Problem Description

In this interactive problem, you are given access to a sorted array of unique elements, but you do not know its size. To interact with this array, you can use the ArrayReader interface, which has a function get(i) that will return the value at the i^th index (0-indexed) if it's within the bounds of the array. If i is outside the bounds of the array, get(i) will return 2^31 - 1, which serves as an indication of being outside the array’s boundaries.

You are also provided an integer target, and your task is to find the index k in the hidden array where secret[k] == target. If the target is not present in the array, you should return -1. The requirement is to accomplish this with a solution that has a runtime complexity of O(log n), which implies that a binary search algorithm should be used.

Intuition

Since we don't know the array's size, we can't directly apply binary search. However, the problem statement helps by stating that the ArrayReader.get(i) function will return a very large number (specifically, 2^31 - 1) if we access an index beyond the array's bounds. This allows us to first locate the possible range where the target could reside.

To find this range, we could intuitively start with an initial guess and then expand our search range exponentially until we find an out-of-bound value. Once we identify an index that returns an out-of-bound value, we know that the array's size is less than this index. We could then perform a binary search between the start of the array and the identified out-of-bound index.

The solution code, however, does not initially find the bounds of the array. Instead, it assumes a range of indexes between 0 and 20,000, which should be more than enough to cover typical constraints of the problem since we are told that get(i) will return 2^31 - 1 for out-of-bound accesses. With this assumption, the solution applies a binary search algorithm right away.

During the binary search, if reader.get(mid) is greater than or equal to target, it means the target is located at mid or to the left of mid (since the array is sorted), so it narrows the search range's upper bound to mid. If reader.get(mid) is less than target, the target can only be to the right of mid, and it narrows the search range's lower bound to mid + 1. This process is repeated until the lower and upper bounds converge to the smallest range possible that might contain the target.

Finally, having honed in on a potential index where target could exist, the solution checks if reader.get(left) is indeed equal to target. If it is, left is returned as the index where target was found. If reader.get(left) does not match target, the function returns -1, indicating that the target is not in the array.

Learn more about Binary Search patterns.

Solution Approach

The solution uses a variation of binary search to efficiently find the target value within a sorted array of unknown size. Binary search is a divide-and-conquer algorithm that operates by repeatedly dividing in half the portion of the list that could contain the item, until there are no more segments left or the item is found.

In the standard binary search, the middle element is typically compared to the target value, and based on that comparison, you can decide if the target would be in the left half or the right half of the current segment.

Here's how the implemented solution strategically applies binary search to the given problem:

1. Initial Range Setting: Given that the exact size of the array is unknown, the solution begins by setting a broad search range for the binary search, with left set to 0 and right set to 20,000. Since we know ArrayReader.get(i) will return a very large number (specifically 2^31 - 1) if i is out of the array bounds, this large upper bound is safe to assume for most inputs as per problem constraints.

2. Binary Search Application: While left is less than right, the solution continuously narrows the search range based on the value returned by the ArrayReader.get(mid):

   • If ArrayReader.get(mid) is greater than or equal to the target, it means the target cannot be to the right of mid (since the array is sorted), so the upper bound of the range (right) is set to mid.
   • If ArrayReader.get(mid) is less than the target, then the target must be to the right of mid, and thus the lower bound of the range (left) is set to mid + 1.

   To find the mid index, the average of left and right is calculated using (left + right) >> 1, which is equivalent to (left + right) / 2 but is often faster in many programming languages due to bit shifting.

3. Target Verification: After exiting the while loop, the left variable should either be at the location of the target value or at the smallest index greater than the target. To verify whether the target has been found, ArrayReader.get(left) is compared to the target:

   • If they match, the index left is returned, indicating the location of the target.
   • If they do not match, it means the target is not present in the array, so -1 is returned.

It's important to understand that binary search reduces the search space by half with each step, leading to the O(log n) runtime complexity, which is a requirement as per the problem.

Moreover, in practice, an initial range could be determined dynamically by starting with a small range and exponentially expanding it (doubling it each time) until an out-of-bound access occurs. This step can optimize the solution further if the potential maximum size of the array is significant compared to the actual size of the array.

Example Walkthrough

Let's say we have an ArrayReader that provides access to a sorted array: [3, 5, 7, 10, 15, 20]. We don't know the size of this array, but we need to find the index of the target value 10 using the solution approach described.

Here's how the solution would work with this example:

1. Initial Range Setting: We start by establishing an initial large search range for the binary search. We set left to 0 and right to 20,000.

2. Binary Search Application: Now we begin the binary search process.

   • Start Loop:

     • Calculate mid as (left + right) >> 1, which is initially (0 + 20,000) >> 1 = 10,000.
     • Check ArrayReader.get(10,000), it returns 2^31 - 1, which means 10,000 is out of the array bounds. Since the value is very high, we set the value of right to mid, which is now 10,000.
     • Recalculate mid as (0 + 10,000) >> 1 = 5,000, and so on, until we get to a mid value inside the bound of the array.

     Continue the loop with left set to 0 and right set now to an index within the array bounds. For this example, let's assume that after several steps (not shown for brevity), we have narrowed down the bounds to left = 2 and right = 5.

     • mid is (2 + 5) >> 1 = 3.
     • ArrayReader.get(3) returns 10, which is the target value.
     • Since ArrayReader.get(mid) == target, the search is over, and we skip to the Target Verification step.

   • End Loop.

3. Target Verification: We verify the value found at the index 3 to ensure it is indeed the target.

   • ArrayReader.get(3) is equal to 10, which matches the target.
   • We return the index 3 because this is where the target was found.

In this example, after several iterations of narrowing the search bounds and recomputing mid, we have successfully found the target value of 10 at index 3 using binary search. This demonstrates the efficiency of binary search in reducing the search range quickly compared to a linear search, meeting the complexity requirement O(log n).

Solution Implementation

Time and Space Complexity

The time complexity of the provided code is O(log n) where n represents the position of the target value or the position where the reader's get method returns INT_MAX signaling the end of the array. This is due to the binary search approach where the space to be searched is halved at each iteration.

The space complexity is O(1) since the algorithm uses a constant amount of extra space, with variables like left, right, and mid that do not depend on the size of the input.

Learn more about how to find time and space complexity quickly using problem constraints.