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# Algorithms | Searching | Question 3

Given a sorted array of integers, what can be the minimum worst-case time complexity to find ceiling of a number x in given array? The ceiling of an element x is the smallest element present in array which is greater than or equal to x. Ceiling is not present if x is greater than the maximum element present in array. For example, if the given array is {12, 67, 90, 100, 300, 399} and x = 95, then the output should be 100.

(A)

O(loglogn)

(B)

O(n)

(C)

O(log(n))

(D)

O(log(n) * log(n))

Explanation:

We modify the standard binary search to find the ceiling. The time complexity T(n) can be written as T(n) <= T(n/2) + O(1) Solution of above recurrence can be obtained by Master Method. It falls in case 2 of the Master Method. The solution is O(Logn).

## C

 `#include ` `/* Function to get index of ceiling of x in arr[low..high]*/``int` `ceilSearch(``int` `arr[], ``int` `low, ``int` `high, ``int` `x)``{``    ``int` `mid;` `    ``/* If x is smaller than or equal to the first element,``      ``then return the first element */``    ``if` `(x <= arr[low])``        ``return` `low;` `    ``/* If x is greater than the last element, then return -1``     ``*/``    ``if` `(x > arr[high])``        ``return` `-1;` `    ``/* get the index of middle element of arr[low..high]*/``    ``mid = (low + high) / 2; ``/* low + (high - low)/2 */` `    ``/* If x is same as middle element, then return mid */``    ``if` `(arr[mid] == x)``        ``return` `mid;` `    ``/* If x is greater than arr[mid], then either arr[mid +``      ``1] is ceiling of x or ceiling lies in``      ``arr[mid+1...high] */``    ``else` `if` `(arr[mid] < x) {``        ``if` `(mid + 1 <= high && x <= arr[mid + 1])``            ``return` `mid + 1;``        ``else``            ``return` `ceilSearch(arr, mid + 1, high, x);``    ``}` `    ``/* If x is smaller than arr[mid], then either arr[mid]``       ``is ceiling of x or ceiling lies in arr[mid-1...high]``     ``*/``    ``else` `{``        ``if` `(mid - 1 >= low && x > arr[mid - 1])``            ``return` `mid;``        ``else``            ``return` `ceilSearch(arr, low, mid - 1, x);``    ``}``}` `/* Driver program to check above functions */``int` `main()``{``    ``int` `arr[] = { 1, 2, 8, 10, 10, 12, 19 };``    ``int` `n = ``sizeof``(arr) / ``sizeof``(arr[0]);``    ``int` `x = 20;``    ``int` `index = ceilSearch(arr, 0, n - 1, x);``    ``if` `(index == -1)``     ``printf``(\"Ceiling of %d doesn\'t exist in array \", x);``   ``else``     ``printf``(\"ceiling of %d is %d\", x, arr[index]);``   ``getchar``();``   ``return` `0;``}`

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