Data Structures and Algorithms in Java[1].pdf - Fulvio Frisone

Example 4.2: We demonstrate below some interesting applications of the
logarithm rules from Proposition 4.1 (using the usual convention that the base of a
logarithm is 2 if it is omitted).
• log(2n) = log2 + log n = 1 + logn, by rule 1
• log(n/2) = logn − log2 = logn − 1, by rule 2
• logn 3 = 3logn, by rule 3
• log2 n = nlog2 =n · 1 = n, by rule 3
• log 4 n = (log n)/ log4 = (logn) /2, by rule 4
• 2 logn = n log2 = n 1 = n, by rule 5.
As a practical matter, we note that rule 4 gives us a way to compute the base-two
logarithm on a calculator that has a base-10 logarithm button, LOG, for
log 2 n = LOGn/LOG2.
4.1.3 The Linear function
Another simple yet important function is the linear function,
f(n)= n.
That is, given an input value n, the linear function f assigns the value n itself.
This function arises in algorithm analysis any time we have to do a single basic
operation for each of n elements. For example, comparing a number x to each
element of an array of size n will require n comparisons. The linear function also
represents the best running time we can hope to achieve for any algorithm that
processes a collection of n objects that are not already in the computer's memory,
since reading in the n objects itself requires n operations.
4.1.4 The N-Log-N function
The next function we discuss in this section is the n-log-n function,
f(n) = nlogn,
that is, the function that assigns to an input n the value of n times the logarithm
base-two of n. This function grows a little faster than the linear function and a lot
slower than the quadratic function. Thus, as we will show on several occasions, if
we can improve the running time of solving some problem from quadratic to n-logn,
we will have an algorithm that runs much faster in general.
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