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Chapter 3: Problem 60

(Requires calculus) Show that if \(c>b>1,\) then \(b^{n}\) is \(O\left(c^{n}\right),\) but \(c^{n}\) is not \(O\left(b^{n}\right) .\)

Short Answer

Expert verified
For c > b > 1, b^n is O(c^n) but c^n is not O(b^n).

Step by step solution

01

Understanding Big-O Notation

In asymptotic analysis, a function f(n) is said to be O(g(n)) if there exist positive constants C and n_0 such that for all n 鈮� n_0, |f(n)| 鈮� C|g(n)|.
02

Expressing the Functions

Here, we have to show that if c > b > 1, then b^n is O(c^n), but c^n is not O(b^n).
03

Showing b^n is O(c^n)

To prove b^n is O(c^n), we need constants C and n_0 such that b^n 鈮� C c^n for all n 鈮� n_0. Since c > b, we can write b = c^(log_c(b)) and since log_c(b) < 1, b^n = c^(n log_c(b)). Hence, we can pick C = 1 and n_0 = 1, making b^n 鈮� c^n.
04

Proving c^n is not O(b^n)

To show that c^n is not O(b^n), assume there exist positive constants C and n_0 such that c^n 鈮� C b^n for all n 鈮� n_0. Taking the logarithm of both sides, n log(c) 鈮� log(C) + n log(b). For sufficiently large n, this would imply log(c) 鈮� log(b), contradicting the assumption c > b. Thus, c^n is not O(b^n).

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Key Concepts

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

Asymptotic Analysis
Asymptotic analysis is a powerful tool in computer science and mathematics. It describes the behavior of functions as inputs become large. This analysis helps us compare the efficiency of algorithms.
Let's break it down:
  • **Big-O Notation**: If we say a function \(f(n)\) is \(O(g(n))\), it means \(f(n)\) grows at a rate no faster than \(g(n)\) up to a constant factor. We express it formally as: there exist constants \(C > 0\) and \(n_0 \) such that \( |f(n)| \leq C|g(n)| \) for all \( n \geq n_0\).
  • **Purpose**: This helps us analyze the upper bound on the running time of an algorithm.
In our exercise, we used asymptotic analysis to compare exponential functions \(b^n\) and \(c^n\) where \(c > b > 1\).
Exponential Functions
Exponential functions are a foundational concept in mathematics. Here's a simple breakdown:
An exponential function with base \(a\) is written as \(a^n\), where \(a\) is a constant and \(n\) is a variable exponent.
Key properties:
  • **Rapid Growth**: These functions grow really quickly. For example, \(2^n\) grows much faster than \(n^3\) as \(n\) becomes large.
  • **Comparisons**: In our exercise, we dealt with \(b^n\) and \(c^n\). If \(c > b > 1\), then \(c^n\) grows faster than \(b^n\).
  • **Big-O Example**: To show \(b^n\) is \(O(c^n)\), we noted that there are constants, like \(C=1\) and \(n_0=1\), making \(b^n \leq C \cdot c^n\).
Such analyses help us see how algorithms fare in large problem sizes.
Logarithmic Functions
Logarithmic functions are the inverse operations of exponential functions.
Let's dive into them:
\( \text{log}_a(x) \) is the exponent to which the base \(a\) must be raised to produce \(x\). For example, \( \text{log}_2(8) = 3 \) because \(2^3 = 8\).
Key points:
  • **Slow Growth**: Logarithms grow very slowly compared to exponential functions.
  • **Use in Proof**: In the exercise, we used the property \( \text{log}(c^n) = n \text{log}(c)\). To show \(c^n\) is not \(O(b^n)\), we assumed \(c^n \leq C b^n\) and took logarithms. This led to a contradiction because \( \text{log}(c) > \text{log}(b)\).
Logs are great for handling large numbers and simplifying multiplicative processes to additive ones.
Proof Techniques
Proof techniques are essential for validating mathematical concepts.
In our exercise, we used two main methods:
  • **Direct Proof**: This is straightforward. We set up the equations and showed using logical steps that for constants \(C\) and \(n_0\), an inequality holds. For instance, we showed \(b^n \leq C c^n\) with appropriately chosen constants.
  • **Proof by Contradiction**: Here, we assume the opposite of what we want to prove. By finding a contradiction, we show that our assumption must be false. We showed that if \(c^n\) were \(O(b^n)\), it would contradict the fact that \(c > b\).
Proofs establish foundational truths in mathematics and computer science. Understanding and using them allows us to confidently work with complex ideas.

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Most popular questions from this chapter

Show that the problem of determining whether a program with a given input ever prints the digit 1 is unsolvable.

The following problems deal with another type of asymptotic notation, called little-o notation. Because little-o notation is based on the concept of limits, a knowledge of calculus is needed for these problems. We say that \(f(x)\) is \(o(g(x))\) [read \(f(x)\) is "little-oh" of \(g(x) ]\) , when $$\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=0$$ (Requires calculus) Show that a) \(x^{2}\) is \(o\left(x^{3}\right)\) b) \(x \log x\) is \(o\left(x^{2}\right)\) c) \(x^{2}\) is \(o\left(2^{x}\right)\) d) \(x^{2}+x+1\) is not \(o\left(x^{2}\right)\)

(Requires calculus) Let \(H_{n}\) be the \(n\) th harmonic number $$H_{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$$ Show that \(H_{n}\) is \(O(\log n)\) [Hint: First establish the inequality $$\sum_{j=2}^{n} \frac{1}{j}<\int_{1}^{n} \frac{1}{x} d x$$ by showing that the sum of the areas of the rectangles of height 1\(/ j\) with base from \(j-1\) to \(j,\) for \(j=2,3, \ldots, n,\) is less than the area under the curve \(y=1 / x\) from 2 to \(n .1\)

a) Suppose we have \(n\) subsets \(S_{1}, S_{2}, \ldots, S_{n}\) of the set \(\\{1,2, \ldots, n\\} .\) Express a brute-force algorithm that determines whether there is a disjoint pair of these sub- sets. [Hint: The algorithm should loop through the subsets; for each subset \(S_{i},\) it should then loop through all other subsets; and for each of these other subsets \(S_{j},\) it should loop through all elements \(k\) in \(S_{i}\) to deter- mine whether \(k\) also belongs to \(S_{j} . ]\) b) Give a big-O estimate for the number of times the algorithm needs to determine whether an integer is in one of the subsets.

Write the selection sort algorithm in pseudocode.
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