Summary: We consider a variety of techniques used to put a list or vector in order, using a binary comparison operation to determine the ordering of pairs of elements.

Sorting a collection of values -- arranging them in a fixed order, usually alphabetical or numerical -- is one of the most common computing applications. When the number of values is even moderately large, sorting is such a tiresome, error-prone, and time-consuming process for human beings that the programmer should automate it whenever possible. For this reason, computer scientists have studied this application with extreme care and thoroughness. One of the clear results of their investigations is that no one algorithm for sorting is best in all cases. Which approach is best depends on whether one is sorting a small collection or a large one, on whether the individual elements occupy a lot of storage (so that moving them around in memory is time-consuming), on how easy or hard it is to compare two elements to figure out which one should precede the other, and so on. In this course we'll be looking at two of the most generally useful algorithms for sorting: insertion sort, which is the subject of this reading, and merge sort, which we will consider in another reading. In our general exploration of sorting, we may also discuss other sorting algorithms. Imagine first that we're given a collection of values and a rule for arranging them. The values might actually be stored in a list, vector, or file. Let's assume first that they are in a list. The rule for arranging them typically takes the form of a predicate with two parameters that can be applied to any two values in the collection to determine whether the first of them could precede the second when the values have been sorted. (For example, if one wants to sort a collection of real numbers into ascending numerical order, the rule should be the predicate <=; if one wants to sort a collection of strings into alphabetical order, ignoring case, the rule should be string-ci<=?; if one wants to sort a collection of real numbers into descending numerical order, the rule should be >=; and so on.)

As one might guess, the key operation in insertion sort is that of insertion. We envision separating the elements to be sorted into two collections: those that are not yet sorted, and those that are already sorted. We repeatedly insert one value from the unsorted collection into the proper place in the sorted collection. The way in which we represent the unsorted and sorted collections has an effect on the way in which we implement the insertion operation. Let's start by representing collections as lists. Midway through the sorting process, we have one list of unsorted values (the values we have not yet processed) and a second, sorted, list. To insert a value into the sorted list, we step over the elements that should precede the value, stopping when we find an element that should follow the value (or when we run out of elements). We then cons the new value onto that section of the list and rebuild the prior elements.

We'll start by considering the specific case of sorting a list of real numbers in ascending order. We begin with the procedure used to insert a new value into a list that is already in order. Because we're inserting the numbers in ascending order, we can use <= as the ordering predicate. In English: If the list into which the new element is to be inserted is empty, return a list containing only the new element. If the new element can precede the first element of the existing list, then, since the existing list is assumed to be sorted already, it must also be able to precede every element of the existing list, so attach the new element onto the front of the existing list and return the result. Otherwise, we haven't yet found the place, so issue a recursive call to insert the new element into the cdr of the current list, then re-attach its car at the beginning of the result.

Now let's return to the overall process of sorting an entire list. The insertion sort algorithm simply takes up the elements of the list to be sorted one by one and inserts each one into a new list, which is initially empty:

Of course, the insertion sort procedure we just wrote works only for lists of real numbers, and only sorts such lists into increasing order. What if we want a more general version? As you may recall, we were able to generalize the process of searching by adding a may-precede? parameter. We can add a similar parameter to our sorting routine. Rather than moving directly to a generalized implementation of insertion-sort, let's first develop the documentation for that procedure. The name, parameters, purpose, and produced value are fairly straightforward. ;;; Procedure: ;;; list-______-sort ;;; Parameters: ;;; lst, a list to be sorted ;;; may-precede?, a binary predicate ;;; Purpose: ;;; Sort lst. ;;; Produces: ;;; sorted, a list. Now, on to the harder part. What conditions must hold in order for us to be able to sort? Well, we need to be able to use may-precede? to compare values. That means that we must be able to apply it to any two elements of the list. We also want to make sure that the may-precede? predicate is sensible. That is, it should be transitive (if a may precede b and b may precede c, then a may precede c) and must order any two elements (for any two elements, a may precede b or b may precede a). What should we do about equal elements? We'll say that each may precede the other. ;;; Preconditions: ;;; may-precede? can be used with the elements of lst. That is for ;;; all values a and b in lst, (may-precede? a b) successfully ;;; returns a truth value. ;;; may-precede? is transitive. That is, for all values a, b, and ;;; c in lst, if (may-precede? a b) and (may-precede? b c), then ;;; (may-precede? a c). ;;; may-precede? is sensible. That is, for all values a and b, ;;; either (may-precede? a b), (may-precede? b a), or both. All that is left is for us to formalize the postconditions. What does it mean to sort a list? The result must be in order (which we specify using may-precede?). In addition, sorting should neither add nor remove values form the list. The easiest way to formalize that idea is to say that the result must be a permutation of the original list. We might also indicate that we do not modify the original list, but that's implicit. (That is, we traditionally only indicate when we modify parameters, not when we fail to modify them.) ;;; Postconditions: ;;; sorted is sorted by may-precede?. That is, for all i such that ;;; 0 <= i < (- (length lst) 1), ;;; (may-precede? (list-ref sorted i) (list-ref sorted (+ i 1))) ;;; sorted is a permutation of lst. As you may recall, we also looked at searching through collections of keyed values for a value with a particular key. Since we search for keyed values, we might also want to sort by key, and use the key as an explicit parameter to the procedure. ;;; Procedure: ;;; list-keyed-_____-sort ;;; Parameters: ;;; lst, a list ;;; get-key, a procedure ;;; may-precede?, a binary predicate ;;; Purpose: ;;; Sort lst. ;;; Produces: ;;; sorted, a list ;;; Preconditions: ;;; get-key? can be applied to each element of lst. ;;; may-precede? can be used with the values returned by get-key. That is ;;; for all values a and b in lst, (may-precede? (get-key a) (get-key b)) ;;; successfully returns a truth value. ;;; may-precede? is transitive. That is, for all keys a, b, and ;;; c, if (may-precede? a b) and (may-precede? b c), then ;;; (may-precede? a c). ;;; may-precede? is sensible. That is, for all keys a and b, ;;; (may-precede? a b) holds, (may-precede? b a) holds, or both ;;; hold. ;;; Postconditions: ;;; sorted is sorted by key using may-precede?. That is, for all i ;;; such that 0 <= i < (- (length lst) 1), ;;; (may-precede? (get-key (list-ref sorted i)) ;;; (get-key (list-ref sorted (+ i 1)))) ;;; sorted is a permutation of lst.

