Informally, an algorithm is any well-defined computational procedure that takes some value, or set of values, as input and produces some value, or set of values, as output. An algorithm is thus a sequence of computational steps that transform the input into the output.
We can also view an algorithm as a tool for solving a well-specified computational problem. The statement of the problem specifies in general terms the desired input/output relationship. The algorithm describes a specific computational procedure for achieving that input/output relationship.
Input: A sequence of n numbers 
a1, a2, . . . , an
.
Output: A permutation (reordering) 
of the input sequence such that 
.
Given an input sequence such as 31, 41, 59, 26, 41, 58, a sorting algorithm returns as output the sequence 26, 31, 41, 41, 58, 59. Such an input sequence is called an instance of the sorting problem. In general, an instance of a problem consists of all the inputs (satisfying whatever constraints are imposed in the problem statement) needed to compute a solution to the problem.
An algorithm is said to be correct if, for every input instance, it halts with the correct output. We say that a correct algorithm solves the given computational problem. An incorrect algorithm might not halt at all on some input instances, or it might halt with other than the desired answer. Contrary to what one might expect, incorrect algorithms can sometimes be useful, if their error rate can be controlled. We shall see an example of this in Chapter 33 when we study algorithms for finding large prime numbers. Ordinarily, however, we shall be concerned only with correct algorithms.
An algorithm can be specified in English, as a computer program, or even as a hardware design. The only requirement is that the specification must provide a precise description of the computational procedure to be followed.
In this book, we shall typically describe algorithms as programs written in a pseudocode that is very much like C, Pascal, or Algol. If you have been introduced to any of these languages, you should have little trouble reading our algorithms. What separates pseudocode from "real" code is that in pseudocode, we employ whatever expressive method is most clear and concise to specify a given algorithm. Sometimes, the clearest method is English, so do not be surprised if you come across an English phrase or sentence embedded within a section of "real" code. Another difference between pseudocode and real code is that pseudocode is not typically concerned with issues of software engineering. Issues of data abstraction, modularity, and error handling are often ignored in order to convey the essence of the algorithm more concisely.
We start with insertion sort, which is an efficient algorithm for sorting a small number of elements. Insertion sort works the way many people sort a bridge or gin rummy hand. We start with an empty left hand and the cards face down on the table. We then remove one card at a time from the table and insert it into the correct position in the left hand. To find the correct position for a card, we compare it with each of the cards already in the hand, from right to left, as illustrated in Figure 1.1.
Our pseudocode for insertion sort is presented as a procedure called INSERTION-SORT, which takes as a parameter an array A[1 . . n] containing a sequence of length n that is to be sorted. (In the code, the number n of elements in A is denoted by length[A].) The input numbers are sorted in place: the numbers are rearranged within the array A, with at most a constant number of them stored outside the array at any time. The input array A contains the sorted output sequence when INSERTION-SORT is finished.
INSERTION-SORT (A)
1 for j2 to length[A]
2 do key
3Insert A[j] into the sorted sequence A[1 . . j - 1].
4 i
5 while i > 0 and A[i] > key
6 do A[i + 1]
7 i
8 A[i + 1]
5, 2, 4, 6, 1, 3. The position of index j is indicated by a circle.We use the following conventions in our pseudocode.
1. Indentation indicates block structure. For example, the body of the for loop that begins on line 1 consists of lines 2-8, and the body of the while loop that begins on line 5 contains lines 6-7 but not line 8. Our indentation style applies to if-then-else statements as well. Using indentation instead of conventional indicators of block structure, such as begin and end statements, greatly reduces clutter while preserving, or even enhancing, clarity.1
2. The looping constructs while, for, and repeat and the conditional constructs if, then, and else have the the same interpretation as in Pascal.
3. The symbol 
indicates that the remainder of the line is a comment.
4. A multiple assignment of the form i j e assigns to both variables i and j the value of expression e; it should be treated as equivalent to the assignment j e followed by the assignment i j.
