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Figure 3.3 Referring to the order of the array elements
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Each element is kept in a numbered component: 22 is in component 3, 33 is in component 5, 44 is in component 1, and so on. So if we save the order of the index numbers (3, 5, 1, 4, 6), then we can access the elements in order: a[3] followed by a[5] followed by a[1], and so forth. An index array is an array whose elements are index values for another array. By storing the index numbers 3, 5, 1, 4, 6 in an index array k[] (shown in Figure 3.4), we can use them to access the data elements 22, 33, 44, 55, 66 in order.
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Figure 3.4 Using an index array
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Now this may be an improvement, but it is not optimal. The reason we wanted to allow the element to be stored in arbitrary positions in the first place was to simplify the insertion and deletion operations. We wanted to avoid having to shift segments of a[] back and forth. But the solution shown in Figure 3.4 merely transfers that obligation from a[] to k[]. If we had to insert the element 50, we could put it at position a[0] or a[2] or any place after a[6], but we would then have to insert its index into the index array k[] between k[2] and k[3] to keep track of the order of the elements. A better solution is to use the same array positions in the index array k[] as we are using in the data array a[]. Since the index array is keeping track of the correct order of the index numbers of the data elements, it can be used to do the same for the index numbers themselves.
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Figure 3.5 Using an index array
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In Figure 3.5, the index array k[] keeps track of the order of the elements in a[]. The starting index 3 is stored in k[0]. That begins the chain of indexes: k[0] = 3, k[3] = 5, k[5] = 1, k[1] = 4, k[4] = 6, k[6] = 0. The index 0 signals the end of the ordered sequence. The index sequence 0, 3, 5, 1, 4, 6 gives us the data elements in order: a[3] = 22, a[5] = 33, a[1] = 44, a[4] = 55, a[6] = 66. The extra variable free, shown Figure 3.5, saves the index of a free location in both the index array k[] and the data array a[]. The value 7 means that k[7] and a[7] should be used next. The implementation of an index array solves the problem of having to shift segments of array elements back and forth during deletions and insertions. For example, to insert x = 50 in Figure 3.5, we first traverse the sequence to find the index i of the largest element that is less than x: i = 1. Then just follow these three steps:
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a[free] = x; k[free] = k[i]; k[i] = free++; // put x into the next free position // store the next index in that position in k[] // store the index of x in k[] and increment free
The results are shown in Figure 3.6.
Figure 3.6 Inserting an element
The Java code for this algorithm is shown in Example 3.2. This improves the insert() method shown in Example 3.2, because its only data movement is the actual insertion of x into the array a[] at line 5. EXAMPLE 3.2 Inserting into an Array that Is Indirectly Ordered
void insert(int x) { int i=0; 3 while (k[i] != 0 && a[k[i]] < x) { 4 i = k[i]; 5 } 6 a[free] = x; 7 k[free] = k[i]; 8 k[i] = free++; 9 } The while loop at lines 3 5 is similar to the while loop at lines 5 7 in Example 3.1 on page 46: it finds the first index i for which a[k[i]] > x. At line 6, x is inserted in the next free location in the array a[]. At line 7, the index of the next location after x is stored in k[free]. At line 8, the index of x is copied into k[i], and then free is incremented to the index of the next free location.
Note that this code assumes that the array is large enough to accommodate all elements that might be inserted. In practice, we would probably include a resize() method.
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