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If the search item is smaller than the element in the middle of the list, the binary search reads the element at the middle of the first half of the list Then, if the search item is larger than that element, the binary search next reads the element at the middle of the second half of the front half of the list Eventually, the search finds the element sought, or concludes that the element is not present in the list Here is pseudocode for a binary search: BinarySearch(list, search_item) begin <-- 1 end <-- length of list match_found <-- false // Repeat search as long as no match has been found // and we have not searched the entire list while match_found = false AND begin <= end // Find the item at the midpoint of the list midpoint <-- (begin + end) / 2.

44 77 55 22 99 88 33 66 22 33 44 55 66 77 88 99 Arrays.binarySearch(a, 44): 2 a[2]: 44 Arrays.binarySearch(a, 45): -4 0 0 0 0 0 0 0 0 55 55 55 55 55 55 55 55 Arrays.equals(a,b): false The array a[] is created and printed at lines 5 6. At line 7, the call Arrays.sort(a) sorts the elements

Fig. 2-37

of the array, putting them in ascending order, as we can see from the output from line 8. At line 9, the Arrays.binarySearch() method is invoked. The second argument, 44, is the search target. The method returns the index 2, which is assigned to k at line 9. Line 11 verifies that 44 is indeed the value of a[2]. The method is invoked again at line 13, this time searching for the target 45. The value is not found in the array, so the method returns a negative number, k = 4. When this happens, the index i = k 1 will be the position in the array where the target element should be inserted to maintain the ascending order of the array. Note that, in this case, i = k 1 = 3, and 45 should be inserted at a[3] since there are three elements in the array that are less than 45. The output from line 17 shows how the Arrays.fill() method works: It filled the eight-element array b[] with the argument 55.

Thus, if V1 ! 15 V, D2 is on and V2 10 V. But, for 0 V1 < 15 V, D2 is o , I2 0, and V2 is given as a function of V1 by (1). Figure 2-38 shows the composite result.

CHAP. 2]

_ 20

Finally, line 18 shows how the Arrays.equals() method works. It will return true only if the two arrays have the same element type (as a[] and b[] do: int[]), the same length (as a[] and b[] do: 8), and the same values at each element (a[] and b[] do not).

_ 10

The java.util.Arrays class is outlined in more detail in page 95. THE SEQUENTIAL SEARCH ALGORITHM The sequential search (also called the linear search) is the simplest search algorithm. It is also the least efficient. It simply examines each element sequentially, starting with the first element, until it finds the key element or it reaches the end of the array. If you were looking for someone on a moving passenger train, you would use a sequential search. Here is the sequential search algorithm: (Postcondition: either the index i is returned where si = x, or 1 is returned.) 1. Repeat steps 2 3, for i = 0 to n 1. 2. (Invariant: none of the elements in the subsequence {s 0...s i 1} is equal to x.) 3. If s i = x, return i . 4. Return 1. It is implemented in Example 2.4. EXAMPLE 2.4 The Sequential Search

// If it s the one we re looking for, we re done if list[midpoint] = search_item match_found = true // If the search item is smaller, the next // list item to check is in the first half else if search_item < list[midpoint] end <-- midpoint - 1 // Otherwise, the next list item to check // is in the back half of the list else begin <-- midpoint + 1 // Return true or false, depending on whether we // found the search_item return match_found With each iteration, the binary search reduces the size of the list to be searched by a factor of 2 So, the binary search generally will find the search item, or conclude that the search item is not in the list, when the algorithm has executed (lg n) iterations or fewer.

Fig. 2-38