Question

Algorithm 1.7 (nth Fibonacci Term, Iterative) is clearly linear in \(n,\) but is it a linear-time algorithm? In Section 1.3 .1 we defined the input size as the size of the input. In the case of the \(nth\) Fibonacci term, \(n\) is the input, and the number of bits it takes to encode \(n\) could be used as the input size. Using this measure the size of 64 is \(\lg 64=6,\) and the size of 1024 is \(\lg 1024=\) 10\. Show that Algorithm 1.7 is exponential-time in terms of its input size. Show further that any algorithm for computing the \(nth\) Fibonacci term must be an exponential-time algorithm because the size of the output is exponential in the input size. See Section 9.2 for a related discussion of the input size.

Short answer

Algorithm 1.7 and any other algorithm for computing the \(nth\) Fibonacci term is an exponential-time algorithm in terms of its input size because the size of the output is exponential in the input size. This is due to the exponential growth of the Fibonacci sequence.

Step-by-step solution

Understand nth Fibonacci Term Computation

The \(nth\) Fibonacci term computation is a repetitive process where the next term is the sum of previous two terms. This results in an exponential growth with respect to the term number \(n\). Consequently, even if we are using an iterative method that seems linear with respect to \(n\), the actual time complexity is bigger due to this size of computation as \(n\) increases.

Understand Input Size as Bit Length

The number of bits it takes to encode a number \(n\) is defined as \(\lg n\) or the log base 2 of \(n\). This is a measure of input size. So the size of 64 is \(\lg 64 = 6\) bits, and likewise the size of 1024 is \(\lg 1024 = 10\) bits.

Relate Fibonacci Term Computation and Input Size

The nth Fibonacci number is approximately equal to \((\phi^n)/\sqrt{5}\), where \(\phi\) is the golden ratio which is about 1.61803. The number of bits in the \(nth\) Fibonacci number is therefore approximately \(n \lg \phi - \frac{1}{2}\lg 5\). As \(n\) increases, this Fibonacci number becomes larger and thus requires more bits. This indicates that the number of bits required scale at best logarithmically with \(n\). Therefore, the algorithm to calculate the \(nth\) Fibonacci term is exponential in terms of input size.

Any Fibonacci Algorithm is Exponential-time

Following the logic above, any algorithm computed for the \(nth\) Fibonacci term must be an exponential-time algorithm. Why? Even if a more effective algorithm is computed, the output size is exponential in the input size. This comes from the fact that the Fibonacci terms grow exponentially. Therefore, the output size, regardless of the algorithm used, will require a significant number of bits to store the larger terms as \(n\) increases.

Key concepts

Time Complexity

When you look at algorithms, one fundamental concept is how efficiently they run, known in computer science as time complexity. This measures the time an algorithm takes to complete as a function of the length of the input. It's crucial this is well understood, as it directly impacts both the practicality and feasibility of software.

Many algorithms may appear straightforward at a glance, but their actual time complexity reveals the intensity of the computation required. For instance, an algorithm that seems to have a linear relationship with the number of input elements could be hiding a more complex reality, especially when the process involves operations that grow rapidly - like in calculating Fibonacci numbers.

Algorithm Input Size

The algorithm input size affects not only the space an algorithm needs to run but also the time it takes to complete its tasks. It's useful to measure input size in bits, especially when dealing with numbers. The bit length of an integer n can be computed as \(\lg n\), the base-2 logarithm of n.

This concept illuminates how even small increases in the numerical input can lead to larger computational complexity. For example, an input with a bit length of 6 is significantly smaller and less complex to handle than one with a bit length of 10. As input sizes expand, their impact on the performance of algorithms becomes more pronounced.

Exponential Growth

Some algorithms have outputs that increase exponentially with respect to the inputs; this tendency is referred to as exponential growth. You've likely heard it outside of the realm of mathematics, characterizing anything that rapidly increases like population growth or viral social media posts.

In the context of computing, exponential growth takes the form of increasingly larger numbers, which means that the computational load intensifies significantly with each step or input increase. An example of this is the Fibonacci sequence where each term is the sum of the two preceding ones, leading to a fast-growing series of numbers.

Fibonacci Sequence

The Fibonacci sequence is a fascinating mathematical series where each number (after the first two) is the sum of the two preceding ones, usually starting with 0 and 1. This sequence isn't just a mathematical marvel; it also appears in different aspects of the natural world such as branching in trees, the arrangement of leaves on a stem, and the fruitlets of a pineapple.

From a computational perspective, the Fibonacci sequence showcases a situation where the output grows at an exponential rate compared to the input size. What initially begins as a seemingly simple series progressively demands more computational resources as the sequence grows, demonstrating the hidden complexity behind the elegant pattern of Fibonacci numbers.