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Example 1.1 We shall assume that the amount a consumer spends (C) is positively related to her disposable income (Yd), i.e., C = f (Yd). Table 1.1 presents data on consumer spending for ve individuals with different levels of income. As seen in the table, consumption and disposable income display a positive relationship. The data from Table 1.1 are plotted in Figure 1-1 and labeled C1. The dependent variable, consumer spending, is plotted on the vertical axis and the independent variable, disposable income, is plotted on the horizontal axis. Graphs are used to present data and the positive or negative relationship of the dependent and independent variables visually.

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}.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

public static void sort(int[] a) { // POSTCONDITION: a[0] <= a[1] <= ... <= 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]; } }

Theorem 14.3 The selection sort is correct. See the solution to Problem 14.19 on page 276 for a proof of this theorem. Theorem 14.4 The selection sort runs in O(n2) time. See the solution to Problem 14.20 on page 276 for a proof of this theorem. Note that even though the bubble sort and the selection sort have the same complexity function, the latter runs quite a bit faster. That fact is suggested by the two traces: The bubble sort made 18 swaps while the selection sort made only 7. The selection sort has the advantage of swapping elements that are far apart, so it makes one swap where the bubble sort could require several. (See Exercise 11.8.) THE INSERTION SORT Like the two previous algorithms, the insertion sort makes n 1 passes through a sequence of n elements. On each pass it inserts the next element into the subarray on its left, thereby leaving that subarray sorted. When the last element is inserted this way, the entire array is sorted. Algorithm 14.3 The Insertion 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 = 1 up to n 1. 2. Hold the element si in a temporary space. 3. Locate the least index j for which sj >= si . 4. Shift the subsequence {sj . . . si 1} up one position to {sj+1 . . . si}. 5. Copy the held value of si into sj. 6. (Invariant: the subsequence { s0..si} is sorted.) EXAMPLE 14.6 The Insertion Sort

public static void sort(int[] a) { // POSTCONDITION: a[0] <= a[1] <= ... <= a[a.length-1];

CHAPTER 1: Introduction to Economics Table 1.1 (in $)

CHAP. 14]