Insertion sort

Our first algorithm, insertion sort, solves the sorting problem introduced in Chap-ter 1:

Input: A sequence of n numbers ha1; a2; : : : ; ani.

Output: A permutation (reordering) ha⁰₁; a⁰₂; : : : ; a_n⁰i of the input sequence such that a⁰₁ a⁰₂     a_n⁰.

The numbers that we wish to sort are also known as the keys. Although conceptu-ally we are sorting a sequence, the input comes to us in the form of an array with n elements.

In this book, we shall typically describe algorithms as programs written in a pseudocode that is similar in many respects to C, C++, Java, Python, or Pascal. If you have been introduced to any of these languages, you should have little trouble

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Figure 2.1 Sorting a hand of cards using insertion sort.

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 hand of playing cards. 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 2.1. At all times, the cards held in the left hand are sorted, and these cards were originally the top cards of the pile on the table.

We present our pseudocode for insertion sort as a procedure called INSERTION -SORT, which takes as a parameter an array AŒ1 : : 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 A: length.) The algorithm sorts the input numbers in place: it rearranges the numbers 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 the INSERTION-SORT procedure is finished.

1 2 3 4 5 6

Figure 2.2 The operation of INSERTION-SORTon the array A D h5; 2; 4; 6; 1; 3i. Array indices appear above the rectangles, and values stored in the array positions appear within the rectangles.

(a)–(e) The iterations of the for loop of lines 1–8. In each iteration, the black rectangle holds the key taken from AŒj , which is compared with the values in shaded rectangles to its left in the test of line 5. Shaded arrows show array values moved one position to the right in line 6, and black arrows indicate where the key moves to in line 8. (f) The final sorted array.

INSERTION-SORT.A/

Loop invariants and the correctness of insertion sort

Figure 2.2 shows how this algorithm works for A D h5; 2; 4; 6; 1; 3i. The in-dex j indicates the “current card” being inserted into the hand. At the beginning of each iteration of the for loop, which is indexed by j , the subarray consisting of elements AŒ1 : : j  1 constitutes the currently sorted hand, and the remaining subarray AŒj C 1 : : n corresponds to the pile of cards still on the table. In fact, elements AŒ1 : : j  1 are the elements originally in positions 1 through j  1, but now in sorted order. We state these properties of AŒ1 : : j  1 formally as a loop invariant:

At the start of each iteration of the for loop of lines 1–8, the subarray AŒ1 : : j 1 consists of the elements originally in AŒ1 : : j  1, but in sorted order.

We use loop invariants to help us understand why an algorithm is correct. We must show three things about a loop invariant:

Initialization: It is true prior to the first iteration of the loop.

Maintenance: If it is true before an iteration of the loop, it remains true before the next iteration.

Termination: When the loop terminates, the invariant gives us a useful property that helps show that the algorithm is correct.

When the first two properties hold, the loop invariant is true prior to every iteration of the loop. (Of course, we are free to use established facts other than the loop invariant itself to prove that the loop invariant remains true before each iteration.) Note the similarity to mathematical induction, where to prove that a property holds, you prove a base case and an inductive step. Here, showing that the invariant holds before the first iteration corresponds to the base case, and showing that the invariant holds from iteration to iteration corresponds to the inductive step.

The third property is perhaps the most important one, since we are using the loop invariant to show correctness. Typically, we use the loop invariant along with the condition that caused the loop to terminate. The termination property differs from how we usually use mathematical induction, in which we apply the inductive step infinitely; here, we stop the “induction” when the loop terminates.

Let us see how these properties hold for insertion sort.

Initialization: We start by showing that the loop invariant holds before the first loop iteration, when j D 2.¹ The subarray AŒ1 : : j  1, therefore, consists of just the single element AŒ1, which is in fact the original element in AŒ1.

Moreover, this subarray is sorted (trivially, of course), which shows that the loop invariant holds prior to the first iteration of the loop.

Maintenance: Next, we tackle the second property: showing that each iteration maintains the loop invariant. Informally, the body of the for loop works by moving AŒj  1, AŒj  2, AŒj  3, and so on by one position to the right until it finds the proper position for AŒj  (lines 4–7), at which point it inserts the value of AŒj  (line 8). The subarray AŒ1 : : j  then consists of the elements originally in AŒ1 : : j , but in sorted order. Incrementing j for the next iteration of the for loop then preserves the loop invariant.

A more formal treatment of the second property would require us to state and show a loop invariant for the while loop of lines 5–7. At this point, however,

1When the loop is a for loop, the moment at which we check the loop invariant just prior to the first iteration is immediately after the initial assignment to the loop-counter variable and just before the first test in the loop header. In the case of INSERTION-SORT, this time is after assigning 2 to the variable j but before the first test of whether j  A: length.

we prefer not to get bogged down in such formalism, and so we rely on our informal analysis to show that the second property holds for the outer loop.

Termination: Finally, we examine what happens when the loop terminates. The condition causing the for loop to terminate is that j > A: length D n. Because each loop iteration increases j by 1, we must have j D n C 1 at that time.

