Question

Explain in English what functions are in the following sets. (a) $n^{C(1)}$ (b) $O\left(n^{O(1)}\right)$ (c) $O\left(O\left(n^{O(1)}\right)\right)$

Short answer

Functions in $n^{C(1)}$ can be raised to a constant power, those in $O(n^{O(1)})$ are polynomial time functions, and functions in $O(O(n^{O(1)}))$ are also polynomial time functions, despite the additional layer of Big O notation.

Step-by-step solution

Interpretation of $n^{C(1)}$

This is a set of all functions that can be raised to the constant power, i.e., the power does not increase with the increase in input size n.

Interpretation of $O(n^{O(1)})$

This is a set of all functions that can be bounded above by a function of n raised to the power of a constant (any constant). This is basically a set of all polynomial time functions.

Interpretation of $O(O(n^{O(1)}))$

This looks a bit convoluted, but it can be understood the same way as the previous one. $O(n^{O(1)})$ is itself a set (as described in Step 2), which again is in O, indicating the function can be bounded above by any other function from this set. Therefore, these are also polynomial time functions.

Key concepts

Polynomial Time

When we consider the efficiency of algorithms, the term 'polynomial time' is crucial. It refers to a class of computational problems that can be solved by an algorithm whose running time grows at a polynomial rate relative to the size of the input.

For instance, if an algorithm takes time proportional to the input size to the second power (^2), we say it runs in polynomial time. Such algorithms are highly valued because their running times are relatively manageable as the input size grows. Problems solvable in polynomial time fall under the complexity class P, which stands for 'polynomial'.

In addition, any function that can be expressed as ^{c}, where 'c' is a constant, signifies a polynomial relationship. The exercise example (a) ^{C(1)} represents such a polynomial function where 'C' is a fixed constant, not affected by the input size 'n'. Consequently, this implies that the function's growth rate will be predictable and bounded by a polynomial.

Asymptotic Notation

Asymptotic notation is a mathematical tool used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity. It is widely used in computational complexity theory to classify algorithms based on their performance and growth rates.

'Big O' notation, denoted by O, is one of the most common forms of asymptotic notation. It provides an upper bound on the growth rate of a function, specifying that a function grows at most as fast as a certain polynomial.

The textbook example (b) O(n^{O(1)}) denotes a set of functions where each function is bounded above by some polynomial in 'n'. Note that O(1) represents a constant, so n^{O(1)} essentially means 'n raised to some constant power'. Big O encapsulates the growth rate without concern for less significant terms or leading coefficients, providing a simplified view of an algorithm's scalability.

Computational Complexity Theory

Computational complexity theory is a boundless field that deals with the resources required to solve computational problems, primarily focusing on time (runtime) and space (memory usage) requirements.

The theory categorizes problems into complexity classes, such as P, NP, NP-complete, and NP-hard, among others, based on how the resource needs scale with input size. 'P' denotes problems that can be solved quickly (in polynomial time) by a deterministic machine. Problems more challenging than those in P, where solutions can be verified in polynomial time but may not be solvable in polynomial time, fall into the NP (nondeterministic polynomial time) class.

Understanding this theory is critical for developing efficient algorithms and determining which problems are tractable. Regarding the previous solutions, both (b) O(n^{O(1)}) and (c) O(O(n^{O(1)})) suggest that the functions relating to these sets can be computed in polynomial time, hence are part of the complexity class P.