Question

Show that if \(d(n)\) is \(O(f(n))\) and \(f(n)\) is \(O(g(n)),\) then \(d(n)\) is \(O(g(n))\).

Short answer

If d(n) is O(f(n)) and f(n) is O(g(n)), then d(n) is O(g(n)) by combining constants from the Big-O definitions.

Step-by-step solution

Understand the Definitions

Begin by recalling the definition of Big-O notation. For a function to be O(h(n)), there must exist positive constants C and n0 such that for all n ≥ n0, the inequality |f(n)| ≤ C|h(n)| holds.

Express Given Relationships with Big-O

Given that d(n) is O(f(n)), there exist constants C1 and n1 such that for all n ≥ n1, |d(n)| ≤ C1|f(n)|. Likewise, since f(n) is O(g(n)), there exist constants C2 and n2 such that for all n ≥ n2, |f(n)| ≤ C2|g(n)|.

Combine the Inequalities

For n large enough (specifically, n ≥ max(n1, n2)), we can combine the two inequalities. First, substitute |f(n)| in the first inequality: |d(n)| ≤ C1|f(n)| ≤ C1(C2|g(n)|) which simplifies to |d(n)| ≤ (C1C2)|g(n)|.

Conclude the Proof

By the definition of Big-O, the combined inequality |d(n)| ≤ (C1C2)|g(n)| demonstrates that d(n) is O(g(n)). Here, the constants are combined into a single constant C = C1C2.

Key concepts

asymptotic analysis

Asymptotic analysis is a technique used to describe the behavior of an algorithm as the input size grows. Its focus is on the growth rate of functions.
This means we analyze how the running time or space requirements of an algorithm increase with the size of the input.
It is particularly useful when comparing different algorithms to understand which one performs better for large inputs.
When performing asymptotic analysis, we use notation such as Big-O, Theta (Θ), and Omega (Ω). These notations help us succinctly describe the upper, tight, and lower bounds of an algorithm's performance.
For example, if an algorithm has a time complexity of O(n), it means that the running time grows linearly with the input size. This analysis abstracts away constant factors and low-order terms, focusing solely on the dominant factors affecting growth.

algorithm complexity

Algorithm complexity refers to the amount of computational resources required by an algorithm to process a given input.
This includes both time complexity (how long the algorithm takes to run) and space complexity (how much memory it requires).

Time complexity is typically expressed using Big-O notation. For instance, a binary search algorithm has a time complexity of O(log n), indicating that its running time increases logarithmically as the input size doubles.

Space complexity is also described using Big-O notation. An algorithm with space complexity O(n) would require memory proportional to the input size.

Understanding an algorithm's complexity helps developers choose the right algorithm for their specific needs, especially when dealing with large datasets or resource constraints. By analyzing both time and space complexity, we can balance processing speed with memory usage, ensuring efficient performance.

transitivity of Big-O

The transitivity property of Big-O notation states that if one function is Big-O of a second function, and the second function is Big-O of a third function, then the first function is Big-O of the third function.
This property allows us to concatenate bounds across multiple functions and is crucial in proving relationships between algorithms.
For example, if we have functions d(n), f(n), and g(n) such that d(n) is O(f(n)) and f(n) is O(g(n)), we can demonstrate that d(n) is O(g(n)).
The step-by-step solution in the exercise breaks this down:

• We start by expressing the given relationships: d(n) is O(f(n)) means there exist constants C1 and n1 such that |d(n)| ≤ C1|f(n)| for all n ≥ n1.
• Similarly, f(n) is O(g(n)) means there exist constants C2 and n2 such that |f(n)| ≤ C2|g(n)| for all n ≥ n2.
• Combining these inequalities for all large enough n (n ≥ max(n1, n2)), we get |d(n)| ≤ C1|f(n)| ≤ C1(C2|g(n)|). This simplifies to |d(n)| ≤ (C1*C2)|g(n)|.

This shows that d(n) is O(g(n)), with the constant factor being the product of C1 and C2.
This understanding of transitivity helps in dissecting complex relationships and proving algorithmic bounds in computational theory.