5. Variables (such as i, j, and key) are local to the given procedure. We shall not use global variables without explicit indication.
6. Array elements are accessed by specifying the array name followed by the index in square brackets. For example, A[i] indicates the ith element of the array A. The notation ". ." is used to indicate a range of values within an array. Thus, A[1. . j] indicates the subarray of A consisting of elements A[1], A[2], . . . , A[j].
7. Compound data are typically organized into objects, which are comprised of attributes or fields. A particular field is accessed using the field name followed by the name of its object in square brackets. For example, we treat an array as an object with the attribute length indicating how many elements it contains. To specify the number of elements in an array A, we write length[A]. Although we use square brackets for both array indexing and object attributes, it will usually be clear from the context which interpretation is intended.
A variable representing an array or object is treated as a pointer to the data representing the array or object. For all fields f of an object x, setting y x causes f[y] = f[x]. Moreover, if we now set f[x] 3, then afterward not only is f[x] = 3, but f[y] = 3 as well. In other words, x and y point to ("are") the same object after the assignment y x.
Sometimes, a pointer will refer to no object at all. In this case, we give it the special value NIL.
8. Parameters are passed to a procedure by value: the called procedure receives its own copy of the parameters, and if it assigns a value to a parameter, the change is not seen by the calling routine. When objects are passed, the pointer to the data representing the object is copied, but the object's fields are not. For example, if x is a parameter of a called procedure, the assignment x y within the called procedure is not visible to the calling procedure. The assignment f[x] 3, however, is visible.
Rewrite the INSERTION-SORT procedure to sort into nonincreasing instead of nondecreasing order.
Consider the searching problem:
Input: A sequence of n numbers A = a1, a2, . . . ,an and a value v.
Output: An index i such that v = A[i] or the special value NIL if v does not appear in A.
Write pseudocode for linear search, which scans through the sequence, looking for v.
Consider the problem of adding two n-bit binary integers, stored in two n-element arrays A and B. The sum of the two integers should be stored in binary form in an (n + 1)-element array C. State the problem formally and write pseudocode for adding the two integers.
Analyzing an algorithm has come to mean predicting the resources that the algorithm requires. Occasionally, resources such as memory, communication bandwidth, or logic gates are of primary concern, but most often it is computational time that we want to measure. Generally, by analyzing several candidate algorithms for a problem, a most efficient one can be easily identified. Such analysis may indicate more than one viable candidate, but several inferior algorithms are usually discarded in the process.
Before we can analyze an algorithm, we must have a model of the implementation technology that will be used, including a model for the resources of that technology and their costs. For most of this book, we shall assume a generic one-processor, random-access machine (RAM) model of computation as our implementation technology and understand that our algorithms will be implemented as computer programs. In the RAM model, instructions are executed one after another, with no concurrent operations. In later chapters, however, we shall have occasion to investigate models for parallel computers and digital hardware.
The time taken by the INSERTION-SORT procedure depends on the input: sorting a thousand numbers takes longer than sorting three numbers. Moreover, INSERTION-SORT can take different amounts of time to sort two input sequences of the same size depending on how nearly sorted they already are. In general, the time taken by an algorithm grows with the size of the input, so it is traditional to describe the running time of a program as a function of the size of its input. To do so, we need to define the terms "running time" and "size of input" more carefully.
The best notion for input size depends on the problem being studied. For many problems, such as sorting or computing discrete Fourier transforms, the most natural measure is the number of items in the input--for example, the array size n for sorting. For many other problems, such as multiplying two integers, the best measure of input size is the total number of bits needed to represent the input in ordinary binary notation. Sometimes, it is more appropriate to describe the size of the input with two numbers rather than one. For instance, if the input to an algorithm is a graph, the input size can be described by the numbers of vertices and edges in the graph. We shall indicate which input size measure is being used with each problem we study.
