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because if the elements are visualized in a vertical column, then each pass appears to bubble up the next largest element by bouncing it off smaller elements, much like the rising bubbles in a carbonated beverage. Algorithm 14.1 The Bubble Sort (Precondition: s = {s0 . . . sn 1} is a sequence of n ordinal values.) (Postcondition: The entire sequence s is sorted.) 1. Do steps 2 4 for i = n 1 down to 1. 2. Do step 3 for j = 0 up to i 1 . 3. If the two consecutive elements sj and sj+1, are out of order, swap them. 4. (Invariants: The subsequence { si . . . sn 1} is sorted, and si = max{ s0 . . . si}.) EXAMPLE 14.4 The Bubble Sort
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public static void sort(int[] a) { // POSTCONDITION: a <= a <= ... <= a[a.length-1]; for (int i = a.length-1; i > 0; i--) { // step 1 for (int j = 0; j < i; j++) { // step 2 if (a[j] > a[j+1]) { swap(a, j, j+1); // step 3 } } // INVARIANTS: a[i] <= a[i+1] <= ... <= a[a.length-1]; // a[j] <= a[i] for all j < i; }
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Theorem 14.1 The Bubble Sort is correct. See the solution to Problem 14.14 on page 276 for a proof of this theorem. Theorem 14.2 The Bubble Sort runs in O(n2) time. See the solution to Problem 14.15 on page 276 for a proof of this theorem. THE SELECTION SORT The selection sort is similar to the bubble sort. It makes the n 1 passes through a sequence of n elements, each time moving the largest of the remaining unsorted elements into its correct position. But it is more efficient than the bubble sort because it doesn t move any elements in the process of finding the largest. It makes only one swap on each pass after it has found the largest. It is called the selection sort because on each pass it selects the largest of the remaining unsorted elements and puts it in its correct position. Algorithm 14.2 The Selection Sort (Precondition: s = {s0 . . . sn 1} is a sequence of n ordinal values.) (Postcondition: The entire sequence s is sorted.) 1. Do steps 2 4 for i = n 1 down to 1. 2. Locate the index m of the largest element among {s0 . . si} . 3. Swap si and sm. 4. (Invariants: the subsequence { si . . . sn 1} is sorted, and si = max{ s0 . . . si}.)
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EXAMPLE 14.5 The Selection Sort
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public static void sort(int[] a) { // POSTCONDITION: a <= a <= ... <= a[a.length-1]; for (int i = a.length-1; i > 0; i--) { // step 1 int m = 0; // step 2 for (int j = 1; j <= i; j++) { if (a[j] > a[m]) { m = j; } } // INVARIANT: a[m] >= a[j] for all j <= i; swap(a, i, m); // step 3 // INVARIANTS: a[j] <= a[i] for all j <= i; // a[i] <= a[i+1] <= ... <= a[a.length-1]; } }
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