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Big-O Simplified!

Big-O notation describes patterns of activity of systems in terms of standard mathematical models in which an amount of resources (time, quantity, cost, etc.) determines scope. The patterns of activity must contain input that can be isolated and modified to be improved without negatively effecting the desired outcomes of their systems. Big-O helps systems analysts describe patterns of activity as best-case, acceptable-case, or worst-case. Big-O is used in business and computer science for documenting system behavior and for arguing system improvements.

 Patterns of activity of systems are typically expressed as algorithms. In the simplest terms, big-O is an estimate of the optimal efficiency of algorithms.

When comparing two or more algorithms for optimal efficiency the following criteria must be met:
  1. Algorithms were designed using the same language.
  2. Algorithms result in the same measure (e.g. running time).
  3. Algorithms use the same set of operations.
  4. Algorithms have an estimated number of operations in common that can represent the best-case of efficiency.
  5. Algorithms have an estimated number of operations in common that can represent the acceptable-case of efficiency (current design being used).
  6. Algorithms have an estimated number of operations in common that can represent the worst-case of efficiency.
After you've decided the scope of optimal efficiency and the standardized model to use, you can equate the estimates to big-O notation. 

Here's an excellent reference to big-O notation:

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