Algorithms Tutorial

Algorithm is a step-by-step procedure for solving a problem or accomplishing a task. In the context of data structures and algorithms, it is a set of well-defined instructions for performing a specific computational task. Algorithms are fundamental to computer science and play a very important role in designing efficient solutions for various problems. Understanding algorithms is essential for anyone interested in mastering data structures and algorithms.

What is an Algorithm?

An algorithm is a finite sequence of well-defined instructions that can be used to solve a computational problem. It provides a step-by-step procedure that convert an input into a desired output.

How do Algorithms Work?

Algorithms typically follow a logical structure:

• Input: The algorithm receives input data.
• Processing: The algorithm performs a series of operations on the input data.
• Output: The algorithm produces the desired output.

Characteristics of an Algorithm:

• Clear and Unambiguous:  The algorithm should be unambiguous. Each of its steps should be clear in all aspects and must lead to only one meaning.
• Well-defined Inputs: If an algorithm says to take inputs, it should be well-defined inputs. It may or may not take input.
• Well-defined Outputs: The algorithm must clearly define what output will be yielded and it should be well-defined as well. It should produce at least 1 output.
• Finiteness: The algorithm must be finite, i.e. it should terminate after a finite time.
• Feasible: The algorithm must be simple, generic, and practical, such that it can be executed using reasonable constraints and resources.
• Language Independent: Algorithm must be language-independent, i.e. it must be just plain instructions that can be implemented in any language, and yet the output will be the same, as expected.

What is the Need for Algorithms?

Algorithms are essential for solving complex computational problems efficiently and effectively. They provide a systematic approach to:

• Solving problems: Algorithms break down problems into smaller, manageable steps.
• Optimizing solutions: Algorithms find the best or near-optimal solutions to problems.
• Automating tasks: Algorithms can automate repetitive or complex tasks, saving time and effort.

Examples of Algorithms

Below are some example of algorithms:

• Sorting algorithms: Merge sort, Quick sort, Heap sort
• Searching algorithms: Linear search, Binary search, Hashing
• Graph algorithms: Dijkstra’s algorithm, Prim’s algorithm, Floyd-Warshall algorithm
• String matching algorithms: Knuth-Morris-Pratt algorithm, Boyer-Moore algorithm

How to Write an Algorithm?

To write an algorithm, follow these steps:

• Define the problem: Clearly state the problem to be solved.
• Design the algorithm: Choose an appropriate algorithm design paradigm and develop a step-by-step procedure.
• Implement the algorithm: Translate the algorithm into a programming language.
• Test and debug: Execute the algorithm with various inputs to ensure its correctness and efficiency.
• Analyze the algorithm: Determine its time and space complexity and compare it to alternative algorithms.

Learn Basics of Algorithms

• What is Algorithm | Introduction to Algorithms
• Definition, Types, Complexity, Examples of Algorithms
• Algorithms Design Techniques
• Why is analysis of an Algorithm important?

Analysis of Algorithms

• Asymptotic Analysis
• Worst, Average and Best Cases
• Asymptotic Notations
• Lower and Upper Bound Theory
• Introduction to Amortized Analysis
• What does ‘Space Complexity’ mean?
• Polynomial Time Approximation Scheme
• Accounting Method | Amortized Analysis
• Potential Method in Amortized Analysis

Types of Algorithms

Algorithms can be different types, depending on what they do and how they’re made. Some common types are:

1. Searching and Sorting Algorithms

• Introduction to Searching Algorithms
• Introduction to Sorting Algorithm
• Stable and Unstable Sorting Algorithms
• Lower bound for comparison based sorting algorithms
• Can Run Time Complexity of a comparison-based sorting algorithm be less than N logN?
• Which sorting algorithm makes minimum number of memory writes?

