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log n n = n log n
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(Here, log means the binary logarithm log2.) Therefore, the algorithm s worst-case complexity function must be O(n log n). Theorem 14.17 applies only to comparison sorts. A comparison sort is an algorithm that sorts elements by comparing their values and then changes their relative positions according to the outcomes of those comparisons. All the sorting algorithms outlined previously are comparison sorts. In contrast, the following sorting algorithms are not comparisons sorts.
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THE RADIX SORT
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SORTING
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The radix sort assumes that the sequence s element type is a lexicographic array of constant size, that is, either a character string type or an integer type. Let r be the array element s radix (e.g., r = 26 for ASCII character strings, r = 10 for decimal integers, r = 2 for bit strings), and let w be the constant width of the lexicographic array. For example, for U.S. Social Security numbers, d = 10 and w = 9. EXAMPLE 14.11 Sorting Books by Their ISBNs
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Every book published since the 1970s has been assigned a unique international standard book number (ISBN). These are usually printed at the bottom of the back cover of the book. For example, the ISBN for this book is 0071476989. (ISBNs are usually hyphenated, like this: 0-07-147698-9, to distinguish the four separate fields that make up the code.) The last digit is a check digit, computed from the other nine digits. Since it can be any of the 10 numeric digits or the letter X, we have that the radix r = 11, while the number of digits d = 10.
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Algorithm 14.10 The Radix Sort (Precondition: s = {s0 . . . sn 1} is a sequence of n integers or character strings with radix r and width w.) (Postcondition: The sequence s is sorted numerically or lexicographically.) 1. Repeat step 2 for d = 0 up to w 1. 2. Apply a stable sorting algorithm to the sequence s, sorting only on digit number d. A sorting algorithm is said to be stable if it preserves the relative order of elements with equal keys. For example, the insertion sort is stable, but the heap sort is not. EXAMPLE 14.12 Sorting ISBNs with the radix sort
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Figure 14.5 shows a sequence of 12 ISBNs and the first four iterations of the radix sort applied to it.
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Figure 14.5 Tracing the radix sort
Note how the stability is needed to conserve the work done by previous iterations. For example, after the first iteration, 8838650527 precedes 0830636528 because 7 < 8. Both of these keys have the same value 2 in their second least significant digit (digit number d = 1). So on the second iteration, which sorts only on digit number 1, these two keys evaluate as being equal. But they should retain their previous relative order because 27 < 28. Stability guarantees that they do. The columns that have been processed are shaded. So after the third iteration, the right-most 3-digit subsequences are sorted: 109 < 13X < 373 < 453. (Note that X stands for the value 10. So 13X numerically means 130 + 10 = 140.)
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EXAMPLE 14.13 The Radix Sort
SORTING
This method assumes that the constants RADIX has WIDTH have been defined. For example, for arrays of ints:
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public static void sort(int[] a) { // POSTCONDITION: a[0] <= a[1] <= ... <= a[a.length-1]; for (int d = 0; d < WIDTH; d++) { // step 1 sort(a, d); // step 2 } } private static void sort(int[] a, int d) { // POSTCONDITION: a[] is sorted stably on digit d; int n = a.length; int[] c = new int[RADIX]; for (int ai : a) { ++c[digit(d,ai)]; // tally the values in a[] } for (int j = 1; j < RADIX; j++) { c[j] += c[j-1]; // c[j] == num elts in a[] that are <= j } int[] tmp = new int[n]; for (int i = n - 1; i >= 0; i--) { tmp[--c[digit(d, a[i])]] = a[i]; } for (int i = 0; i < n; i++) a[i] = tmp[i]; } private static int digit(int d, int x) { // returns digit number d of integer x // e.g., digit(2, 1234567890) returns 8; return x/(int)Math.pow(10,d)%RADIX; }
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