Introduction to Algorithms

Definition

• An algorithm is a step-by-step procedure or formula for solving a problem.

Characteristics

1. Finiteness: Must complete after a finite number of steps.
2. Well-defined Inputs: Should specify inputs clearly.
3. Well-defined Outputs: Should produce outputs based on the inputs.
4. Effectiveness: Each step must be basic enough to be performed precisely.

Types of Algorithms

• Sorting Algorithms: Organize data in a specific order (e.g., Quick Sort, Merge Sort).
• Search Algorithms: Retrieve information from data structures (e.g., Binary Search).
• Greedy Algorithms: Make local optimal choices at each stage (e.g., Kruskal's Algorithm).
• Dynamic Programming: Solve complex problems by breaking them down into simpler sub-problems (e.g., Fibonacci sequence calculation).
• Backtracking: Explore all potential solutions and backtrack to find the correct one (e.g., N-Queens problem).

Complexity Analysis

• Time Complexity: Measures the time an algorithm takes to complete as a function of input size.
  • Common notations: O(1), O(n), O(n log n), O(n^2).
• Space Complexity: Measures the total space used by the algorithm as a function of input size.

Common Applications

• Data processing and analysis.
• Artificial intelligence and machine learning.
• Operating systems.
• Cryptography and security.

Pseudocode

• Helps in designing algorithms before actual coding.
• Structured with clear operational steps without specific programming syntax.

Performance

• Efficiency is key: time and space considerations guide algorithm choice.
• Trade-offs: Faster algorithms may use more memory, and vice versa.

Practical Considerations

• Consider real-world constraints like hardware limits and user needs when implementing algorithms.

Algorithm Definition

• A step-by-step procedure or formula for solving a problem.

Algorithm Characteristics

• Must complete after a finite number of steps.
• Clearly defined inputs.
• Produces outputs based on the inputs.
• Each step must be basic enough to be performed precisely.

Algorithm Types

• Sorting Algorithms organize data in a specific order (e.g., Quick Sort, Merge Sort).
• Search Algorithms retrieve information from data structures (e.g., Binary Search).
• Greedy Algorithms make local optimal choices at each stage (e.g., Kruskal's Algorithm).
• Dynamic Programming solves complex problems by breaking them down into simpler sub-problems (e.g., Fibonacci sequence calculation).
• Backtracking explores all potential solutions and backtracks to find the correct one (e.g., N-Queens problem).

Complexity Analysis

• Time Complexity measures the time an algorithm takes to complete as a function of input size.
• Common notations include O(1), O(n), O(n log n), O(n^2).
• Space Complexity measures the total space used by the algorithm as a function of input size.

Common Algorithm Applications

• Data processing and analysis.
• Artificial intelligence and machine learning.
• Operating systems.
• Cryptography and security.

Pseudocode

• Helps in designing algorithms before actual coding.
• Structured with clear operational steps without specific programming syntax.

Algorithm Performance

• Efficiency is key: time and space considerations guide algorithm choice.
• Trade-offs: Faster algorithms may use more memory, and vice versa.

Practical Algorithm Considerations

• Consider real-world constraints like hardware limits and user needs when implementing algorithms.