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Chapter 5: Problem 10

. a) Prove that \(f \in O(f)\) for all \(f: \mathbf{Z}^{+} \rightarrow \mathbf{R}\). b) Let \(f, g: \mathbf{Z}^{+} \rightarrow \mathbf{R}\). If \(f \in O(g)\) and \(g \in O(f)\), prove that \(O(f)=O(g) .\) That is, prove that for all \(h: \mathbf{Z}^{+} \rightarrow \mathbf{R}\), if \(h\) is dominated by \(f\), then \(h\) is dominated by \(g\), and conversely. c) If \(f, g: \mathbf{Z}^{+} \rightarrow \mathbf{R}\), prove that if \(O(f)=O(g)\), then \(f \in\) \(O(g)\) and \(g \in O(f)\)

Short Answer

Expert verified
In terms of Big O notation, it has been proven that: 1) Any function \(f\) is in \(O(f)\), 2) If a function \(f\) is in \(O(g)\) and a function \(g\) is in \(O(f)\), then \(O(f) = O(g)\),3) If \(O(f) = O(g)\), then a function \(f\) is in \(O(g)\) and a function \(g\) is in \(O(f)\).

Step by step solution

01

Prove that \(f \in O(f)\)

According to the definition of Big O notation, we say that \(f \in O(g)\) if there exist positive constants \(M\) and \(n_0\) such that \(|f(n)| \leq M * |g(n)|\) for all \(n \geq n_0\). For \(f \in O(f)\), select \(M = 1\) and any positive integer for \(n_0\), we know that \(|f(n)| \leq 1 * |f(n)|\) is true for all \(n \geq n_0\). So, we can conclude that \(f \in O(f)\).
02

Prove that if \(f \in O(g)\) and \(g \in O(f)\) then \(O(f)=O(g)\)

Suppose that \(f \in O(g)\) and \(g \in O(f)\). This means there are constants \(M1\), \(M2\), \(n_0_1\), and \(n_0_2\) such that \(|f(n)| \leq M1 * |g(n)|\) and \(|g(n)| \leq M2 * |f(n)|\) for all \(n \geq n_0_1\) and \(n \geq n_0_2\) respectively. Now, let \(h\) be a function such that \(h \in O(f)\). This means that there exist constants \(M3\) and \(n_0_3\) such that \(|h(n)| \leq M3 * |f(n)|\) for all \(n \geq n_0_3\). Because \(f \in O(g)\), we can replace \(f\) in the inequality with \(M1 * g(n)\) giving us \(|h(n)| \leq M3 * M1 * |g(n)|\) for all \(n \geq n_0\). This means \(h \in O(g)\). Because \(h\) is an arbitrary function, we have proven that if \(h \in O(f)\) then \(h \in O(g)\). Similarly, if \(h \in O(g)\), then \(h \in O(f)\). This proves that \(O(f) = O(g)\).
03

Prove that if \(O(f) = O(g)\), then \(f \in O(g)\) and \(g \in O(f)\)