The running time of an algorithm on a particular input is the number of primitive operations or "steps" executed. It is convenient to define the notion of step so that it is as machine-independent as possible. For the moment, let us adopt the following view. A constant amount of time is required to execute each line of our pseudocode. One line may take a different amount of time than another line, but we shall assume that each execution of the ith line takes time ci, where ci is a constant. This viewpoint is in keeping with the RAM model, and it also reflects how the pseudocode would be implemented on most actual computers.2
We start by presenting the INSERTION-SORT procedure with the time "cost" of each statement and the number of times each statement is executed. For each j = 2, 3, . . . , n, where n = length[A], we let tj be the number of times the while loop test in line 5 is executed for that value of j. We assume that comments are not executable statements, and so they take no time.
T(n) = c1n + c2 (n - 1) + c4 (n - 1) + c5 (n - 1) + c8 (n - 1)
= (c1 + c2 + c4 + c8)n - (c2 + c4 + c5 + c8).
This running time can be expressed as an + b for constants a and b that depend on the statement costs ci; it is thus a linear function of n.
This worst-case running time can be expressed as an2 + bn+ c for constants a, b, and c that again depend on the statement costs ci; it is thus a quadratic function of n.
In some particular cases, we shall be interested in the average-case or expected running time of an algorithm. One problem with performing an average-case analysis, however, is that it may not be apparent what constitutes an "average" input for a particular problem. Often, we shall assume that all inputs of a given size are equally likely. In practice, this assumption may be violated, but a randomized algorithm can sometimes force it to hold.
Consider sorting n numbers stored in array A by first finding the smallest element of A and putting it in the first entry of another array B. Then find the second smallest element of A and put it in the second entry of B. Continue in this manner for the n elements of A. Write pseudocode for this algorithm, which is known as selection sort. Give the best-case and worst-case running times of selection sort in 
-notation.
Consider linear search again (see Exercise 1.1-3). How many elements of the input sequence need to be checked on the average, assuming that the element being searched for is equally likely to be any element in the array? How about in the worst case? What are the average-case and worst-case running times of linear search in -notation? Justify your answers.
Consider the problem of evaluating a polynomial at a point. Given n coefficients a0, a1, . . . , an - 1 and a real number x, we wish to compute 
. Describe a straightforward (n2)-time algorithm for this problem. Describe a (n)-time algorithm that uses the following method (called Horner's rule) for rewriting the polynomial:
Express the function n3/1000 - 100n2 - 100n + 3 in terms of -notation.
How can we modify almost any algorithm to have a good best-case running time?
There are many ways to design algorithms. Insertion sort uses an incremental approach: having sorted the subarray A[1 . . j - 1], we insert the single element A[j] into its proper place, yielding the sorted subarray A[1 . . j].
Many useful algorithms are recursive in structure: to solve a given problem, they call themselves recursively one or more times to deal with closely related subproblems. These algorithms typically follow a divide-and-conquer approach: they break the problem into several subproblems that are similar to the original problem but smaller in size, solve the subproblems recursively, and then combine these solutions to create a solution to the original problem.
The divide-and-conquer paradigm involves three steps at each level of the recursion:
Divide the problem into a number of subproblems.
Combine the solutions to the subproblems into the solution for the original problem.
The merge sort algorithm closely follows the divide-and-conquer paradigm. Intuitively, it operates as follows.
Divide: Divide the n-element sequence to be sorted into two subsequences of n/2 elements each.
Conquer: Sort the two subsequences recursively using merge sort.
Combine: Merge the two sorted subsequences to produce the sorted answer.
The key operation of the merge sort algorithm is the merging of two sorted sequences in the "combine" step. To perform the merging, we use an auxiliary procedure MERGE(A,p,q,r), where A is an array and p, q, and r are indices numbering elements of the array such that p 
q < r. The procedure assumes that the subarrays A[p. .q] and A[q + 1. .r] are in sorted order. It merges them to form a single sorted subarray that replaces the current subarray A[p. .r].
