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Arrays work well for unordered sequences, and even for ordered sequences if they don t change much. But if you want to maintain an ordered list that allows quick insertions and deletions, you should use a linked data structure. This chapter shows how to do that. MAINTAINING AN ORDERED ARRAY 2 outlines how the binary search can find elements very quickly in an array that is sorted. This suggests that we should keep our arrays in sorted order. But inserting new elements in an ordered array is difficult. The main problem is that we have to shift all the larger elements forward to make room for the new element to be placed in its correct ordered position. This can be done by the insert() method shown in Example 3.1. EXAMPLE 3.1 Inserting into an Ordered Array
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void insert(int[] a, int n, int x) { // preconditions: a[0] <= ... <= a[n-1], and n < a.length; 3 // postconditions: a[0] <= ... <= a[n], and x is among them; 4 int i = 0; 5 while (i < n && a[i] <= x) { 6 ++i; 7 } 8 System.arraycopy(a, i, a, i+1, n-i); // copies a[i..n) into a[i+1..n+1) 9 a[i] = x; 10 } The insert() method takes three arguments: the array a[], the number n of elements that are already sorted in the array, and the new element x to be inserted among them. The preconditions at line 2 specify that the first n elements of the array are in ascending order and that the array has room for at least one more element. The postconditions at line 3 specify that the array is still in ascending order and that x has
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been successfully inserted among them. The code at lines 4 7 searches the array for the correct position for x to be inserted. It should be the smallest index i for which a[i] > x. For example, if x = 50 for the array shown in Figure 3.1, then the correct position for x is at index i = 1, because a[0] <= x < a[1]. After the correct position i has been located for x, the insert() method shifts the elements that are greater than x one position to the right. This is accomplished by the call
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System.arraycopy(a, i, a, i+1, n-i);
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Figure 3.1 Making room for the new element
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at line 8. The arraycopy() method is a static method in the System class. It is usually the most efficient way to copy elements between arrays or within a single array. Its five arguments are: the source array, the index of the first element to be copied from the source array, the destination array, the index in the destination array where the first element is to be copied, and the number of elements to be copied. If n = 4 and i = 1, as shown in Figure 3.1, then the call is
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System.arraycopy(a, 1, a, 2, 3);
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This shifts elements {a[1], a[2], a[3]} = {66, 88, 99} into elements {a[2], a[3], a[4]}. Finally, x is inserted into a[i] at line 9, as shown in Figure 3.2.
Figure 3.2 Copying x into its correct position
The insert() method may have to move a lot of data. For example, if n = 1000 and x is less than all of those elements, then the method will move all 1000 elements. On average, inserting into a sorted array of n elements will move n/2 elements. So this is a (n) operation. Deleting an element is simply the reverse of the insertion process. It too will have to move n/2 elements, on average. So deletion is also a (n) operation. INDIRECT REFERENCE One solution to the data movement problem that is intrinsic to dynamic ordered arrays is to use an auxiliary index array to keep track of where the elements actually are. This solution requires more space (a second array) and makes the code a bit more complicated. But it eliminates the need to move the elements. It allows the elements to be stored at an arbitrary position in the array, using the auxiliary index array to locate them for ordered access. The main idea is shown in Figure 3.3. The elements {22, 33, 44, 55, 66} are kept in arbitrary positions in the array a[], and their order is determined by some auxiliary mechanism.
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