Okay, let's now implement the two forms of generalized insertion sort. This time, we'll make the insert procedure local, since it need not be used outside of insertion sort. What changes do we need to make for keyed insertion sort? Not many. The only differences are (a) we need to add the get-key parameter and (b) every time we used may-precede?, we must now add calls to get-key. Fortunately, there's only one call to may-precede?, so the update is small. Of course, this code is quite close to that of list-insertion-sort. Hence, rather than doing a cut-paste-edit on that code, we might instead call list-insertion-sort, providing it with a predicate that extracts keys and then compares. Why use the second strategy? One obvious reason is that it's smaller. A second is that it may actually be less work to write this version than to figure out what to change in list-insertion-sort. But the best reason to use this strategy (that is, calling another function with a well-designed parameter, rather than cutting, pasting, and changing) is that it makes updates much easier. If we realize that we made a mistake in list-insertion-sort or simply find a better way to do some part of it, we only have one procedure to update, rather than two.

Of course, as we saw in our exploration of binary search, in order to do binary search, we need a sorted vector, rather than a sorted list. To sort a vector, we could turn that vector into a list, sort the list, and then build a new vector. However, in sorting a vector, the goal of sorting is often to overwrite the old arrangement of those values with a new, sorted arrangement of the same values. This type of sorting is often called in-place sorting. Let's consider how we might use the ideas of insertion sort to sort a vector in place. As you may recall, insertion sort requires us to have two collections: one of which is sorted and the other of which is unsorted. We will partition the original vector into two sub-vectors: a sorted sub-vector, in which all of the elements are in the correct order relative to one another, and an unsorted sub-vector in which the elements are still in their original positions. The two sub-vectors are not actually separated; instead, we just keep track of a boundary between them inside the original vector. Items to the left of the boundary are in the sorted sub-vector; items to its right, in the unsorted one. Initially the boundary is at the left end of the vector. The plan is to shift the boundary, one position at a time, to the right end. When it arrives, the entire vector has been sorted. Here's the plan for the main algorithm. The insert! procedure takes four parameters: an element to be inserted into the sorted part of the vector, the vector itself, the current boundary position, and the comparison procedure. The new element can be inserted at any position up to and including the current boundary position, but it must be placed in the correct order relative to elements to the left of that boundary. This means that any elements that should follow the new one should be shifted one position to the right in order to make room for the new one. (Elements that precede the new one can keep their current positions.) How does this work? We assume that there's a space at position pos of the vector. (That is, that we can safely insert something there without removing anything from the vector.) We know that the condition holds at the beginning from the description. That is, the postcondition specifically ignores the value that was in the boundary position. We also know that the conditional holds from the way insert! was called from insertion-sort!, since the boundary is initially the position of the value we insert. Now, what do we do? If the position is at the left end of the vector, there's nothing smaller in the vector, so we just put the new value there. If the thing to the left of the current position is smaller, we know we've reached the right place, so we put the value there. In every other case, the value to the left is larger than the value we want to insert, so we shift that value right (into the pos position) and continue working one position to the left. Since we've copied the value to the right, it is remains safe to insert something in the position just vacated (that is, pos-1). You will have a chance to explore this procedure further in the laboratory.