Batcher's Odd–Even Mergesort

To begin, we shall consider a sorting method that is based on just two abstract operations: the compare–exchange operation and the perfect shuffle operation (along with its inverse, the perfect unshuffle). The algorithm, developed by Batcher in 1968, is known as Batcher's odd–even mergesort. It is a simple task to implement the algorithm using shuffles, compare–exchanges, and double recursion, but it is more challenging to understand why the algorithm works and to untangle the shuffles and recursion to see how it operates at a low level.

We encountered the compare–exchange operation briefly in , where we noted that some of the elementary sort methods discussed there could be expressed more concisely in terms of this abstract operation. Now, we are interested in methods that examine the data exclusively with compare–exchange operations. Standard comparisons are ruled out: The compare–exchange operation does not return a result, so there is no way for a program to take action that depends on data values.

Definition 11.1 A nonadaptive sorting algorithmis one where the sequence of operations performed depends on only the number of the inputs, rather than on the values of the keys.

In this section, we do allow operations that unilaterally rearrange the data, such as exchanges and perfect shuffles, but they are not essential, as we shall see in . Nonadaptive methods are equivalent to straight-line programs for sorting: They can be expressed simply as a list of the compare–exchange operations to be performed. For example, the sequence

compexch(a[0], a[1]) compexch(a[1], a[2]) compexch(a[0], a[1])


is a straight-line program for sorting three elements. We use loops, shuffles, and other high-level operations for convenience and economy in expressing algorithms, but our goal in developing an algorithm is to define, for each N, a fixed sequence of compexch operations that can sort any set of N keys. We can assume without loss of generality that the key values are the integers 1 through N (see Exercise 11.4); to know that a straight-line program is correct, we have to prove that it sorts each possible permutation of these values (see, for example, Exercise 11.5).

Perfect shuffle and perfect unshuffle

The shuffle function rearranges a subarray a[l], ..., a[r] by splitting that subarray in half, then alternating elements from each half: Elements in the first half go in the even-numbered positions in the result, and elements in the second half go in the odd-numbered positions in the result. The unshuffle function does the opposite: Elements in the even-numbered positions go in the first half of the result, and elements in the odd-numbered positions go in the second half of the result.

We use these functions only for subarrays with an even number of elements. Also, while both use an auxiliary array, initialized as for mergesort (see Program 8.3), it is not difficult to do these permutations without using such an array (see Exercise 11.42).

static ITEM[] aux; static void shuffle(ITEM a[], int l, int r) { int i, j, m = (l+r)/2; for (i = l, j = 0; i <= r; i+=2, j++) { aux[i] = a[l+j]; aux[i+1] = a[m+1+j]; } for (i = l; i <= r; i++) a[i] = aux[i]; } static void unshuffle(ITEM a[], int l, int r) { int i, j, m = (l+r)/2; for (i = l, j = 0; i <= r; i+=2, j++) { aux[l+j] = a[i]; aux[m+1+j] = a[i+1]; } for (i = l; i <= r; i++) a[i] = aux[i]; }


Few of the sorting algorithms that we considered in Chapters 6 through 10 are nonadaptive—they all use less or examine the keys in other ways, then take differing actions depending on key values. One exception is bubble sort (see ), which uses only compare– exchanges. Pratt's version of shellsort (see ) is another nonadaptive method.

Program 11.1 gives an implementation of the other abstract operations that we shall be using—the perfect shuffle and the perfect unshuffle—and Screenshot gives an example of each. The perfect shuffle rearranges an array in a manner corresponding to the way that a deck of cards might be rearranged when shuffled by an expert: It is split precisely in half, then the cards are taken alternately from each half to make the shuffled deck. We always take the first card from the top half of the deck. If the number of cards is even, the two halves have the same number of cards; if the number of cards is odd, the extra card ends up in the top half. The perfect unshuffle does the opposite: We make the unshuffled deck by putting cards alternately in the top half and the bottom half.

Screenshot Perfect shuffle and perfect unshuffle

To perform a perfect shuffle (left), we take the first element in the file, then the first element in the second half, then the second element in the file, then the second element in the second half, and so forth. Consider the elements to be numbered starting at 0, top to bottom. Then, elements in the first half go to even-numbered positions, and elements in the second half go to odd-numbered positions. To perform a perfect unshuffle (right), we do the opposite: Elements in even-numbered positions go to the first half, and elements in oddnumbered positions go to the second half.

Java graphics 11fig01.gif


Batcher's sort is exactly the top-down mergesort of ; the difference is that instead of one of the adaptive merge implementations from , it uses Batcher's odd-even merge, a nonadaptive top-down recursive merge. Program 8.3 itself does not access the data at all, so our use of a nonadaptive merge implies that the whole sort is nonadaptive.

We shall implicitly assume in the text throughout this section and that the number of items to be sorted is a power of 2. Then, we can always refer to "N/2" without a caveat about N being odd, and so forth. This assumption is impractical, of course—our programs and examples involve other file sizes—but it simplifies the discussion considerably. We shall return to this issue at the end of .

Batcher's merge is itself a divide-and-conquer recursive method. To do a 1-by-1 merge, we use a single compare–exchange operation. Otherwise, to do an N-by-N merge, we unshuffle to get two N/2-by-N/2 merging problems, and then solve them recursively to get two sorted files. Shuffling these files, we get a file that is nearly sorted—all that is needed is a single pass of N/2 - 1 independent compare–exchange operations: between elements 2i and 2i +1for i from 1 to N/2 - 1. An example is depicted in Screenshot. From this description, the implementation in Program 11.2 is immediate.

