Algorithm Method - Orrisa PCS Exam Notes

Algorithm method – meaning and quality, indefinate, definite

In mathematics, computing, linguistics and related subjects, an algorithm is a sequence of finite instructions, often used for calculation and data processing. It is formally a type of effective method in which a list of well-defined instructions for completing a task will, when given an initial state, proceed through a well-defined series of successive states, eventually terminating in an end-state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as probabilistic algorithms, incorporate randomness. A step-by-step method of solving a problem or making decisions, as in making a diagnosis. An established mechanical procedure for solving certain mathematical problems.

Properties of algorithm

Input

The input is the data to be transformed during the computation to produce the output.
Input precision requires that you know what kind of data, how much and what form the data should be.

Output

The output is the data resulting from the computation.
Output precision also requires that you know what kind of data, how much and what form the output should be (or even if there will be any output at all!)

Finiteness

Algorithms must terminate after a finite number of steps.

Definiteness

Each instruction of the algorithm should be clear and unambiguous.

Effectiveness

Every instruction must be basic enough to be carried out theoretically or by using paper and pencil.

Example of an algorithm: Design an algorithm to add two numbers and display the result.

Step 1 − START

Step 2 − declare three integers a, b & c

Step 3 − define values of a & b

Step 4 − add values of a & b

Step 5 − store output of step 4 to c

Step 6 − print c

Step 7 − STOP

Algorithms tell the programmers how to code the program. Alternatively, the algorithm can be written as :

Step 1 − START ADD

Step 2 − get values of a & b

Step 3 − c ← a + b

Step 4 − display c

Step 5 − STOP

Algorithm Analysis Efficiency of an algorithm can be analyzed at two different stages, before implementation and after implementation. They are the following:

A Priori Analysis

This is a theoretical analysis of an algorithm. Efficiency of an algorithm is measured by assuming that all other factors, for example, processor speed, are constant and have no effect on the implementation.

A Posterior Analysis

This is an empirical analysis of an algorithm. The selected algorithm is implemented using programming language. This is then executed on target computer machine. In this analysis, actual statistics like running time and space required, are collected.

Algorithm Complexity

Suppose X is an algorithm and n is the size of input data, the time and space used by the algorithm X are the two main factors, which decide the efficiency of X.

Time Factor

Time is measured by counting the number of key operations such as comparisons in the sorting algorithm.

Space Factor

Space is measured by counting the maximum memory space required by the algorithm.

Space Complexity

Space complexity of an algorithm represents the amount of memory space required by the algorithm in its life cycle. The space required by an algorithm is equal to the sum of the following two components :

A fixed part that is a space required to store certain data and variables, that are independent of the size of the problem. For example, simple variables and constants used, program size, etc.
A variable part is a space required by variables, whose size depends on the size of the problem. For example, dynamic memory allocation, recursion stack space, etc.

Space complexity S(P) of any algorithm P is S(P) = C + SP(I), where C is the fixed part and S(I) is the variable part of the algorithm, which depends on instance characteristic I. Following is a simple example that tries to explain the concept :

Algorithm: SUM(A, B)

Step 1 – START

Step 2 – C ← A + B + 10

Step 3 – Stop

Here we have three variables A, B, and C and one constant. Hence S(P) = 1 + 3.

Time Complexity

Time complexity of an algorithm represents the amount of time required by the algorithm to run to completion. Time requirements can be defined as a numerical function T(n), where T(n) can be measured as the number of steps, provided each step consumes constant time.

For example, addition of two n-bit integers takes n steps. Consequently, the total computational time is T(n) = c ∗ n, where c is the time taken for the addition of two bits. Here, we observe that T(n) grows linearly as the input size increases.

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