2. Greedy Algorithms

• Introduction to Greedy Algorithms
• Activity Selection Problem
• Huffman Coding
• Job Sequencing Problem
• Quiz on Greedy Algorithms
• Minimum Number of Platforms Required for a Railway/Bus Station

3. Dynamic Programming Algorithms

• Introduction to Dynamic Programming
• Overlapping Subproblems Property
• Optimal Substructure Property
• Longest Increasing Subsequence
• Longest Common Subsequence
• Min Cost Path
• Coin Change
• Matrix Chain Multiplication
• 0-1 Knapsack Problem
• Longest Palindromic Subsequence
• Palindrome Partitioning

4. Pattern Searching Algorithms

• Introduction to Pattern Searching
• Naive Pattern Searching
• KMP Algorithm
• Rabin-Karp Algorithm
• Pattern Searching using a Trie of all Suffixes
• Aho-Corasick Algorithm for Pattern Searching
• Z algorithm (Linear time pattern searching Algorithm)

5. Backtracking Algorithm

• Introduction to Backtracking
• Print all permutations of a given string
• The Knight’s tour problem
• Rat in a Maze
• N Queen Problem
• Subset Sum
• m Coloring Problem
• Hamiltonian Cycle
• Sudoku

6. Divide and Conquer Algorithm

• Introduction to Divide and Conquer
• Merge Sort
• Write your own pow(x, n) to calculate x*n
• Count Inversions
• Closest Pair of Points
• Strassen’s Matrix Multiplication

7. Geometric Algorithm

• Introduction to Geometric Algorithms
• Closest Pair of Points | O(nlogn) Implementation
• How to check if a given point lies inside or outside a polygon?
• How to check if two given line segments intersect?
• Given n line segments, find if any two segments intersect
• How to check if given four points form a square
• Convex Hull using Jarvis’ Algorithm or Wrapping

8. Mathematical Algorithms

• Introduction to Mathematical Algorithms
• Write an Efficient Method to Check if a Number is Multiple of 3
• Write a program to add two numbers in base 14
• Program for Fibonacci numbers
• Average of a stream of numbers
• Multiply two integers without using multiplication, division and bitwise operators, and no loops
• Babylonian method for square root
• Sieve of Eratosthenes
• Pascal’s Triangle
• Given a number, find the next smallest palindrome
• Program to add two polynomials
• Multiply two polynomials
• Count trailing zeroes in factorial of a number

9. Bitwise Algorithms

• Introduction to Bitwise Algorithms
• Little and Big Endian
• Detect opposite signs
• Swap bits
• Turn off the rightmost set bit
• Rotate bits
• Next higher number with same number of set bits
• Swap two nibbles in a byte

10. Graph Algorithms

• Introduction to Graph Algorithms
• BFS, DFS
• Cycles in Graph
• Shortest paths
• MST
• Topological Sorting
• Connectivity
• Max Flow

11. Randomized Algorithms

• Introduction to Randomized Algorithms
• Linearity of Expectation
• Expected Number of Trials until Success
• Randomized Algorithms | Set 0 (Mathematical Background)
• Randomized Algorithms | Set 1 (Introduction and Analysis)
• Randomized Algorithms | Set 2 (Classification and Applications)
• Randomized Algorithms | Set 3 (1/2 Approximate Median)
• Reservoir Sampling

12. Branch and Bound Algorithms

• Branch and Bound | Set 1 (Introduction with 0/1 Knapsack)
• Branch and Bound | Set 2 (Implementation of 0/1 Knapsack)
• Branch and Bound | Set 3 (8 puzzle Problem)
• Branch And Bound | Set 4 (Job Assignment Problem)
• Branch and Bound | Set 5 (N Queen Problem)
• Branch And Bound | Set 6 (Traveling Salesman Problem)

Quizzes:

• Analysis of Algorithms
• Sorting
• Divide and Conquer
• Greedy Algorithms
• Dynamic Programming
• Backtracking
• NP Complete
• Searching
• Recursion
• Bit Algorithms