This follows directly from the definition of \(O(f) = O(g)\). If \(O(f) = O(g)\), then by definition for any function \(h\) in \(O(f)\), \(h\) is also in \(O(g)\), and vice versa. Since \(f\) is in \(O(f)\), by definition \(f\) should also be in \(O(g)\). Similarly, since \(g\) is in \(O(g)\), \(g\) must also be in \(O(f)\). We can thereby conclude that \(f \in O(g)\) and \(g \in O(f)\) if \(O(f) = O(g)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Asymptotic Notation
Asymptotic notation is a mathematical tool used in computer science and mathematics to describe the limiting behavior of functions. It provides a way to categorize functions based on their growth rates when the input progresses towards infinity. The most common asymptotic notations are Big O, Big Theta, and Big Omega.
Big O notation, denoted as \(O(g(n))\), describes an upper bound of a function. It tells us that a function \(f(n)\) will not grow faster than \(g(n)\) up to a constant factor, for large enough \(n\). That means there exists some positive constant \(M\) and \(n_0\) such that:
\[|f(n)| \leq M \cdot |g(n)| \text{ for } n \geq n_0\]
This notation is particularly useful for analyzing algorithms to understand their time or space complexity.
  • Big Theta (\(\Theta\)): This provides a tight bound, implying the function grows asymptotically as \(g(n)\).
  • Big Omega (\(\Omega\)): Describes a lower bound, indicating that \(f(n)\) grows at least as fast as \(g(n)\).
Understanding asymptotic notation helps to evaluate the efficiency of algorithms, ensuring better optimization and performance.
Dominance Relation
The dominance relation plays a crucial role in determining how functions compare to each other in terms of their growth rates. When one function dominates another, it implies that its growth rate is higher. Understanding dominance relations can be crucial in determining which algorithm is more efficient under specific conditions.
In the concept of Big O notation, if we say \(f \in O(g)\), it implies that the function \(f\) is dominated by the function \(g\). This notation indicates that there is a constant \(M\) and a starting point \(n_0\) above which \(f(n)\) will always be less than or equal to \(M\cdot g(n)\).
Sometimes dominance relations can be mutual. For example, if \(f \in O(g)\) and \(g \in O(f)\), it can be inferred that both functions grow at comparable rates. Thus, their asymptotic notations are considered equivalent. This equivalence is presented as \(O(f) = O(g)\).
Summing up, knowing the dominance relation helps programmers and mathematicians choose or design efficient algorithms, focusing on function behavior for substantial input sizes.
Mathematical Proofs
Mathematical proofs serve as a tool for establishing the truth behind mathematical statements and relations. They provide a logical foundation that ensures the rigor and correctness of mathematical concepts.
In the context of Big O, proofs validate the properties and relationships between functions. For instance, to show \(f \in O(g)\), a proof would demonstrate the existence of constants \(M\) and \(n_0\) such that the inequality \(|f(n)| \leq M \cdot |g(n)|\) holds for all \(n \geq n_0\).
Mathematical proofs make use of assumptions, logical deductions, and conclusions. They often begin by assuming a set condition and then logically working towards proving (or disproving) the statement.
Utilizing mathematical proofs ensures that algorithms not only appear efficient but are also guaranteed to be optimal under specified conditions. They form the backbone of computational theory, providing clarity and assurance in the reliability of algorithms.

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Most popular questions from this chapter

. Let \(f, g: \mathbf{R} \rightarrow \mathbf{R}\) where \(f(x)=a x+b\) and \(g(x)=c x+d\) for all \(x \in \mathbf{R}\), with \(a, b, c, d\) real constants. What relationship(s) must be satisfied by \(a, b, c, d\) if \((f \circ g)(x)=(g \circ f)(x)\) for all \(x \in \mathbf{R} ?\)

\text { If } n \in \mathbf{Z}^{+} \text {with } n \geq 4 \text {, verify that } S(n, n-2)=\left(\begin{array}{l} n \\ 3 \end{array}\right)+3\left(\begin{array}{l} n \\ 4 \end{array}\right) \text {. }

A lock has \(n\) buttons labeled \(1,2, \ldots, n\). To open this lock we press each of the \(n\) buttons exactly once. If no two or more buttons may be pressed simultaneously, then there are \(n !\) ways to do this. However, if one may press two or more buttons siTultaneously, then there are more than \(n !\) ways to press all of te buttons. For instance, if \(n=3\) there are six ways to press te buttons one at a time. But if one may also press two or more uttons simultaneously, then we find 13 cases - namely, (1) \(1,2,3\) (2) \(1,3,2\) (3) \(2,1,3\) (4) \(2,3,1\) (5) \(3,1,3\) (6) \(3,2,1\) (7) \(\\{1,2\\}, 3\) (8) \(3,\\{1,2\\}\) (9) \(\\{1,3\\}, 2\) (10) \(2,\\{1,3\\}\) (11) \(\\{2,3\\}, 1\) (12) \(1,\\{2,3\\}\) (13) \(\\{1,2,3\\}\), [Here, for example, case (12) indicates that one presses button 1 first and then buttons 2,3 (together) second.] (a) How many ways are there to press the buttons when \(n=4 ? n=5 ?\) How many for \(n\) in general? (b) Suppose a lock has 15 buttons. To open this lock one must press 12 different buttons (one at a time, or simultaneously in sets of two or more). In how many ways can this be done?

\text { Let } n \in \mathbf{N}, n \geq 2 \text {. Show that } S(n, 2)=2^{n-1}-1 \text {. }

. Let \(f, g: \mathbf{Z}^{+} \rightarrow \mathbf{R}\) where $$ f(n)=\left\\{\begin{array}{ll} 2, & \text { for } n \text { even } \\ 1, & \text { for } n \text { odd } \end{array} \quad g(n)= \begin{cases}3, & \text { for } n \text { even } \\ 4, & \text { for } n \text { odd }\end{cases}\right. $$ Prove or disprove each of the following: (a) \(f \in O(g)\); and (b) \(g \in O(f)\).
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