Question

Show that \(n\) is \(O(n \log n)\)

Short answer

n is O(n log n) with constants c = 1 and n0 = 2.

Step-by-step solution

Define Big-O notation

Big-O notation is used to describe the upper bound of the growth rate of a function. If a function f(n) is O(g(n)), it means there exist positive constants c and n0 such that for all n ≥ n0, f(n) ≤ c * g(n).

Write the inequality for Big-O

To show that a function f(n) is O(g(n)), we need to find constants c and n0 such that for all n ≥ n0, f(n) ≤ c * g(n). In this case, we need to show that n ≤ c * n log n for some constants c and n0.

Rearrange the inequality

Rearrange the inequality n ≤ c * n log n. Divide both sides by n (assuming n > 0) to obtain 1 ≤ c * log n.

Find appropriate constants

We need to find constants c and n0 such that for all n ≥ n0, the inequality 1 ≤ c * log n holds true. Select c = 1 and n0 = 2 since for n ≥ 2, log n ≥ 1. Therefore, 1 ≤ log n .

Conclude the proof

Since we found constants c = 1 and n0 = 2 such that for all n ≥ 2, the inequality 1 ≤ log n is satisfied, we have shown that n is O(n log n).

Key concepts

asymptotic analysis

Asymptotic analysis is a method used to describe the behavior of functions as inputs get very large. This concept evaluates the efficiency and performance of an algorithm or function by focusing on its growth rate rather than exact outcomes. It helps in determining which algorithm will be more efficient when dealing with large amounts of data.
For instance, given two functions, we analyze how they perform as they approach infinity. It’s important because it allows us to predict trends and make decisions based on long-term behavior.
Using asymptotic analysis, we can determine if one function grows faster, slower, or at the same rate compared to another function. This forms the foundation of understanding Big-O notation, which makes it easier to compare different algorithms.

upper bound

In Big-O notation, the term 'upper bound' refers to the maximum rate at which a function's execution time increases as its input grows. To put it simply, if a function's running time is O(g(n)), the function will not grow faster than g(n) multiplied by a constant factor.
Finding the upper bound involves:

• Identifying the dominant term in the function.
• Representing other lower-order terms and constants.

For example, in the function f(n) = 2n^2 + 3n + 4, the term 2n^2 will dominate as n becomes very large. Hence, the upper bound or Big-O of the function is O(n^2).
This is crucial for algorithm analysis as it helps determine the worst-case scenario performance, ensuring efficiency and reliability.

growth rate

The growth rate of a function describes how quickly it increases as the input size grows. This is a pivotal concept in understanding how algorithms scale and perform with larger datasets.
Common growth rates include:

• Constant - O(1): The runtime remains the same regardless of input size.
• Logarithmic - O(log n): Increases slowly as input grows.
• Linear - O(n): Grows proportionately with input size.
• Polynomial - O(n^k): Grows faster than linear, where k is a positive integer.
• Exponential - O(2^n): Grows extremely fast, doubling with each increment of n.

Understanding growth rates allows developers to select appropriate algorithms by evaluating how they scale, ensuring optimal performance for varying input sizes.