Question

Match the Big-O notation with its definition or use. A. $\mathrm{O}(1)$ B. $\mathrm{O}({\log }_{2}N)$ C. $\mathrm{O}(N)$ D. $\mathrm{O}(N{\log }_{2}N)$ E. $\mathrm{O}\left(N^2\right)$ F. $\mathrm{O}\left(2^N\right)$ G. O $(N!)$ Factorial time

Short answer

Factorial time is O(N!). Match it with G.

Step-by-step solution

Identify Factorial Time Complexity

Factorial time complexity is represented by the function $O(N!)$, which describes an algorithm whose performance will grow very quickly as the size of the input increases. This is the most time-consuming of the options provided and is commonly found in problems involving permutations.

Match with Big-O Notation

From the list provided, $G.O(N!)$ is the option that matches the description of factorial time complexity. This means any question involving factorial time complexity will be represented by option G.

Key concepts

Algorithm Efficiency

Algorithm efficiency is a measure of how well an algorithm performs, particularly in terms of time and space. Efficiency evaluates how quickly an algorithm can solve a problem and how much space it requires.

Understanding the efficiency of an algorithm is crucial because it affects the performance of software programs, especially as input sizes grow. A more efficient algorithm will require less time and resources to perform the same task.

Key factors in assessing algorithm efficiency include:

• Time Complexity: How long an algorithm takes to run with a given input size.
• Space Complexity: How much memory the algorithm uses during execution.

Algorithm efficiency helps in comparing different approaches to solving a problem, helping developers choose the most optimal one based on the constraints and requirements.

Time Complexity

Time complexity is a metric used to determine how the runtime of an algorithm grows as the input size increases. It gives a high-level perspective of the algorithm's performance by abstracting away hardware specifics.

Time complexity is commonly expressed using Big-O notation, which describes the upper bound of the algorithm's growth rate. Big-O notation helps to categorize algorithms by their worst-case scenarios, making it easier to predict performance.

Some common time complexities include:

• O(1): The algorithm has constant time complexity, meaning it takes the same amount of time to execute regardless of input size.
• O(N): The algorithm's time complexity scales linearly with the input size, doubling the input doubles the execution time.
• O(N²): The execution time increases quadratically as input size increases.
• O(N!): Factorial time complexity, where the time to execute grows very quickly with input size, often used in algorithms that perform exhaustive searches.

Factorial Time Complexity

Factorial time complexity, denoted as $O(N!)$, represents one of the most time-intensive classes in computational complexity.

This complexity arises in algorithms that consider every possible arrangement or permutation of a given dataset, such as the Traveling Salesman Problem. As the input size increases, the number of permutations grows factorially, leading to rapid increases in computation time.

A key point to understand about factorial time complexity is:

• With $N!$, as $N$ grows, the number of permutations multiplies by each successive integer. For example, $5!=5\times 4\times 3\times 2\times 1=120$, showing the explosive growth rate.

Due to this rapid growth, algorithms with $O(N!)$ are inefficient for large datasets and are often impractical, necessitating more efficient approximations or heuristics for real-world applications.