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Chapter 3: Problem 65

(Requires calculus) Represent pictorially that \(x \log x\) is \(o\left(x^{2}\right)\) by graphing \(x \log x, x^{2},\) and \(x \log x / x^{2}\) . Explain how this picture shows that \(x \log x\) is \(o\left(x^{2}\right) .\)

Short Answer

Expert verified
The graph of \( h(x) = \frac{x \log x}{x^2} \) shows that it tends to zero as \( x \to \infty \), indicating \( x \log x \) grows much slower than \ x^2 \.

Step by step solution

01

Title - Define the Functions

Consider the functions: 1. \(f(x) = x \log x\).2. \(g(x) = x^2\).3. \(h(x) = \frac{x \log x}{x^2}\).\
02

Title - Graph the Functions

Plot the graphs of the functions \(f(x) = x \log x \), \(g(x) = x^2 \), and \( h(x) = \frac{x \log x}{x^2} \).
03

Title - Analyze the Graphs

Observe the behavior of \(f(x) = x \log x \) and \(g(x) = x^2 \) as \(x\) grows larger.- Notice that \(f(x)\) grows slower than \(g(x)\) as \(x \to \infty\).- The function \(h(x) = \frac{x \log x}{x^2} \) tends towards zero as \ x \ tends to infinity. This suggests that \( x \log x \) grows slower compared to \( x^2 \) .
04

Title - Conclusion

From the graph of \( h(x) = \frac{x \log x}{x^2} \) tending towards zero as \(x \to \infty\), we see that the growth rate of \( x \log x \) is much smaller compared to \( x^2 \). Thus, \( x \log x \) is \( o ( x^2 ) \).

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Key Concepts

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

Big-O Notation
To understand asymptotic analysis, we need to start with Big-O notation. This notation helps us describe the upper bound of an algorithm鈥檚 running time. It tells us how the runtime of an algorithm increases as the input size grows.
This is crucial for comparing the efficiency of different algorithms.
In mathematical terms, if a function \(f(x)\) is \(O(g(x))\), it means there are constants \(c\) and \(n_0\) such that for all \(x \textgreater n_0\), \(|f(x)| \textleq c|g(x)|\).
Here are some common examples:
  • \textbf{O(1)}: Constant time - the operation does not depend on the size of the input.
  • \textbf{O(n)}: Linear time - the operation grows directly in proportion to the input size.
  • \textbf{O(n^2)}: Quadratic time - the operation time is proportional to the square of the input size.
Knowing this helps us classify algorithms based on their time complexity and determine the most efficient one for a given problem.
Asymptotic Behavior
Asymptotic behavior examines how a function behaves as its input becomes very large. It's like looking far into the horizon to see the overall trend, ignoring small details.
We use this to compare functions and see which one grows faster in the long run.
For example, let鈥檚 look at \(f(x) = x \text log x\) and \(g(x) = x^2\). As \(x\) gets very large, even though both functions grow, one grows much faster than the other.
When considering the function \(h(x) = \frac{x \text log x}{x^2}\):
  • As \(x\) approaches infinity, the numerator \(x \text log x\) grows slower compared to the denominator \(x^2\).
  • This means the overall fraction tends towards zero.
  • We say \(x \text log x\) is 'little-o' of \(x^2\), written as \(x \text log x = o(x^2)\).
This notation indicates that \(x \text log x\) grows much slower than \(x^2\) as \(x\) becomes very large.
Growth Rates
Growth rates help us understand how the runtime or complexity of an algorithm increases with the size of the input. Identifying growth rates allows us to predict performance and efficiency.
Some common growth rates you鈥檒l encounter:
  • \textbf{Constant (O(1))}: The time taken is constant, regardless of input size.
  • \textbf{Logarithmic (O(log n))}: The time increases logarithmically as input size increases.
  • \textbf{Linear (O(n))}: The time increases linearly with the input size.
  • \textbf{Quadratic (O(n^2))}: The time increases proportionally to the square of the input size.
Now, relating to the given exercise:
We compared two functions \(x \text log x\) and \(x^2\). By analyzing their graphs:
  • \(x \text log x\) grows much slower than \(x^2\) as \(x \rightarrrow \textinfty\).
  • The function \(\frac{x \text log x}{x^2}\) approaching zero shows that \(x \text log x\) is asymptotically less dominant than \(x^2\).
Understanding these various growth rates helps you judge the efficiency of different algorithms and choose the best one for your needs.

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Most popular questions from this chapter

What is the largest \(n\) for which one can solve within a day using an algorithm that requires \(f(n)\) bit operations, where each bit operation is carried out in \(10^{-11}\) seconds, with these functions \(f(n) ?\) $$ \begin{array}{llll}{\text { a) } \log n} & {\text { b) } 1000 n} & {\text { c) } n^{2}} \\ {\text { d) } 1000 n^{2}} & {\text { e) } n^{3}} & {\text { f) } 2^{n}} \\ {\text { g) } 2^{2 n}} & {\text { h) } 2^{2^{n}}}\end{array} $$

Give a big- \(O\) estimate of the product of the first \(n\) odd positive integers.

The conventional algorithm for evaluating a polynomial \(a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\) at \(x=c\) can be expressed in pseudocode by $$ \begin{array}{c}{\text { procedure polynomial }\left(c, a_{0}, a_{1}, \ldots, a_{n} : \text { real numbers) }\right.} \\ {\text { power } :=1} \\ {\quad y :=a_{0}} \\ {\text { for } i :=1 \text { to } n} \\ {\text { power } :=\text { power } * c} \\ {\quad y :=y+a_{i} * \text { power }} \\ {\text { return } y\left\\{y=a_{n} c^{n}+a_{n-1} c^{n-1}+\cdots+a_{1} c+a_{0}\right\\}}\end{array} $$ where the final value of \(y\) is the value of the polynomial at \(x=c .\) a) Evaluate \(3 x^{2}+x+1\) at \(x=2\) by working through each step of the algorithm showing the values assigned at each assignment step. b) Exactly how many multiplications and additions are used to evaluate a polynomial of degree \(n\) at \(x=c\) ? (Do not count additions used to increment the loop variable.)

(Requires calculus) The two parts of this exercise describe the relationship between little- \(o\) and big- \(O\) notation. a) Show that if \(f(x)\) and \(g(x)\) are functions such that \(f(x)\) is \(o(g(x)),\) then \(f(x)\) is \(O(g(x))\) . b) Show that if \(f(x)\) and \(g(x)\) are functions such that \(f(x)\) is \(O(g(x)),\) then it does not necessarily follow that \(f(x)\) is \(o(g(x))\)

Describe an algorithm to find the longest word in an English sentence (where a sentence is a sequence of symbols, either a letter or a blank, which can then be broken into alternating words and blanks).
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