Question

Let \(p(n)=a_{4} n^{4}+a_{k-1} n^{k-1}+\dots a_{1} n+a_{0},\) where \(a_{k}>0 .\) Using the Properties of Order in Section \(1.4 .2,\) show that \(p(n) \in \Theta\left(n^{k}\right)\)

Short answer

\(p(n) \in \Theta(n^{k})\)

Step-by-step solution

State the Given Polynomial

We are given the polynomial \(p(n)=a_{4} n^{4}+a_{k-1} n^{k-1}+\dots a_{1} n +a_{0}\) with the coefficients all greater than zero. We are asked to show that this polynomial is big Theta of \(n^{k}\).

Apply the Properties of Order

Using the properties of order in algorithm analysis, we know that if there are positive constants \(c_{1}, c_{2}\) and \(n_{0}\) such that \(c_{1} n^{k} ≤ p(n) ≤ c_{2} n^{k}\) for all \(n ≥ n_{0}\), then we can say that \(p(n) \in \Theta(n^{k})\). Each term of \(p(n)\) is less than or equal to \(a_{4}n^{4}\), when \(n \geq 1\). Additionally, since \(a_{4} n^{4} \) is a term of \(p(n)\), \(a_{4} n^{4} ≤ p(n)\). Thus, the inequalities hold with \(c_{1} = a_{4}\), \(c_{2} = (a_{4} + a_{k-1} + \dots + a_{1}+ a_{0})\), and \(n_{0} = 1\).

Conclusion

Therefore, by the properties of order theory in algorithm analysis, it can be concluded that \(p(n) \in \Theta(n^{k})\). The exercise did not specify the exact value of \(k\), meaning that this conclusion holds for any positive integer \(k\).

Key concepts

Big Theta Notation

Understanding Big Theta Notation is essential for students studying algorithm analysis, as it provides a framework for classifying the running time of an algorithm. Big Theta Notation, denoted as \(\Theta\), describes a function's growth rate in terms of upper and lower bounds. To say that a function \(f(n)\) is \(\Theta(g(n))\) means there are positive constants \(c_1\), \(c_2\), and \(n_0\) such that for all \(n \geq n_0\), the function \(f(n)\) is squeezed between \(c_1 \cdot g(n)\) and \(c_2 \cdot g(n)\):

\[ c_1 \cdot g(n) \leq f(n) \leq c_2 \cdot g(n) \]
This notation indicates that \(f(n)\) grows at the same rate as \(g(n)\) when \(n\) becomes large. It tells us that for sufficiently large values of \(n\), \(f(n)\) and \(g(n)\) are essentially equivalent in terms of their growth rates, up to constant factors. This stability is crucial for evaluating algorithm efficiency, as it holds regardless of the precise constants involved.

Properties of Order

The Properties of Order form the bedrock of comparing functions in algorithm analysis. They enable us to determine mathematical inequalities that are key to understanding an algorithm's behavior for large values of \(n\). In our context, we utilize these properties to show that a given polynomial \(p(n)\) is bounded above and below by a function of the form \(n^k\).

By establishing that there are specific constants that create bounds for \(p(n)\), we can use the Properties of Order to affirm our Big Theta notation. Juxtaposing \(p(n)\) with \(n^k\) allows us to see that the polynomial has an upper limit, dominated by its highest degree term, and a lower limit, substantiated by \(n^k\) itself. The beauty of this approach is that it distills a complex polynomial to its essence, focusing on the term with the highest power of \(n\) to assess its growth. The exercise illustrates with \(p(n)\) that regardless of the number of terms or coefficients, the leading term \(a_k n^k\) dictates the polynomial's order magnitude for large \(n\).

Polynomial Time Complexity

Polynomial Time Complexity lies at the heart of algorithmic efficiency discussions. When an algorithm's running time can be expressed as a polynomial function of the size of the input, \(n\), we describe it as having polynomial time complexity. This means that the time it takes to complete a computation is a polynomial function of the input size.

Algorithms with polynomial time complexity are considered efficient and tractable, especially when compared with those having exponential time complexities. In practice, polynomials with lower exponents are more desirable; for instance, an algorithm with a complexity of \(O(n^2)\) is more efficient than one with \(O(n^4)\) as \(n\) grows large. The exercise highlights a polynomial of degree \(k\), suggesting that the operation count for a hypothetical algorithm increases at a rate proportional to \(n^k\) as the size of the input increases. Understanding this concept prepares students to distinguish between efficient algorithms and those that may not scale well with larger inputs.