Problem 45 - Discrete Mathematics with Applications

Question

Let \(f_{1}(n)=\mathrm{O}(g(n))\) and \(f_{2}(n)=k f_{1}(n),\) where \(k\) is a positive constant. Show that \(f_{2}(n)=\mathrm{O}(g(n))\).

Step-by-Step Solution

Answer

Since \(f_{1}(n)=\mathrm{O}(g(n))\), there exist constants \(c_{1}>0\) and \(n_{1}\) such that \(f_{1}(n) \leq c_{1} * g(n)\) for all \(n > n_{1}\). Then, since \(f_{2}(n)=k f_{1}(n)\), we have \(f_{2}(n) \leq k(c_{1} * g(n))\). Let \(c_{2} = k * c_{1}\), then we get \(f_{2}(n) \leq c_{2} * g(n)\). Since we found constants \(c_{2}>0\) and \(n_{1}\) such that this inequality holds, we conclude that \(f_{2}(n)=\mathrm{O}(g(n))\).

Step 1: Use the Definition of Big-O for \(f_1(n)\)

Since \(f_1(n) = O(g(n))\), by definition, there exists constants \(c_1 > 0\) and \(n_1\) such that \(f_1(n) \leq c_1 * g(n)\) for all \(n > n_1\).

Step 2: Rewrite \(f_2(n)\) in Terms of \(f_1(n)\)

We are given that \(f_2(n) = k f_1(n)\). Now, we need to find an upper bound for \(f_2(n)\) in terms of \(g(n)\).

Step 3: Substitute \(f_1(n)\) Bound into \(f_2(n)\)

Since \(f_1(n) \leq c_1 * g(n)\) for all \(n > n_1\), we can replace \(f_1(n)\) in the expression for \(f_2(n)\) as follows: \[f_2(n) = k f_1(n) \leq k(c_1 * g(n))\].

Step 4: Simplify and Find a Big-O Bound for \(f_2(n)\)

Now, we can simplify the inequality: \[f_2(n) \leq (k * c_1) * g(n)\]. Since k is a positive constant, \(k * c_1\) is also a positive constant. Let \(c_2 = k * c_1\). Then, we get: \[f_2(n) \leq c_2 * g(n)\].

Step 5: Conclude that \(f_2(n)\) is \(\mathrm{O(g(n))}\)

Since we found constants \(c_2 > 0\) and \(n_1\) such that \(f_2(n) \leq c_2 * g(n)\) for all \(n > n_1\), we can conclude that \(f_2(n) = O(g(n))\).

Key Concepts

Asymptotic Analysis

Asymptotic analysis is a method of describing the behavior of functions as inputs become indefinitely large. It's mainly used in computer science to analyze the efficiency and limitations of algorithms.
In the context of this exercise, we are concerned with the asymptotic behavior of two functions, \(f_1(n)\) and \(f_2(n)\), in relation to \(g(n)\).

• Purpose of Asymptotic Analysis: This type of analysis helps us understand the long-term growth rates of algorithms more clearly. Instead of focusing on exact measures like execution time, we look at trends.
• Relative Comparisons: By comparing \(f_1(n)\) and \(f_2(n)\) to \(g(n)\), we are determining the efficiency of these functions in relation to each other, especially when \(n\) becomes very large.

Through finding \(f_2(n) = O(g(n))\), the analysis allows us to predict how \(f_2(n)\) will scale and behave as \(n\) grows.

Algorithm Efficiency

Algorithm efficiency refers to how effectively an algorithm performs, often in terms of time and space complexity. Big O notation is instrumental in expressing space and time complexities, which are key aspects of algorithm efficiency.
For example, in this exercise, we begin with the assumption that \(f_1(n) = O(g(n))\). This implies that the efficiency of \(f_1(n)\) can be bounded by \(g(n)\), highlighting its time complexity.

• Bounding Time Complexity: When we say \(f_1(n)\) is \(O(g(n))\), we're indicating that in terms of time, it won't grow faster than a certain bound when \(n\) gets large.
• Constant Multiplicative Factor: In the context here, multiplying \(f_1(n)\) by a constant \(k\) to form \(f_2(n)\) doesn't change the overall efficiency level. The constant factor doesn't alter the broad scaling pattern.

This understanding helps us assess whether an algorithm will be practical for large inputs and where it might have inefficiencies.

Upper Bounds

Upper bounds are used to describe the worst-case scenario of an algorithm's growth rate. In the Big O notation context, an upper bound is a ceiling beyond which the function’s growth will not exceed.
For our problem, the task was to show that \(f_2(n)\) is \(O(g(n))\), thereby determining an upper bound for \(f_2(n)\).

• Finding Upper Bounds: By demonstrating \(f_2(n) \leq c_2 \cdot g(n)\), with \(c_2\) as a positive constant, we effectively stated that \(f_2(n)\) cannot grow faster than a multiplied form of \(g(n)\).
• Importance in Analysis: Establishing a clear upper bound provides insights into the maximum resource consumption, be it time or space, providing a robust indication of the worst-case performance.

Understanding upper bounds allows algorithm designers to set realistic expectations and create more efficient solutions, especially as systems scale.