THE ROLE OF ALGORITHMS IN

COMPUTING

Instructor: Dr. Faisal Anwer, DCS, AMU

SOURCE:

Introduction to Algorithm (3rd Edition) by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein + Freely Accessible Web Resources

ALGORITHM

An Algorithm is any well defined computational procedure that

takes some value, or set of values, as input and produces some

value, or set of values as output.

An algorithm is thus a sequence of computational steps that

transform the input into the output.

We can also view an algorithm as a tool for solving a well-

specified computational problem.

Al-Khwarizmi referred algorism as the rules of performing

arithmetic using Hindu–Arabic numerals and the systematic

solution of linear and quadratic equations.

An algorithm is said to be correct if, for every input instance, it

halts with the correct output.

Algorithms are the threads that tie together most of the subfields

of computer science.

Calling something an algorithm means that the

following properties are all true:

An algorithm is an unambiguous description

An algorithm expects a defined set of inputs.

An algorithm produces a defined set of outputs.

An algorithm is guaranteed to terminate and produce

a result, always stopping after a finite time.

Most algorithms are guaranteed to produce the correct

result.

If an algorithm imposes a requirement on its inputs

(called a precondition), that requirement must be met.

KINDS OF PROBLEMS ARE SOLVED BY

ALGORITHMS

Finding good routes on which the data will travel ininternet enabled world.

Search engine to quickly find pages on which particularinformation resides.

We are given two ordered sequences of symbols, X =(x1,x2,…..) and Y(y1, y,….) and we wish to find alongest common subsequence of X and Y .

Finding route with minimum distance from one node toother node in a weighted graph.

Manufacturing and other commercial enterprises oftenneed to allocate resources in the most beneficial way. Anoil company may wish to know where to place its wellsin order to maximize its expected profit.

EXAMPLES OF ALGORITHM

Simple algorithm for finding maximum in a list ofpositive nos.

Problem: Given a list of positive numbers, return the largestnumber on the list.

Inputs: A list L of positive numbers. This list must contain atleast one number.

Outputs: A number n, which will be the largest number of thelist.

Algorithm

1. Start

2. Declare max

3. Set max to 0.

4. For each number x in the list L perform following

1. compare x to max.

2. If x is larger, set max to x.

5. max is now set to the largest number in the list.

6. Stop

EXAMPLES OF ALGORITHM(CONT…)

Does this meet the criteria for being an

algorithm?

Is it unambiguous? Yes.

Each step of the algorithm consists of primitive operations,

and translating each step into any language is very easy.

Does it have defined inputs and outputs? Yes.

Is it guaranteed to terminate? Yes.

The list L is of finite length, so after looking at every

element of the list the algorithm will stop.

Does it produce the correct result? Yes.

In a formal setting you would provide a careful proof of

correctness.

PROBLEMS VS ALGORITHMS VS PROGRAMS

For each problem or class of problems, there may

be many different algorithms.

For each algorithm, there may be many different

implementations (programs).

EXPRESSING ALGORITHMS

An algorithm may be expressed in a number of

ways, including:

natural language: usually verbose and ambiguous

flow charts: avoid most issues of ambiguity; difficult

to modify without specialized tools; largely

standardized.

pseudo-code: also avoids most issues of ambiguity;

vaguely resembles common elements of programming

languages; no particular agreement on syntax

programming language: tend to require expressing

low-level details.

PSEUDOCODE

High-level description of an algorithm.

More structured than plain English.

Less detailed than a program.

Preferred notation for describing algorithms.

Hides program design issues.

PSEUDOCODE

Control flow

if … then … [else …]

while … do …

repeat … until …

for … do …

Method declaration

Algorithm method (arg [, arg…])

Input …

Output

Method call

method (arg [, arg…])

Return value

return expression

Expressions

Assignment (equivalent to )

Equality testing(equivalent to )

n2 Superscripts and other mathematical formatting allowed

EXAMPLE: FIND THE MAX ELEMENT OF AN ARRAY

Algorithm arrayMax(A, n)

Input array A of n integers

Output maximum element of A

Max A[0]

for i 1 to n 1 do

if A[i] Max then

Max A[i]

return Max

ALGORITHMS AS A TECHNOLOGY

Suppose computers are infinitely fast and computer memory

are inexpensive.

Do we still need to study Algorithm and its

analysis ?

Yes, at least to check that the algorithm terminates and

produce correct result.

Of course, computers may be fast, but they are not

infinitely fast.

And memory may be inexpensive, but it is not free.

Computing time and memory are therefore bounded

resources.

EXERCISES

Suppose we are comparing implementations of insertion

sort and merge sort on the same machine. For inputs of

size n, insertion sort runs in 8n2 steps, while merge sort

runs in 64nlgn steps. For which values of n does

insertion sort beat merge sort?

What is the smallest value of n such that an algorithm

whose running time is 100n2 runs faster than an

algorithm whose running time is 2n on the same

machine?

SUMMARY

Knowledge of algorithm distinguishes the truly

skilled programmers from the novices.

Some tasks can be accomplished without knowing

much about algorithms, but with a good

background in algorithms, you can do much, much

more.

Next Task

Computing time and space in memory are bounded

resource.

We should use the resources wisely, and algorithms that are

efficient in terms of time or space will help you do so.