Suppose we have an array A with 33,554,431 elements. We want to apply binary search to look for some element k. A test of the form "is k = A[i]?" is a probe. How many probes will be performed in the worst case?
Question
Suppose we have an array A with 33,554,431 elements. We want to apply binary search to look for some element k. A test of the form "is k = A[i]?" is a probe. How many probes will be performed in the worst case?
Solution 1
Binary search is a divide-and-conquer algorithm that halves the search space at each step. It starts with the middle element and if the target value is equal to the middle element, it returns the index. If the target value is less than the middle element, it performs the operation on the left half of the array. If the target value is greater than the middle element, it performs the operation on the right half.
The worst-case scenario for binary search is when the target value is not in the array, which would require searching the entire array. The number of probes in the worst case is equal to the height of the binary search tree, which is log2(n) + 1, where n is the number of elements in the array.
So, for an array with 33,554,431 elements, the number of probes in the worst case would be log2(33,554,431) + 1.
Using a calculator, log2(33,554,431) is approximately 25. Therefore, the number of probes in the worst case would be 25 + 1 = 26.
Solution 2
Binary search is a divide-and-conquer algorithm that halves the search space at each step. It starts with the middle element and if the target value is equal to the middle element, it returns the index. If the target value is less than the middle element, it performs the operation on the left half of the array. If the target value is greater than the middle element, it performs the operation on the right half.
The worst-case scenario for binary search is when the target value is not in the array, which would require searching the entire array. The number of probes in the worst case is equal to the height of the binary search tree, which is log2(n) + 1, where n is the number of elements in the array.
So, for an array with 33,554,431 elements, the number of probes in the worst case would be log2(33,554,431) + 1.
Using a calculator, log2(33,554,431) is approximately 25. Therefore, the number of probes in the worst case would be 25 + 1 = 26.
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