Algorithms Fundamentals

Algorithms

Definition

• A set of instructions to solve a specific problem or perform a particular task
• A well-defined procedure that takes some input and produces a corresponding output

Characteristics

• Finiteness: Algorithms must terminate after a finite number of steps
• Definiteness: Each step of the algorithm must be precisely defined
• Effectiveness: The algorithm must be capable of being performed exactly and in a finite amount of time
• Generality: The algorithm should be applicable to a wide range of inputs

Types of Algorithms

• Recursive algorithms: Solve a problem by breaking it down into smaller instances of the same problem
• Dynamic programming algorithms: Break down a problem into smaller sub-problems and solve each only once
• Greedy algorithms: Make the locally optimal choice at each step, hoping to find a global optimum
• Backtracking algorithms: Explore all possible solutions and backtrack when a dead end is reached

Algorithm Design Techniques

• Divide and Conquer: Divide the problem into smaller sub-problems, solve each, and combine the solutions
• Dynamic Programming: Break down the problem into smaller sub-problems, solve each, and store the solutions
• Greedy Method: Make the locally optimal choice at each step
• Brute Force: Try all possible solutions and select the best one

Algorithm Analysis

• Time complexity: The amount of time an algorithm takes to complete, usually measured in Big O notation
• Space complexity: The amount of memory an algorithm uses, usually measured in Big O notation
• Trade-offs: Often, there is a trade-off between time and space complexity in algorithm design

Algorithms

Definition and Characteristics

• An algorithm is a set of instructions to solve a specific problem or perform a particular task, with a well-defined procedure that takes input and produces output
• Algorithms have four main characteristics:
  • Finiteness: They must terminate after a finite number of steps
  • Definiteness: Each step is precisely defined
  • Effectiveness: They can be performed exactly and in a finite amount of time
  • Generality: They are applicable to a wide range of inputs

Algorithm Classification

Recursive Algorithms

• Solve a problem by breaking it down into smaller instances of the same problem

Dynamic Programming Algorithms

• Break down a problem into smaller sub-problems and solve each only once

Greedy Algorithms

• Make the locally optimal choice at each step, hoping to find a global optimum

Backtracking Algorithms

• Explore all possible solutions and backtrack when a dead end is reached

Algorithm Design Techniques

Divide and Conquer

• Divide the problem into smaller sub-problems, solve each, and combine the solutions

Dynamic Programming

• Break down the problem into smaller sub-problems, solve each, and store the solutions

Greedy Method

• Make the locally optimal choice at each step

Brute Force

• Try all possible solutions and select the best one

Algorithm Analysis

Time and Space Complexity

• Time complexity: The amount of time an algorithm takes to complete, usually measured in Big O notation
• Space complexity: The amount of memory an algorithm uses, usually measured in Big O notation

Trade-offs

• Often, there is a trade-off between time and space complexity in algorithm design