MERGE-SORT(A,p,r)
1 if p < r
2 then q(p + r)/2
3 MERGE-SORT(A,p,q)
4 MERGE-SORT(A, q + 1, r)
5 MERGE(A,p,q,r)
To sort the entire sequence A = A[1],A[2], . . . ,A[n], we call MERGE-SORT(A, 1, length[A]), where once again length[A] = n. If we look at the operation of the procedure bottom-up when n is a power of two, the algorithm consists of merging pairs of 1-item sequences to form sorted sequences of length 2, merging pairs of sequences of length 2 to form sorted sequences of length 4, and so on, until two sequences of length n/2 are merged to form the final sorted sequence of length n. Figure 1.3 illustrates this process.
When an algorithm contains a recursive call to itself, its running time can often be described by a recurrence equation or recurrence, which describes the overall running time on a problem of size n in terms of the running time on smaller inputs. We can then use mathematical tools to solve the recurrence and provide bounds on the performance of the algorithm.
5, 2, 4, 6, 1, 3, 2, 6. The lengths of the sorted sequences being merged increase as the algorithm progresses from bottom to top.
In Chapter 4, we shall see how to solve common recurrences of this form.
Write pseudocode for MERGE(A,p,q,r).
Use mathematical induction to show that the solution of the recurrence
Referring back to the searching problem (see Exercise 1.1-3), observe that if the sequence A is sorted, we can check the midpoint of the sequence against v and eliminate half of the sequence from further consideration. Binary search is an algorithm that repeats this procedure, halving the size of the remaining portion of the sequence each time. Write pseudocode, either iterative or recursive, for binary search. Argue that the worst-case running time of binary search is (lg n).
Observe that the while loop of lines 5-7 of the INSERTION-SORT procedure in Section 1.1 uses a linear search to scan (backward) through the sorted subarray A[1 . . j - 1]. Can we use a binary search (see Exercise 1.3-5) instead to improve the overall worst-case running time of insertion sort to (n lg n)?
Suppose we are comparing implementations of insertion sort and merge sort on the same machine. For inputs of size n, insertion sort runs in 8n2 steps, while merge sort runs in 64n 1g n steps. For which values of n does insertion sort beat merge sort? How might one rewrite the merge sort pseudocode to make it even faster on small inputs?
1-1 Comparison of running times
1-2 Insertion sort on small arrays in merge sort
Although merge sort runs in (n lg n) worst-case time and insertion sort runs in (n2) worst-case time, the constant factors in insertion sort make it faster for small n. Thus, it makes sense to use insertion sort within merge sort when subproblems become sufficiently small. Consider a modification to merge sort in which n/k sublists of length k are sorted using insertion sort and then merged using the standard merging mechanism, where k is a value to be determined.
b. Show that the sublists can be merged in (n lg(n/k)) worst-case time.
d. How should k be chosen in practice?
Let A[1 . . n] be an array of n distinct numbers. If i < j and A[i] > A[j], then the pair (i, j) is called an inversion of A.
a. List the five inversions of the array 2, 3, 8, 6, 1.
Knuth [123] provides an encyclopedic treatment of many sorting algorithms. His comparison of sorting algorithms (page 381) includes exact step-counting analyses, like the one we performed here for insertion sort. Knuth's discussion of insertion sort encompasses several variations of the algorithm. The most important of these is Shell's sort, introduced by D. L. Shell, which uses insertion sort on periodic subsequences of the input to produce a faster sorting algorithm.
The worst-case running time of an algorithm is an upper bound on the running time for any input. Knowing it gives us a guarantee that the algorithm will never take any longer. We need not make some educated guess about the running time and hope that it never gets much worse.
r, the subarray has at most one element and is therefore already sorted. Otherwise, the divide step simply computes an index q that partitions A[p. .r] into two subarrays: A[p. .q], containing n/2] elements, and A[q + 1. .r], containing 
elements.4
x
denotes the least integer greater than or equal to x, and