You are given an infinite array A[·] in which the first n...

You are given an infinite array $A[\cdot]$ in which the first $n$ cells contain integers in sorted order and the rest of the cells are filled with $\infty .$ You are not given the value of $n .$ Describe an algorithm that takes an integer $x$ as input and finds a position in the array containing $x,$ if such a position exists, in $O(\log n)$ time. (If you are disturbed by the fact that the array $A$ has infinite length, assume instead that it is of length $n,$ but that you don't know this length, and that the implementation of the array data type in your programming language returns the error message oo whenever elements $A[i]$ with $i>n$ are accessed.

Key Concepts

Handling Unknown Array Size

When the size of a data structure is unknown, standard searching techniques must be adapted. The strategy involves first finding an approximate boundary—typically using an exponential search—to determine a plausible interval where the target might reside, and then applying binary search within that interval to find the target value.

Time Complexity Analysis (O(log n))

Time complexity analysis is essential for evaluating the efficiency of an algorithm. In this context, both the exponential search to determine the search boundaries and the subsequent binary search operate in logarithmic time relative to the number of elements, ensuring that the overall algorithm remains efficient even when dealing with large or unknown data sizes.

Exponential Search

Exponential search is an algorithm used to find the range in which a target value might lie in a sorted sequence with unknown bounds. It works by checking elements at exponentially increasing indices, such as 1, 2, 4, 8, and so on, until a value greater than or equal to the target is found. This efficiently identifies a subarray within which a conventional search method can then be applied.

Binary Search

Binary search is a classic algorithm for locating an element in a sorted array by repeatedly dividing the search interval in half. Once a suitable search interval is determined, binary search compares the middle element of the interval with the target value and eliminates half of the interval each time, leading to an efficient O(log n) time complexity.

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