Question

Show that for any two bases \(a\) and \(b\) for logarithms, if \(a\) and \(b\) are both greater than \(1,\) then there is a constant \(c\) such that \(\log _{a} N \leq c\left(\log _{b} N\right)\) Thus, there is no need to specify a base in \(\mathrm{O}(\log N) .\) That is, \(\mathrm{O}\left(\log _{a} N\right)\) and \(\mathrm{O}\left(\log _{b} \mathrm{N}\right)\) mean the same thing.

Short answer

Answer: There is no need to specify a base in O(log N) notation because, for any two bases a and b greater than 1, there exists a constant c such that log_a N ≤ c(log_b N). This implies that the growth rates of log_a N and log_b N are asymptotically the same, even though the bases are different. Thus, O(log_a N) and O(log_b N) mean the same thing, and specifying a base is not necessary in O(log N) notation.

Step-by-step solution

Use the change of base formula

We are given that \(a > 1\) and \(b > 1\). Using the change of base formula, we get: $$\log_a N = \frac{\log_b N}{\log_b a}$$

Determine an inequality for the constant \(c\)

Now we need to find an inequality for the constant \(c\) such that \(\log_a N \leq c(\log_b N)\). Divide both sides of the equality we got in step 1 by \(\log_b N\): $$\frac{\log_a N}{\log_b N} = \frac{1}{\log_b a}$$ We can multiply both sides by \(\log_b N\) to get the desired inequality form: $$\log_a N \leq c(\log_b N) \iff 1 \leq c \log_b a$$ For the inequality to hold, \(c \geq \frac{1}{\log_b a}\).

Conclude that there is no need to specify a base in \(O(\log N)\)

From the inequality we derived in step 2, we see that there exists a constant \(c \geq \frac{1}{\log_b a}\) such that \(\log_a N \leq c(\log_b N)\). This means that, asymptotically, the growth rate of \(\log_a N\) and \(\log_b N\) is the same, even though the bases are different. Thus, there is no need to specify a base in \(O(\log N)\), and \(O(\log_a N)\) and \(O(\log_b N)\) mean the same thing.

Key concepts

Logarithm Properties

Understanding the properties of logarithms is essential in both mathematics and computer science. Logarithms, like the ones we discuss in our exercise, exhibit particular characteristics that allow for various manipulations and simplifications. For example, the change of base formula used in the exercise solution is an invaluable logarithmic property.

This formula is expressed mathematically as \[\log_a N = \frac{\log_b N}{\log_b a}\] It shows us how to convert a logarithm to a new base by dividing the logarithm in the new base by the logarithm of the old base in terms of the new base. Understanding this property helps you compare and equate logarithmic expressions that have different bases, which is precisely what's needed to show that different logarithmic bases do not affect asymptotic complexity, as seen in the exercise.

Other key properties of logarithms include the product rule \(\log_b (MN) = \log_b M + \log_b N\), the quotient rule \(\log_b (M / N) = \log_b M - \log_b N\), and the power rule \(\log_b (M^p) = p\cdot\log_b M\). These rules simplify dealing with logarithmic terms and are fundamental in various fields like computer science, where they assist in algorithm analysis.

Asymptotic Notation

Asymptotic notation is a way of describing the limit behavior of functions, often used in the context of algorithm analysis to classify and compare the relative performance of algorithms as the input size becomes large. The specific notation mentioned in the exercise is called 'Big O' notation, symbolized as \(O(f(N))\), where \(f(N)\) is some function of \(N\).

The key intuition behind Big O is to capture the upper bound on the growth rate of an algorithm's running time or space usage. For instance, saying an algorithm has a time complexity of \(O(\text{log} N)\) suggests that, as the input size grows, the time it takes to complete rises at most logarithmically. One crucial point highlighted by the exercise is the base independence in Big O notation; that is, the choice of the logarithm base does not affect the classification because logarithms of different bases differ only by a constant factor, which Big O notation ignores.

There are also other forms of asymptotic notations, including 'Big Theta' \(\Theta\) and 'Big Omega' \(\Omega\), which are used to denote the tight and lower bounds of an algorithm's time or space complexity, respectively.

Computational Complexity

Computational complexity refers to the amount of resources an algorithm requires to solve a problem as a function of the size of the input to the problem. It's a critical area in computer science because it helps us understand and predict how an algorithm will scale with increasingly larger inputs.

There are two main types of complexity: 'time complexity', which concerns the number of steps taken to complete an algorithm, and 'space complexity', which concerns the amount of memory needed. The exercise we’ve been examining deals with time complexity, specifically the complexity class \(O(\text{log} N)\), which characterizes algorithms that scale logarithmically.

Examples of logarithmic time algorithms include binary search and certain tree operations. The exercise's conclusion, that specific logarithm bases do not alter the complexity class in Big O notation, underscores that constant factors and lower order terms are discarded when expressing computational complexity. This abstraction allows us to focus on the most significant aspect of an algorithm's growth behavior as input sizes grow large.