The efficiency of algorithms is paramount, especially when working with large data sets. There are often trade-offs between computational speed and space utilization in a program. Precomputing results can speed up programs significantly but increase the amount of disk space used.

### Orders of Growth

To evaluate an algorithm’s efficiency, we can use the concept of “order of growth” (OOG). It assesses how the algorithm scales when the size of input increases. We are typically interested in the worst-case scenario and the upper bound of the algorithm’s efficiency.

Different types of order of growth include:

**Constant**: The efficiency remains the same regardless of the number of inputs.**Linear**: The efficiency decreases linearly as the number of inputs increases.**Quadratic**: The efficiency decreases dramatically as the number of inputs increases (proportional to the square of the input size).**Logarithmic**: The efficiency decreases as the logarithm of the number of inputs, so the impact of increasing inputs diminishes over time.**n log n**: The efficiency falls between quadratic and linear growth.**Exponential**: The worst case, where the efficiency decreases drastically (faster than quadratic growth) as the number of inputs increases.

In practice, the goal is to aim for logarithmic growth, but this is often challenging. Linear or n log n growth are more realistic targets for efficient algorithms.

### Big O Notation

The Big O notation describes the upper bound of the time complexity in the worst-case scenario. It aims to express the rate of growth of an algorithm relative to the input size. The notation counts the number of operations or “steps” in an algorithm, ignoring constants and focusing on the dominant term.

### Complexity Classes

Complexity classes categorize algorithms based on their time and space complexity. These include:

**Constant Complexity (O(1))**: The complexity does not depend on the input size.**Logarithmic Complexity (O(log n))**: The complexity grows logarithmically with the input size. Examples include algorithms that split large problems into smaller ones, such as binary search.**Linear Complexity (O(n))**: The complexity grows linearly with the input size, as seen in algorithms that must process each input individually.**Log-linear Complexity (O(n log n))**: A middle-ground efficiency class seen in algorithms like merge sort.**Polynomial Complexity (O(n^c))**: The complexity grows with the square of the input size. Common in algorithms with nested loops or recursive calls.**Exponential Complexity (O(c^n))**: The complexity grows exponentially with the input size, making these algorithms very inefficient.

### Search and Sort Algorithms

Different search and sort algorithms fall into different complexity classes:

**Linear search**: A brute force method that checks each item one by one. It is an O(n) algorithm with linear complexity.**Bisection search**: An efficient method that halves the search space with each step. It is an O(log n) algorithm with logarithmic complexity. It requires the input data to be sorted.

Sorting algorithms include:

**BOGO sort**: A method that randomly shuffles the elements until they are sorted. It has an unbounded time complexity.**Bubble sort**: A method that compares adjacent elements and swaps them if they are in the wrong order, gradually “bubbling” the largest element to the end. This method has a polynomial time complexity, specifically O(n^2).

The efficiency of an algorithm greatly depends on the data set and the specific use case. Understanding the trade-offs between different algorithms can help make more informed decisions when implementing solutions to programming problems.