Substituting n C 1 for j in the wording of loop invariant, we have that the subarray AŒ1 : : n consists of the elements originally in AŒ1 : : n, but in sorted order. Observing that the subarray AŒ1 : : n is the entire array, we conclude that the entire array is sorted. Hence, the algorithm is correct.

We shall use this method of loop invariants to show correctness later in this chapter and in other chapters as well.

Pseudocode conventions

We use the following conventions in our pseudocode.

 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-else statements²as 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.³

 The looping constructs while, for, and repeat-until and the if-else conditional construct have interpretations similar to those in C, C++, Java, Python, and Pascal.⁴ In this book, the loop counter retains its value after exiting the loop, unlike some situations that arise in C++, Java, and Pascal. Thus, immediately after a for loop, the loop counter’s value is the value that first exceeded the for loop bound. We used this property in our correctness argument for insertion sort. The for loop header in line 1 is for j D 2 to A: length, and so when this loop terminates, j D A: length C 1 (or, equivalently, j D n C 1, since n D A:length). We use the keyword to when a for loop increments its loop

2In an if-else statement, we indent else at the same level as its matching if. Although we omit the keyword then, we occasionally refer to the portion executed when the test following if is true as a then clause. For multiway tests, we use elseif for tests after the first one.

3Each pseudocode procedure in this book appears on one page so that you will not have to discern levels of indentation in code that is split across pages.

4Most block-structured languages have equivalent constructs, though the exact syntax may differ.

Python lacks repeat-until loops, and its for loops operate a little differently from the for loops in this book.

counter in each iteration, and we use the keyword downto when a for loop decrements its loop counter. When the loop counter changes by an amount greater than 1, the amount of change follows the optional keyword by.

 The symbol “//” indicates that the remainder of the line is a comment.

 A multiple assignment of the form i D j D e assigns to both variables i and j the value of expression e; it should be treated as equivalent to the assignment j D e followed by the assignment i D j .

 Variables (such as i , j , and key) are local to the given procedure. We shall not use global variables without explicit indication.

 We access array elements by specifying the array name followed by the in-dex in square brackets. For example, AŒi  indicates the i th element of the array A. The notation “: :” is used to indicate a range of values within an ar-ray. Thus, AŒ1 : : j  indicates the subarray of A consisting of the j elements AŒ1; AŒ2; : : : ; AŒj .

 We typically organize compound data into objects, which are composed of attributes. We access a particular attribute using the syntax found in many object-oriented programming languages: the object name, followed by a dot, followed by the attribute name. 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 A: length.

We treat a variable representing an array or object as a pointer to the data rep-resenting the array or object. For all attributes f of an object x, setting y D x causes y: f to equal x: f . Moreover, if we now set x: f D 3, then afterward not only does x: f equal 3, but y: f equals 3 as well. In other words, x and y point to the same object after the assignment y D x.

Our attribute notation can “cascade.” For example, suppose that the attribute f is itself a pointer to some type of object that has an attribute g. Then the notation x:f :g is implicitly parenthesized as .x:f /:g. In other words, if we had assigned y D x:f , then x:f :g is the same as y:g.

Sometimes, a pointer will refer to no object at all. In this case, we give it the special valueNIL.

 We pass parameters 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 procedure. When objects are passed, the pointer to the data representing the object is copied, but the object’s attributes are not. For example, if x is a parameter of a called procedure, the assignment x D y within the called procedure is not visible to the calling procedure. The assignment x:f D 3, however, is visible. Similarly, arrays are passed by pointer, so that

a pointer to the array is passed, rather than the entire array, and changes to individual array elements are visible to the calling procedure.

 A return statement immediately transfers control back to the point of call in the calling procedure. Most return statements also take a value to pass back to the caller. Our pseudocode differs from many programming languages in that we allow multiple values to be returned in a single return statement.

 The boolean operators “and” and “or” are short circuiting. That is, when we evaluate the expression “x and y” we first evaluate x. If x evaluates toFALSE, then the entire expression cannot evaluate toTRUE, and so we do not evaluate y.

If, on the other hand, x evaluates toTRUE, we must evaluate y to determine the value of the entire expression. Similarly, in the expression “x or y” we eval-uate the expression y only if x evaleval-uates to FALSE. Short-circuiting operators allow us to write boolean expressions such as “x ¤ NILand x: f D y” without worrying about what happens when we try to evaluate x: f when x is NIL.

 The keyword error indicates that an error occurred because conditions were wrong for the procedure to have been called. The calling procedure is respon-sible for handling the error, and so we do not specify what action to take.

Exercises

2.1-1

Using Figure 2.2 as a model, illustrate the operation of INSERTION-SORT on the array A D h31; 41; 59; 26; 41; 58i.

2.1-2

Rewrite the INSERTION-SORTprocedure to sort into nonincreasing instead of non-decreasing order.

2.1-3

Consider the searching problem:

Input: A sequence of n numbers A D ha1; a2; : : : ; ani and a value .

Output: An index i such that  D AŒi  or the special value NIL if  does not appear in A.

Write pseudocode for linear search, which scans through the sequence, looking for . Using a loop invariant, prove that your algorithm is correct. Make sure that your loop invariant fulfills the three necessary properties.

2.1-4

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 C 1/-element array C . State the problem formally and write pseudocode for adding the two integers.