Algorithm Analysis and Complexity

Study Notes

Benchmarking: Running algorithms on a real machine to measure speed and memory consumption. Limited by specific conditions (machine, language, input data) and less useful for generalized analysis.
Asymptotic Analysis (Frequency Count Method): Analyzes algorithms' behavior as input size grows. Focuses on counting machine instructions based on input size. Efficient for predicting performance with large datasets, independent of specific implementation and inputs.

Definition: Describes how the execution time of an algorithm grows with the problem size.
Importance: Measures how fast an algorithm runs.

Definition: Mathematical notation used in complexity theory to describe the order of growth (asymptotic behavior) of functions.
- It focuses on the dominant term of a function as input size increases.
- Used to measure the performance of algorithms, providing an upper bound of the function's growth (worst-case scenario).
Importance:
- Describes an algorithm's efficiency, time complexity, and space complexity.
- Provides a standardized way to compare algorithms.

Computer speed: Processing power directly affects execution time.
Programming language: Different languages can have varying execution speeds.
Compiler: The compiler translates the code into machine instructions, influencing efficiency.
Input: The size and characteristics of the input data significantly affect the algorithm's run time.

Input size: Directly impacts how long an algorithm takes.
Rate of growth: Measures how fast the running time increases with the input size. This is crucial for determining how efficient the algorithm is.
Simplifying Functions: Focus on the dominant term (the term that grows fastest) in a function to understand its growth rate, ignoring constants and less significant terms.
Example: An algorithm with a running time of [6n^2 + 100n + 300] machine instructions will be dominated by the [6n^2] term as [n] becomes large, indicating a time complexity of O(n^2).
Best Algorithm: O(log2 n) - Logarithmic growth, where the function increases slowly.
Worst Algorithm: O(2n) - Exponential growth, where the function increases rapidly.
Asymptotic Notation (in simplifying rate of growth): Streamlines functions by removing constant coefficients and less significant terms, focusing on the overall growth trend for large inputs.