A Practical Introduction to Data Structures and Algorithm Analysis

32 Chap. 2 Mathematical PreliminariesIn this book, nearly all logarithms used have a base of two. This is becausedata structures and algorithms most often divide things in half, or store codes withbinary bits. Whenever you see the notation log n in this book, either log 2 n is meantor else the term is being used asymptotically and the actual base does not matter. Ifany base for the logarithm other than two is intended, then the base will be shownexplicitly.Logarithms have the following properties, for any positive values of m, n, andr, and any positive integers a and b.1. log(nm) = log n + log m.2. log(n/m) = log n − log m.3. log(n r ) = r log n.4. log a n = log b n/ log b a.The first two properties state that the logarithm of two numbers multiplied (ordivided) can be found by adding (or subtracting) the logarithms of the two numbers.4 Property (3) is simply an extension of property (1). Property (4) tells us that,for variable n and any two integer constants a and b, log a n and log b n differ bythe constant factor log b a, regardless of the value of n. Most runtime analyses inthis book are of a type that ignores constant factors in costs. Property (4) says thatsuch analyses need not be concerned with the base of the logarithm, because thiscan change the total cost only by a constant factor. Note that 2 log n = n.When discussing logarithms, exponents often lead to confusion. Property (3)tells us that log n 2 = 2 log n. How do we indicate the square of the logarithm(as opposed to the logarithm of n 2 )? This could be written as (log n) 2 , but it istraditional to use log 2 n. On the other hand, we might want to take the logarithm ofthe logarithm of n. This is written log log n.A special notation is used in the rare case where we would like to know howmany times we must take the log of a number before we reach a value ≤ 1. Thisquantity is written log ∗ n. For example, log ∗ 1024 = 4 because log 1024 = 10,log 10 ≈ 3.33, log 3.33 ≈ 1.74, and log 1.74 < 1, which is a total of 4 log operations.4 These properties are the idea behind the slide rule. Adding two numbers can be viewed as joiningtwo lengths together and measuring their combined length. Multiplication is not so easily done.However, if the numbers are first converted to the lengths of their logarithms, then those lengths canbe added and the inverse logarithm of the resulting length gives the answer for the multiplication (thisis simply logarithm property (1)). A slide rule measures the length of the logarithm for the numbers,lets you slide bars representing these lengths to add up the total length, and finally converts this totallength to the correct numeric answer by taking the inverse of the logarithm for the result.