CSC331 Week 2 Topic D

8/13/2019 CSC331 Week 2 Topic D

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CSC331 Discrete Structuresand Logic

October 2012

Topic D

Algorithms


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Section 4.1 Introduction

Analgorithm is a finite set of instructions with

the following characteristics:

Precision: steps are precisely stated

Uniqueness: Results of each step of execution are

uniquely defined. They depend only on inputs and

results of preceding steps

Finiteness: the algorithm stops after finitely many

steps


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More characteristics of algorithms

Input: the algorithm receives input

Output: the algorithm produces output

Generality: the algorithm applies to various sets of

inputs


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Example: a simple algorithm

Algorithm to find the largest of three

numbers a, b, c:

Assignment operator

s := k means copy the value of k into s

1. x:= a

2. If b > x then x:= b

3. If c > x then x:= cA traceis a check of the algorithm for

specific values of a, b and c


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Section 4.2 Examples of Algorithms

Algorithm 4.2.1Text Search:


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Traces of algorithms

In simple words, a trace is a test run of an

algorithm for some specific inputs.

We will do an example of trace for the Text

Search algorithm together.


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Section 4.3 Analysis of algorithms

Complexity: the amount of time and/or

space needed to execute the algorithm.

Complexity depends on many factors: data

representation type, kind of computer,

computer language used, etc.


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Types of complexity

Best-case time = minimum time needed to

execute the algorithm for inputs of size n Worst-case time = maximum time needed

to execute the algorithm for inputs of size n

Average-case time = average time needed


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Order of an algorithm

Let f and g be functions with domain Z+= {1, 2, 3,}

f(n) = O(g(n)): f(n) is of order at most g(n)

if there exists a positive constant C1such that |f(n)| < C1|g(n)|

for all but finitely many n

f(n) = (g(n)): f(n) is of order at least g(n)

if there exists a positive constant C2such that |f(n)| > C2|g(n)|

for all but finitely many n

f(n) = (g(n)): f(n) is or order g(n) if it is O(g(n)) and

(g(n)).


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Section 4.4 Recursive algorithms

A recursive procedureis a procedure that invokes

itself

Example: given a positive integer n, factorial of nisdefined as the product of n by all numbers less than n

and greater than 0. Notation: n! = n(n-1)(n-2)3.2.1

Observe that n! = n(n-1)! = n(n-1)(n-2)!, etc.

A recursive algorithmis an algorithm that containsa recursive procedure


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Fibonacci sequence

Leonardo Fibonacci (Pisa, Italy, ca. 1170-1250)

Fibonacci sequence f1, f2, defined recursively

as follows:

f1= 1

f2= 2

fn= fn-1+ fn-2 for n > 3

First terms of the sequence are: 1, 2, 3, 5, 8, 13,21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,

CSC331 Week 2 Topic D

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