When analyzing algorithms, it is common to consider the worst-case, best-case, and average-case scenarios. Let’s explore each of these types of analysis:

## 1. Worst-Case Analysis:

- The worst-case analysis determines the upper bound on the running time or space requirements of an algorithm for the input that results in the maximum execution time or space usage.
- It assumes that the input is chosen in such a way that it leads to the most inefficient behavior of the algorithm.
- It provides a guarantee on the performance of the algorithm, ensuring that it will not exceed a certain time complexity or space complexity for any input of size ‘n’.
- The worst-case analysis helps identify scenarios in which the algorithm may perform poorly and assists in assessing its efficiency in the worst possible scenario.

## 2. Best-Case Analysis:

- The best-case analysis determines the lower bound on the running time or space requirements of an algorithm for the input that results in the minimum execution time or space usage.
- It assumes that the input is chosen in such a way that it leads to the most efficient behavior of the algorithm.
- The best-case analysis is often used to showcase the algorithm’s potential when the input is already in an ideal or sorted state.
- However, the best-case scenario is not always realistic or representative of the algorithm’s typical performance.

## 3. Average-Case Analysis:

- The average-case analysis considers the expected or average running time or space requirements of an algorithm for all possible inputs of size ‘n’.
- It takes into account the likelihood of different inputs occurring and assigns probabilities to them.
- It requires a probability distribution of the inputs and calculates the average running time or space usage by considering the weighted sum of the running times or space usages for different inputs.
- The average-case analysis is generally more informative and practical than the worst-case analysis alone, as it provides insight into the expected behavior of the algorithm under typical conditions.

## Difference between Worst-case, best-case, and average-case analysis:

Worst-Case Analysis | Best-Case Analysis | Average-Case Analysis | |

Purpose | Determines the upper bound on running time or space requirements. | Determines the lower bound on running time or space requirements. | Considers the expected or average performance under typical conditions. |

Input | Input that leads to the most inefficient behavior. | Input that leads to the most efficient behavior. | Probability distribution of inputs. |

Assumption | Input is chosen to maximize algorithm inefficiency. | Input is chosen to minimize algorithm inefficiency. | Considers a distribution of inputs based on probability. |

Guarantees | Provides a guarantee on algorithm performance. | Not a guarantee but showcases algorithm potential. | Provides insight into the expected behavior under typical conditions. |

Practicality | Helps identify scenarios where the algorithm may perform poorly. | Not always realistic or representative of actual performance. | More informative and practical, reflects real-world scenarios. |

Decision-Making | Helps in assessing algorithm efficiency and worst-case behavior. | Useful in understanding algorithm capabilities in ideal scenarios. | Helps in comparing algorithms’ performance under typical conditions. |

Performance Range | Upper bound on time complexity or space requirements. | Lower bound on time complexity or space requirements. | Average running time or space usage based on probability distribution. |