algorithm
among all inputs of size n. This time is called the best-case time for
inputs of size n. We can also ask for the maximum time needed to
execute the algorithm among all inputs of size n. This time is called the
worst-case time for inputs of size n. Another important case is
average-case timethe average time needed to execute the algorithm
over some finite set of inputs all of size n.

Example
In practise, we are less interested in the exact best-case or worst-case
time required for an algorithm to execute than we are in how the
best-case or worst-case time grows as the size of the input increases
Example 3
Suppose that the worst-case time of an algorithm is
t(n) = 60n2
+ 5n + 1
for input of size n. For large n, the term 60n2 is approximately equal to
t(n)

Figure: Comparing growth of t(n) with 60n2
Under these assumptions, t(n) grows like n2 as n increases. We say that
t(n) is of order n2 and write t(n) = Θ(n2), which is read “t(n) is theta of
n2.” The basic idea is to replace an expression, such as
t(n) = 60n2 + 5n + 1, with a simpler expression, such as n2, that grows
at the same rate as t(n). The formal definitions follow.

Definition
Definition 4
Let f and g be functions with domain {1, 2, 3, ...}.
We write
f(n) = O(g(n))
and say that f(n) is of order at most g(n) or f(n) is big oh of g(n)
if there exists a positive constant C1 such that
|f(n)| ≤ C1|g(n)|
for all but finitely many positive integers n.

Definition
Definition
We write
f(n) = Ω(g(n))
and say f(n) is of order at least g(n) or f(n) is omega of g(n) if
there exists a positive constant C2 such that
|f(n)| ≥ C2|g(n)|
for all but finitely many positive integers n.

Definition
Definition
We write
f(n) = Θ(g(n))
and say f(n) is of order g(n) or f(n) is theta of g(n) if
f(n) = O(g(n)) and f(n) = Ω(g(n))

Example 5
Since
60n2 + 5n + 1 ≤ 60n2 + 5n2 + n2 = 66n2for all n ≥ 1, we may take
C1 = 66 , then
60n2
+ 5n + 1 = O(n2
)
60n2 + 5n + 1 ≥ 60n2 for all n ≥ 1, we may take C2 = 60, then
60n2
+ 5n + 1 = Ω(n2
)
Since 60n2 + 5n + 1 = O(n2) and 60n2 + 5n + 1 = Ω(n2), then
60n2
+ 5n + 1 = Θ(n2
)

Theorem
Theorem 6
Let
p(n) = aknk
+ ak−1nk−1
+ · · · + a1n + a0
be a polynomial in n of degree k, where each ai is nonnegative. Then
p(n) = Θ(nk
)

Contents
1 Introduction

A recursive function (pseudocode) is a function that invokes itself. A
recursive algorithm is an algorithm that contains a recursive function.
Recursion is a powerful, elegant, and natural way to solve a large class
of problems. A problem in this class can be solved using a
divide-and-conquer technique in which the problem is decomposed
into problems of the same type as the original problem. Each
subproblem, in turn, is decomposed
further until the process yields subproblems that can be solved in a
straightforward way. Finally, solutions to the subproblems are
combined to obtain a solution to the original problem.

Example
Example 7 (Example)
Recall that if n ≥ 1, n! = n(n − 1) · · · 2 · 1, and 0! = 1. Notice that if
n ≥ 2, n factorial can be written “in terms of itself” since, if we “peel
off” n, the remaining product is simply (n − 1)!; that is,
n! = n(n − 1)(n − 2) · · · 2 · 1 = n · (n − 1)!
Figure: Decomposing the Factorial
Problem
Figure: Combining Subproblems of
the Factorial Problem

Algorithms
Algorithm
Figure: recursive algorithm

Example
Example 8
A robot can take steps of 1 meter or 2 meters. We write an algorithm to
calculate the number of ways the robot can walk n meters. As
examples:
Figure: walk(n) = walk(n − 1) + walk(n − 2)

Robot walk

Robot Walking

Chapter IV Algorithm for I3ab Ams at itc

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