Best, Average, and Worst Case Scenarios

When analyzing an algorithm’s time and space complexity, we distinguish between best case, average case, and worst case scenarios. Although Big O notation typically focuses on the worst case to provide an upper-bound guarantee, understanding all three is crucial for a complete picture of an algorithm's performance.

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Table of Contents

• Best Case (Ω-Notation)
• Worst Case (O-Notation)
• Average Case (Θ-Notation)
• Summary and Practical Implications

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Best Case (Ω-Notation)

The best-case scenario describes the minimum possible time (or space) an algorithm takes to complete, given the most favorable input. This lower bound is represented by Omega notation (Ω).

• When does it occur? When the input data is arranged in the most favorable way for the algorithm.
• Relevance: It indicates an algorithm's maximum possible speed for specific inputs, but it is less useful for practical guarantees since ideal conditions are often rare.

Example: Linear Search

function linearSearch(arr, target) {
  for (let i = 0; i < arr.length; i++) {
    if (arr[i] === target) {
      return i; // Found it!
    }
  }
  return -1; // Not found
}

• Best Case Input: The target element is the first one in the array (arr[0]).
• Time Complexity (Best Case): Ω(1)

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Worst Case (O-Notation)

The worst-case scenario describes the maximum possible time (or space) an algorithm might take. This is the most common focus of complexity analysis and is represented by Big O notation (O).

• When does it occur? When the input data is arranged in the least favorable way, forcing the maximum number of operations.
• Relevance: This is the most important metric for guarantees. It ensures that the algorithm will never perform worse, which is critical for system reliability and resource planning.

Example: Linear Search (continued)

• Worst Case Input: The target element is the last element, or it is not in the array at all.
• Time Complexity (Worst Case): O(n)

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Average Case (Θ-Notation)

The average-case scenario describes the expected time (or space) an algorithm takes on a "typical" input. This is represented by Theta notation (Θ).

• When does it occur? This is the performance averaged over all possible inputs of a given size n, often requiring assumptions about the input's statistical distribution.
• Relevance: It provides a more realistic measure of day-to-day performance, but it can be much harder to calculate precisely.

Example: Linear Search (continued)

• Average Case Input: The target element is found somewhere in the middle of the array. On average, the loop runs n/2 times.
• Time Complexity (Average Case): Θ(n) (constant factors like 1/2 are dropped)

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Summary and Practical Implications

Scenario	Notation	Description	Practical Use
Best Case	Ω(g(n))	Minimum operations (lower bound).	Shows best-case speed, but rarely used for guarantees.
Average Case	Θ(g(n))	Expected operations on typical inputs.	Realistic but harder to calculate; needs assumptions.
Worst Case	O(g(n))	Maximum operations (upper bound).	Most common; provides performance guarantees.

When discussing algorithm complexity, a question like "What's the Big O?" almost always refers to the worst-case time complexity. This is because the worst-case provides a reliable upper bound, which is essential for designing robust and predictable systems.