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# Algorithms in Programming

## Introduction

Welcome, In this hub I will discuss about good algorithms design and software Development. Not only good programming can make good software, the choice of the algorithm is an important aspect of good software design.

To analyze an Algorithm we take two factors into consideration, they are the **Time** and **Space**, they are measured in respect to __size of the input__ into the algorithm. Computers are supposed to solve complex problems with relatively large inputs.

I will discuss the importance of Algorithm Design for a computer program.

## Computer Algorithm

An algorithm is a description of a sequence of activities that constitute a process of getting the desired outputs from the given inputs. An algorithm when expressed in a language understood by a computer is a computer program.

An algorithm should have the following *characteristics*:

1. **Finiteness** : it should have terminate in a finite number of steps

2. **Definite**: Each step should be precisely defined.

3. **Effective**: Each step should be sufficiently basic to understand.

4 **Input** and 5. **Output**: for a given input the algorithm produces a precise output.

## Why study Algorithm

Let us consider an algorithm to compute the *GCD / HCF* of two numbers.

__Algo1__. using the basic definition

Input n1,n2; Min=iif(n1<n2,n1,n2) for i=Min to 1 step -1 if n1 mod i ==0 and n2 mod i ==0 then return i; break else continue

__Algo2__: the Euclid algorithm, the intelligent method.

__Algo3__: factorizing and grouping we studied at school.

There are many more algorithms iterative and recursive, we can code to make the work done. We know a few Sorting algorithms; Now, the question arises __Which algorithm will be better__ __than the other__, this comes to the study on *Analysis of Algorithms*.

The algorithm which produces its output in less time for all expected inputs on a standard computer is obviously better. But, how shall me measure it?

Generally, the resources that are taken into consideration for analyzing algorithm include:

1. **Time** expected to be taken in executing the instances of the problem generally as a function of size of the instance. we say the Time Complexity of the algorithm.

2. **Memory space** expected to be required by the computer system in executing the instances of the problem generally as a function of the size of the instances. we say the Space Complexity of the algorithm.

We can write algorithms for our problems in recursive manner or using the iterative approach. There are many *well known techniques* for designing an algorithm say 1. **Divide & Conquer**, 2. **Dynamic Programming**, 3. **Greedy approach**, 4. **Search and traversals**, 5. **Back Tracking**, 6. **Branch and Bound**.

After the study of an ** algorithm**, good

**choices makes the efficient computer program.**

*Data Structure*## Asympototic Bounds

These are the mathematical functions and notations for the Analysis. I will not be discussing the Asymptotic Analysis nor defining them in this blog, but I shall be using the Asymptotic analysis to analyze the algorithms discussed later. Please check the web for well known asymptotic functions and notations.

The __approaches for determining the complexity__ for executing an algorithm can be **empirical** or **theoretical**. In the view of the advantages of the theoretical approach we shall use it as the only approach for computing complexities of algorithms.

## Analysis of Algorithms

As, I do not want to be much theoretical, I shall not write the complete algorithms to explain the concepts discussed, and I did not earlier either; but I will discuss the main concept behind the actual algorithm. *I would therefore expect everyone reading this blog having some programming experience with the knowledge of computing*.

**Problem1**. Consider a program to find, if the given input number * n* is a Prime number.

to solve the problem we normally use a

*loop starting from*, computing the

__2__to the number__n-1____mod__at each iteration and

*break the loop*if

*is found to be a composite number. Clearly, the for-loop executes*

__n__**which is obvious in the worst case.**

*at-most*__n__times*Do we need at-most*We can first check if the number is

__n__iterations in average case?__even__, and increment the

*loop step by*, thus checking for divisibility by

__2__instead of__1__step*4,6,8,10...*even numbers would not be necessary. Now, making this slight modification to our algorithm would execute in

*steps even in worst case which is better.*

__n/2 +1__*This can be improved further?*We know that if a number

*is divisible by*

__n__*(integer of course) then there also exists another integer*

__a__*by which*

__b__*is divisible, and the relation is given by*

__n__*; if*

__a * b =n__*are distinct then the relation between*

__a , b__*is either*

__a & b__*. Now, we can easily*

__a>b__or__a<b__*terminate the loop when the loop-counter reaches square-root-of number*. Thus our algorithm performs much better

__n__**O(√n)**even at worst case compared to

**O(n)**.

int number, factor; printf("Input a number"); scanf("%d", &number); if (number==2 || number==3) {printf("The number is a Prime number"); getch(); exit(0);} else if (number%2==0) {printf("The number is a even number"); getch(); exit(0);} for (factor=3;;factor+=2) { if (number%factor==0) {printf("The number is a composite number"); break;} else if (factor*factor>number) {printf("The number is Prime"); break;} }

## more examples on algorithm Analysis

In the next example, I will discuss the problem of computing __prefix average__.

The input is a single dimension array *A* of *n* elements, the output is the array *B* where for every element *i* in *B* is the average of the elements __1__ to *i* of array *A*.

this is : B[i]=(1/i)∑A[k] *for k=1 to i*.

We can compute *B[i]* for every *i*, that is at every step of *i* the array *A* from index *1* to *i* is computed, and assigned to B[i]. here we will need two nested loops which results the total operations is a *quadratic function of input size n*. We can modify the algorithm to a

*linear algorithm by simply avoiding to compute the partial sums repeatedly*.

for i=1 to n sum=sum+A[i] B[i]=Sum/i end for

**Problem 3.** Computation of the Exponent of a number

lastly explaining this problem using an example rather than theoretically defining it.

Suppose I want to compute 5 to the power 15, So instead of computing 5*5*5*5*5*5*5*5*5*5*5*5*5*5*5 * 14* multiplication steps, ((((5^2)^2)^2)^2)/5 we need only

*multiplication steps, squaring at each iteration.*

__5__In the ** next **problem on Integer multiplication the Karatsuba algorithm is faster than the known methods.

## Algorithm Design

There are many more topics of interest in designing efficient algorithms, of which the following topics of interest shall be discussed later on this blog.

The topis of Interest I would like to discuss are, *Please check the web for further information*.**1**. Divide and Conquer**2**. Randomization**3**. Greedy Technique**4**. Dynamic Programming**5**. Graph Algorithms

and finally **6**. Algorithmically Unsolvable problems.

I hope I was able to give an introduction regarding the importance of algorithm analysis and design of good computer programs. Hope you enjoyed the reading.

**© 2012 Dipankar Basu**