Question

Use induction to show that \(W(n) \in \Omega(n \lg n)\) for the following recurrence. This is Recurrence 8.2 in Section 8.5 .4 where \(m\) (group size) is 3. \\[ W(n)=W\left(\frac{2 n}{3}\right)+W\left(\frac{n}{3}\right)+\frac{5 n}{3} \\]

Short answer

The result of the proof by induction shows that the function \(W(n)\) is in \(\Omega(n \log n)\) for the given reccurence relation, as we have shown that the property holds for the base case \(n=1\) and if it holds for \(n\), it also holds for \(n+1\).

Step-by-step solution

Basis step

Firstly, it needs to be shown that the property holds true for the smallest possible value(s). Here, the basis of induction will be \(W(1) = 1\). It is clear that \(1 = \Omega(1 \log 1)\) because \(1 \log 1 = 0\) and \(\Omega\) is a lower bound. So, the base case holds true.

Inductive step

Now that the base case has been established, the property needs to be shown to hold true for \(n+1\) when it holds true for \(n\). Thus, the inductive hypothesis is that \(W(n) = \Omega(n \log n)\) for all \(n\). This means that there is some \(c > 0\) and \(n_0\) such that for all \(n > n_0, W(n) \geq c n \log n\). Now, consider \(W(n+1)\) and substitute \(W(n)\) with \(\Omega(n \log n)\), \(W\left(\frac{2 n}{3}\right)+W\left(\frac{n}{3}\right)+\frac{5 n}{3} \geq c \cdot \frac{2n}{3} \log \frac{2n}{3} + c \cdot \frac{n}{3} \log \frac{n}{3} + \frac{5 n}{3}\).

Simplifying

The statement from the previous step can be simplified by applying the logarithmic properties. Then, you get \(W(n + 1) \geq c \cdot \frac{2n}{3} (\log n + \log \frac{2}{3}) + c \cdot \frac{n}{3} \log n + \frac{5 n}{3} \geq c \cdot n \log n - c \cdot n + \frac{5 n}{3}\), since \(\log a + \log b = \log ab\) and \(-c \cdot n + \frac{5n}{3} > 0\) for a suitably large \(n\). So, the statement holds for \(n + 1\) as well.

Key concepts

Recurrence Relations

Recurrence relations are equations that define sequences of values using recursion. Essentially, they describe a function in terms of its past values, often making them beneficial for analyzing algorithm efficiency or problem-solving in computer science.
Consider the recurrence relation from the exercise:

• \[ W(n) = W\left(\frac{2n}{3}\right) + W\left(\frac{n}{3}\right) + \frac{5n}{3} \]
• This equation states that the value of \(W(n)\) depends on the values of \(W\) for smaller input sizes.
• Recurrence relations are crucial for studying divide-and-conquer algorithms, where problems are broken down into subproblems of similar form.

The provided relation is similar to those found in many sorting and optimization problems. Repeatedly breaking down a problem and combining results inherently forms a tree of recursive calls, with each level contributing to the overall complexity.

Asymptotic Analysis

Asymptotic analysis offers a way to describe the behavior of functions as the input size grows towards infinity. It is particularly useful in computer science to understand algorithm performance in terms of time or space.

• Using notations like \(\Omega\), we can express lower bounds of an algorithm.
• In this exercise, \(W(n) \in \Omega(n \log n)\) implies that the minimum growth rate of \(W(n)\) is similar to \(n \log n\).
• This is significant for determining the efficiency and speed of algorithms, especially when dealing with large inputs.

The asymptotic lower bound tells us that no matter how fast the function \(W(n)\) grows, it cannot grow slower than \(n \log n\).

Logarithmic Functions

Logarithmic functions are an integral part of mathematical analysis and recur frequently in computer science, especially in algorithms involving sorting, searching, and data structures.

• The logarithm function, \(\log n\), increases very slowly compared to linear and polynomial functions.
• Properties of logarithms, such as \(\log ab = \log a + \log b\) and \(\log\left(\frac{a}{b}\right) = \log a - \log b\), are vital for simplifying complex recursive functions.
• In the given problem, breaking down \(\log\left(\frac{2n}{3}\right)\) and \(\log\left(\frac{n}{3}\right)\) helps in managing the recurrence relation.

Logarithms allow us to interpret data in algorithms where the input size is exponentially large, providing insights into the necessary computational resources.