Screenshot Top-down Batcher's odd–even merge example

To merge A G I N O R S T with A E E L M P X Y, we begin with an unshuffle operation, which creates two independent merging problems of about one-half the size (shown in the second line): we have to merge A I O S with A E M X (in the first half of the array) and G N R T with E L P Y (in the second half of the array). After solving these subproblems recursively, we shuffle the solutions to these problems (shown in the next-to-last line) and complete the sort by compare–exchanging E with A, G with E, L with I, N with M, P with O, R with S, and T with X.

Java graphics 11fig02.gif


Why does this method sort all possible input permutations? The answer to this question is not at all obvious—the classical proof is an indirect one that depends on a general characteristic of nonadaptive sorting programs.

Batcher's odd–even merge (recursive version)

This recursive program implements an abstract inplace merge, using the shuffle and unshuffle operations from Program 11.1, although they are not essential—Program 11.3 is a bottom-up nonrecursive version of this program with shuffling removed. Our primary interest here is that this implementation provides a compact description of Batcher's algorithm, when the file size is a power of 2.

static void merge(ITEM[] a, int l, int m, int r) { if (r == l+1) compExch(a, l, r); if (r < l+2) return; unshuffle(a, l, r); merge(a, l, (l+m)/2, m); merge(a, m+1, (m+1+r)/2, r); shuffle(a, l, r); for (int i = l+1; i < r; i+=2) compExch(a, i, i+1); }


Property 11.1

(0–1 principle) If a nonadaptive program produces sorted output when the inputs are all either 0 or 1, then it does so when the inputs are arbitrary keys.

See Exercise 11.7. Screenshot


Property 11.2

Batcher's odd–even merge (Program 11.2) is a valid merging method.

Using the 0–1 principle, we check only that the method properly merges when the inputs are all either 0 or 1. Suppose that there are i 0s in the first subfile and j 0s in the second subfile. The proof of this property involves checking four cases, depending on whether i and j are odd or even. If they are both even, then the two merging subproblems each involve one file with i/2 0s and one file with j/2 0s, so both results have (i + j)/2 0s. Shuffling, we get a sorted 0–1 file. The 0–1 file is also sorted after shuffling in the case that i is even and j is odd and the case that i is odd and j is even. But if both i and j are odd, then we end up shuffling a file with (i + j)/2 + 1 0s with a file with (i + j)/2 - 1 0s, so the 0–1 file after shuffling has i + j - 1 0s, a 1, a 0, then N - i - j - 1 1s (see Screenshot), and one of the comparators in the final stage completes the sort. Screenshot

Screenshot Four cases for 0-1 merging

These four examples consist of five lines each: a 0-1 merging problem; the result of an unshuffle operation, which gives two merging problems; the result of recursively completing the merges; the result of a shuffle; and the result of the final odd–even compares. The last stage performs an exchange only when the number of 0s in both input files is odd.

Java graphics 11fig03.gif


We do not need actually to shuffle the data. Indeed, we can use Programs 11.2 and 8.3 to output a straight-line sorting program for any N, by changing the implementations of compexch and shuffle to maintain indices and to refer to the data indirectly (see Exercise 11.12). Or, we can have the program output the compare–exchange instructions to use on the original input (see Exercise 11.13). We could apply these techniques to any nonadaptive sorting method that rearranges the data with exchanges, shuffles, or similar operations. For Batcher's merge, the structure of the algorithm is so simple that we can develop a bottom-up implementation directly, as we shall see in .

Exercises

Java graphics icon01.gif 11.1 Give the result of shuffling and unshuffling the keys E A S Y Q U E S T I O N.

Generalize Program 11.1 to implement h-way shuffle and unshuffle. Defend your strategy for the case that the file size is not a multiple of h.

Java graphics roundbullet.gif 11.3 Implement the shuffle and unshuffle operations without using an auxiliary array.

Java graphics roundbullet.gif 11.4 Show that a straight-line program that sorts N distinct keys will sort N keys that are not necessarily distinct.

Java graphics icon01.gif 11.5 Show how the straight-line program given in the text sorts each of the six permutations of the integers 1, 2, and 3.

ScreenshotGive a straight-line program that sorts four elements.

Java graphics roundbullet.gif 11.7 Prove Property 11.1. Hint: Show that if the program does not sort some input array with arbitrary keys, then there is some 0–1 sequence that it does not sort.

Java graphics icon01.gif 11.8 Show how the keys A E Q S U Y E I N O S T are merged using Program 11.2, in the style of the example diagrammed in Screenshot.

Java graphics icon01.gif 11.9 Answer Exercise 11.8 for the keys A E S Y E I N O Q S T U.

ScreenshotAnswer Exercise 11.8 for the keys 1 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0.

Empirically compare the running time of Batcher's mergesort with that of standard top-down mergesort (Programs 8.3 and 8.2) for N = 103, 104, 105, and 106.

Give implementations of compexch, shuffle, and unshuffle that cause Programs 11.2 and 8.3 to compute, given N, an array p such that p[i] is the index of the ith smallest element, for 0 Screenshot i < N.

ScreenshotGive implementations of compexch, shuffle, andunshuffle that cause Programs 11.2 and 8.3 to print, given N, a straight-line program for sorting N elements.

If we put the second file for the merge in reverse order, we have a bitonic sequence, as defined in . Changing the final loop in Program 11.2 to start at l instead of l+1 turns the program into one that sorts bitonic sequences. Show how the keys A E S Q U Y T S O N I E are merged using this method, in the style of the example diagrammed in Screenshot.

Java graphics roundbullet.gif 11.15 Prove that the modified Program 11.2 described in Exercise 11.14 sorts any bitonic sequence.


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