In this section, we review some mathematical notations and tools used throughout this book, including logarithms, big-Oh notation, and probability theory. This review will be brief and is not intended as an introduction. Readers who feel they are missing this background are encouraged to read, and do exercises from, the appropriate sections of the very good (and free) textbook on mathematics for computer science .
The expression denotes the number raised to the power of . If is a positive integer, then this is just the value of multiplied by itself times:
In this book, the expression denotes the base- logarithm of . That is, the unique value that satisfies
An informal, but useful, way to think about logarithms is to think of as the number of times we have to divide by before the result is less than or equal to 1. For example, when one does binary search, each comparison reduces the number of possible answers by a factor of 2. This is repeated until there is at most one possible answer. Therefore, the number of comparison done by binary search when there are initially at most possible answers is at most .
Another logarithm that comes up several times in this book is the natural logarithm. Here we use the notation to denote , where -- Euler's constant -- is given by
In one or two places in this book, the factorial function is used. For a non-negative integer , the notation (pronounced `` factorial'') is defined to mean
The quantity can be approximated using Stirling's Approximation:
Related to the factorial function are the binomial coefficients. For a non-negative integer and an integer , the notation denotes:
When analyzing data structures in this book, we want to talk about the running times of various operations. The exact running times will, of course, vary from computer to computer and even from run to run on an individual computer. When we talk about the running time of an operation we are referring to the number of computer instructions performed during the operation. Even for simple code, this quantity can be difficult to compute exactly. Therefore, instead of analyzing running times exactly, we will use the so-called big-Oh notation: For a function , denotes a set of functions,
We generally use asymptotic notation to simplify functions. For example, in place of we can write . This is proven as follows:
A number of useful shortcuts can be applied when using asymptotic notation. First:
Continuing in a long and distinguished tradition, we will abuse this notation by writing things like when what we really mean is . We will also make statements like ``the running time of this operation is '' when this statement should be ``the running time of this operation is a member of .'' These shortcuts are mainly to avoid awkward language and to make it easier to use asymptotic notation within strings of equations.
A particularly strange example of this occurs when we write statements like
The expression also brings up another issue. Since there is no variable in this expression, it may not be clear which variable is getting arbitrarily large. Without context, there is no way to tell. In the example above, since the only variable in the rest of the equation is , we can assume that this should be read as , where .
Big-Oh notation is not new or unique to computer science. It was used by the number theorist Paul Bachmann as early as 1894, and is immensely useful for describing the running times of computer algorithms. Consider the following piece of code: One execution of this method involves
Big-Oh notation allows us to reason at a much higher level, making it possible to analyze more complicated functions. If two algorithms have the same big-Oh running time, then we won't know which is faster, and there may not be a clear winner. One may be faster on one machine, and the other may be faster on a different machine. However, if the two algorithms have demonstrably different big-Oh running times, then we can be certain that the one with the smaller running time will be faster for large enough values of .
An example of how big-Oh notation allows us to compare two different functions is shown in Figure 1.5, which compares the rate of growth of versus . It might be that is the running time of a complicated linear time algorithm while is the running time of a considerably simpler algorithm based on the divide-and-conquer paradigm. This illustrates that, although is greater than for small values of , the opposite is true for large values of . Eventually wins out, by an increasingly wide margin. Analysis using big-Oh notation told us that this would happen, since .
In a few cases, we will use asymptotic notation on functions with more than one variable. There seems to be no standard for this, but for our purposes, the following definition is sufficient:
Some of the data structures presented in this book are randomized; they make random choices that are independent of the data being stored in them or the operations being performed on them. For this reason, performing the same set of operations more than once using these structures could result in different running times. When analyzing these data structures we are interested in their average or expected running times.
Formally, the running time of an operation on a randomized data structure is a random variable, and we want to study its expected value. For a discrete random variable taking on values in some countable universe , the expected value of , denoted by , is given by the formula
One of the most important properties of expected values is linearity of expectation. For any two random variables and ,
A useful trick, that we will use repeatedly, is defining indicator random variables. These binary variables are useful when we want to count something and are best illustrated by an example. Suppose we toss a fair coin times and we want to know the expected number of times the coin turns up as heads. Intuitively, we know the answer is , but if we try to prove it using the definition of expected value, we get
Using indicator variables and linearity of expectation makes things much easier. For each , define the indicator random variable