What is an Algorithm | Introduction to Algorithms

Algorithm is a set of finite, well-defined steps or instructions designed to solve a problem or perform a computation. It can also be defined as a procedure for solving a mathematical or computational problem in a finite number of steps, often involving repetitive or recursive operations.

Need for Algorithms:

• Solve complex problems efficiently and effectively.
• Automate processes, making them reliable, faster, and easier.
• Enable computers to perform tasks difficult or impossible for humans.
• Widely used in mathematics, computer science, engineering, finance, and data analysis.

Use of the Algorithms

Algorithms are fundamental in solving problems efficiently across various fields:

• Computer Science: Basis of programming, from simple sorting and searching to AI and machine learning.
• Mathematics: Solve problems like linear equations, shortest paths, and optimization.
• Operations Research: Optimize logistics, transportation, and resource allocation.
• Artificial Intelligence: Power intelligent systems for tasks like image recognition, NLP, and decision-making.
• Data Science: Analyze and extract insights from large datasets in marketing, healthcare, finance, etc.

Algorithms can be simple or complex, depending on the task. Think of it like following a recipe: step-by-step instructions lead to the desired outcome.

Properties of an Algorithm

An algorithm is a precise set of instructions to solve a problem. For a set of instructions to qualify as an algorithm, it must have the following properties:

Clear and Unambiguous:

• Each step should be precise and lead to only one interpretation.

Well-Defined Inputs:

• An algorithm may take zero or more inputs, which must be clearly specified.

Well-Defined Outputs:

• It must produce at least one output, clearly defined.

Finiteness:

• The algorithm must terminate after a finite number of steps.
• Infinite loops or recursion without base conditions violate this property.

Effectiveness:

• Steps should be simple, feasible, and executable with available resources.

Deterministic:

• For the same input, the algorithm should always produce the same output.

Language Independent:

• Algorithms are plain instructions and can be implemented in any programming language, producing the same result.

How to express an Algorithm?

• Natural Language: Written in plain English. Easy to describe, but can be unclear for complex problems.
• Flowchart: Graphical representation of steps. Easier to visualize than natural language.
• Pseudocode: Text-based, code-like instructions without language syntax.
  • Best way to express an algorithm.
  • Understandable even to beginners with basic knowledge.

Steps to Design an Algorithm

To design an algorithm, the following prerequisites must be considered:

1. Problem Definition: Clearly define the problem to be solved.
2. Constraints: Identify any limitations or rules.
3. Inputs: Determine what data will be provided.
4. Outputs: Specify the expected results.
5. Solution Feasibility: Ensure the solution works within the given constraints.

Example: Consider the example find the Largest of Three Numbers.

Step 1: Fulfilling Prerequisites

• Problem: Find the largest of three numbers.
• Constraints: Only numeric inputs are allowed.
• Inputs: Three numbers (num1, num2, num3).
• Output: The largest number among the three.
• Solution: Compare the numbers using conditional statements to determine the largest.

Step 2: Designing the algorithm

Now let's design the algorithm with the help of the above pre-requisites:

START

1. Read three numbers: num1, num2, num3

2. If num1 > num2 and num1 > num3, then

largest = num1

Else if num2 > num3, then

largest = num2

Else

largest = num3

3. Print largest

END

Step 3: Implementation

Program:

#include <iostream>
using namespace std;

int main() {
    int num1, num2, num3, largest;

    cout << "Enter three numbers: ";
    cin >> num1 >> num2 >> num3;

    if (num1 > num2 && num1 > num3)
        largest = num1;
    else if (num2 > num3)
        largest = num2;
    else
        largest = num3;

    cout << "Largest number: " << largest << endl;
    return 0;
}

import java.util.Scanner;

public class LargestNumber {
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        System.out.print("Enter three numbers: ");
        int num1 = sc.nextInt();
        int num2 = sc.nextInt();
        int num3 = sc.nextInt();
        int largest;

        if (num1 > num2 && num1 > num3)
            largest = num1;
        else if (num2 > num3)
            largest = num2;
        else
            largest = num3;

        System.out.println("Largest number: " + largest);
        sc.close();
    }
}

# Example inputs
num1 = 10
num2 = 25
num3 = 15

if num1 > num2 and num1 > num3:
    largest = num1
elif num2 > num3:
    largest = num2
else:
    largest = num3

print("Largest number:", largest)

Output

Largest number: 25

Time Complexity: O(1)

• The algorithm performs a fixed number of comparisons, regardless of the input values, so the execution time is constant.

Auxiliary Space: O(1)

• Only a fixed number of variables are used (num1, num2, num3, largest), so memory usage is constant.

Note:

• A problem can have multiple algorithmic solutions.
• For example, finding the largest of three numbers can be done using:
  • Nested if-else comparisons (as shown)
  • Python’s max() function (largest = max(num1, num2, num3))
  • Ternary operators or conditional expressions
  • Sorting the numbers and picking the last element

The method chosen depends on readability, simplicity, or performance considerations.

How to Analyze an Algorithm

To determine if an algorithm is efficient, its performance is analyzed in two ways:

1. Priori Analysis (Before Implementation):
   • Evaluates the algorithm theoretically, without running it.
   • Assumes factors like processor speed are constant.
   • Provides an approximate measure of time and space complexity.
2. Posterior Analysis (After Implementation):
   • Evaluates the algorithm practically, by executing it.
   • Measures real performance: correctness, time taken, and memory used.
   • Dependent on hardware and compiler.

Read Also:
Complete Data Structures & Algorithms Tutorial for topic-wise guide, practice problems and interview questions.