Question

Show that: (a) \(\sin (n) \in O(1)\) - that is, the sine function is asymptotically dominated by the constant function \(1 .\) (b) \(n \sin (n) \in \mathbf{O}(n)\)

Short answer

(a) \(\sin(n) \in O(1)\), (b) \(n \sin(n) \in O(n)\).

Step-by-step solution

Understanding Big O Notation

Big O notation, denoted as \(O(f(n))\), is used to describe an upper bound on the time complexity of an algorithm as a function of the input size. Specifically, \(g(n) \in O(f(n))\) means there exist positive constants \(c\) and \(n_0\) such that \(0 \leq g(n) \leq c \cdot f(n)\) for all \(n \geq n_0\).

Show that \(\sin(n) \in O(1)\)

The sine function \(\sin(n)\) is bounded between -1 and 1 for all real numbers \(n\). Therefore, we can choose \(c = 1\) and \(n_0 = 1\). This means \(|\sin(n)| \leq 1\) for all \(n \geq n_0\), which satisfies the definition of big O with \(f(n) = 1\). Thus, \(\sin(n) \in O(1)\).

Show that \(n \sin(n) \in O(n)\)

We are given the function \(n \sin(n)\) and need to show that it is asymptotically bounded above by \(n\). Since \(|\sin(n)| \leq 1\) for all \(n\), we have \(|n \sin(n)| \leq n\). Thus, we can choose \(c = 1\) and \(n_0 = 1\) to satisfy \(0 \leq n|\sin(n)| \leq c \cdot n\) for all \(n \geq n_0\). Hence, \(n \sin(n) \in O(n)\).

Key concepts

Asymptotic Analysis

Asymptotic analysis is a method used in computer science and mathematics to describe the behavior of functions as the input size tends to infinity. This type of analysis helps us to characterize the efficiency of algorithms by focusing on their time or space complexity in terms of input size.

One of the most common tools used for this purpose is Big O notation. It provides an upper bound, encapsulating the worst-case growth rate of a function as the input size grows.

Understanding Big O allows developers to compare different algorithms or programs by estimating how they will perform as data sets increase considerably. Key terms in asymptotic analysis include:

• Upper Bound: A function grows at most as quickly as the given complexity in the worst case.
• Tight Bound: When the upper and lower bounding functions are equivalent.
• Loose Bound: An upper bound greater than necessary.

By using asymptotic analysis, we can make educated guesses about resource usage, like time or memory allocation, during the execution of an algorithm.

Bounded Functions

Bounded functions are those that do not exceed a certain upper boundary in absolute terms, no matter what values are given as inputs. This concept was crucial in showing that \(\sin(n) \in O(1)\) , as the sine function oscillates between -1 and 1.

In the context of big O notation, a bounded function can be compared to a constant function over its input domain.

• For instance, \(\sin(n)\) , whether \(n\) is large or small, remains confined within this range of -1 to 1.
• Using the constant function \(f(n) = 1\), it's proved that \(\sin(n)\)'s variations never exceed a constant \(c=1\), thereby falling under the category of \(O(1)\).

This bounded nature emphasizes that certain operations or calculations remain consistent and efficient without escalating resource demands as input sizes grow. Understanding bounded functions provides insight into stability aspects of algorithms and systems.

Time Complexity

Time complexity is a vital measure in computational theory, indicating how the runtime algorithm grows as its input size increases. By examining an algorithm’s time complexity, we gain insights into its potential performance and scalability, especially as inputs become large.

Big O notation is a widely used method to express time complexity, helping in identifying the upper bound of an algorithm's growth rate.

• In the previous example, \(n \sin(n)\) was asserted to be \(O(n)\).
• This signifies that even if \(\sin(n)\) fluctuates, multiplying by \(n\) keeps the growth rate linear.
• It signifies a linear time complexity, implying that as we increase data, the processing time rises proportionately with the amount \(n\).

While viewing time complexity, it's essential to identify variations in execution behavior, thus allowing developers to make intelligent choices about which algorithms to apply in specific situations. An optimal time complexity can lead to efficient performance, saving valuable computational resources.