How to Analyze Algorithms: Big-O, Omega, and Theta for Newbies

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

• It helps ensure correctness and efficiency in problem-solving.

• To find an optimized algorithm, we need to measure its performance, typically by analyzing its time or space (memory) complexity.

• The method of measuring performance must be independent of specific computer systems and input data, which leads to the field of Algorithm Analysis.

Algorithm Analysis

A problem can be solved using different algorithms.

• The goal is to find the optimized algorithm that solves the problem in the least time and with the least memory consumption.

• Asymptotic analysis helps determine the efficiency of an algorithm as the input size grows, typically focusing on large inputs.

Types of Algorithm Analysis:

To analyze an algorithm, we evaluate how its performance varies with different inputs, particularly considering how long it takes to execute.

1. Worst Case:

   The longest time an algorithm will take to solve the problem. This time typically increases as the input size grows (e.g., O(n²) for an algorithm with quadratic time complexity). Worst-case analysis provides an upper bound.

2. Best Case:

   The shortest time an algorithm will take to solve the problem. This happens in the most favorable situation (e.g., when the input is already sorted for a sorting algorithm). Best-case analysis gives a lower bound.

3. Average Case:

   The expected time the algorithm takes for a random input. This is usually more practical but harder to calculate than worst-case and best-case analysis.

Asymptotic Notations

Big-O Notation (O):

Big-O describes the upper bound of the algorithm’s running time. It gives the worst-case time complexity and ensures that the algorithm does not grow faster than a certain rate for large input sizes.

Mathematically:

f(n) = O(g(n))

This means that f(n) grows at most as fast as g(n) for large n.

• Big-O focuses on the worst-case scenario.

Omega Notation (Ω):

Omega describes the lower bound of the algorithm’s running time. It gives the best-case time complexity, ensuring that the algorithm does not grow slower than a certain rate for large inputs.

Mathematically:

f(n) = Ω(g(n))

This means that f(n) grows at least as fast as g(n) for large n.

• Omega focuses on the best-case scenario.

Theta Notation (Θ):

Theta provides a tight bound on the running time. It gives both the upper and lower bounds, meaning that the algorithm’s time complexity grows at the same rate in both the best and worst cases.

Mathematically:

f(n) = Θ(g(n))

This means that f(n) grows at the same rate as g(n) for large n.

• Theta is used to describe the average case when the upper and lower bounds are the same.

Order of Growth

The order of growth describes how the time or space complexity of an algorithm scales with the size of the input (n). The most common growth rates include:

• Constant: O(1)

• Logarithmic: O(log n)

• Linear: O(n)

• Quadratic: O(n²)

• Exponential: O(2ⁿ)