Algorithm Analysis Toolkit

ABSTRACT

This toolkit provides the formal framework for measuring the “cost” of an algorithm. It focuses on asymptotic analysis—determining how execution time and memory requirements scale as the input size ($n$) grows.

Foundational Metrics

The core methods for assessing performance before and during execution.

• Time Analysis
  • Definition: Estimating the number of elementary operations (additions, assignments, etc.) an algorithm performs.
  • Purpose: Allows for hardware-independent performance comparisons.
• Runtime of Algorithms
  • Focus: Measuring the actual wall-clock time required for execution.
  • Variables: Covers how environment factors (CPU, RAM, Background tasks) can influence raw timing data.

Formal Notation & Logic

The mathematical language used to categorize growth rates and prove correctness.

• Asymptotic Notation
  • Big-O ($O$): The upper bound; the “worst-case” scenario for growth.
  • Big-Omega ($\Omega$): The lower bound; the “best-case” scenario.
  • Big-Theta ($\Theta$): The tight bound; where growth is exactly defined.
• Loop Invariants
  • Definition: A property of a program loop that is true before (and after) each iteration.
  • Why it matters: A critical tool for formal verification; used to mathematically prove that an algorithm will produce the correct result.

Complexity Cheat Sheet