Problem Description

In this problem, we are given an array nums that contains a sequence of unique integers. The sequence may be sorted or not sorted. We are tasked with implementing an algorithm that simulates a binary search-like process, but instead of choosing the middle element as a pivot, it chooses a random pivot during each iteration. If the pivot matches the target, the function returns true. If the pivot is less than the target, all elements to the left of the pivot (including the pivot itself) are removed from the sequence, and the search continues with the remaining elements. Similarly, if the pivot is greater than the target, all elements to the right of the pivot are removed from the sequence. This continues until either the target is found (and true is returned) or the sequence is empty (and false is returned).

The crux of the problem is to find out how many values in the nums array are guaranteed to be found using this function, regardless of which pivots are randomly selected in the process. A "guaranteed" value here means that no matter how the elements are chosen as pivots in each iteration, it will always end up being found if it is indeed in the sequence.

Intuition

To solve this problem, we need to identify which elements in the sequence would always be found, regardless of the random selection of the pivot. To do this, we need to find out whether choosing any pivot would lead to accidentally discarding the target element when it shouldn't have been discarded. To be a 'binary search-able' number, a target value in the sequence must never be in the same subsequence as any incorrect pivot (an incorrect pivot being any element that would not be the target and would mistakenly remove the target from consideration).

For example, let's say our target is the number 3 in an unsorted sequence. If at any point during the function execution, the number 3 is in a subsequence with a pivot greater than 3, it might get discarded even though it hasn't been evaluated yet.

Hence, the process involves two checks:

1. From left to right, we check if any element is smaller than any element to its left. If it is, then in some random pivot selection, it could be incorrectly disregarded as a pivot.
2. From right to left, we check if any element is greater than any element to its right. If it is, then it could also lead to the same problem but in the opposite direction.

We maintain an array, ok, to keep track of whether each element in nums is binary search-able. We initially set all elements in this array to 1 (representing 'true'), indicating that we assume all elements are binary search-able.

During the execution, we utilize two variables mx and mi to keep track of the maximum value encountered so far from the left and the minimum value from the right, respectively. Starting from the left of the array, for each element x, if it is smaller than mx, then it cannot be binary search-able as there exists a larger value to the left. Thus, we set ok[i] to 0 for such an element. We then update mx with the maximum value seen so far.

We perform a similar procedure starting from the right of the array, using mi to track the minimum value seen so far. If any element nums[i] is greater than mi, we set ok[i] to 0, indicating that the element can't be guaranteed to be searchable, and then we update mi.

At the end of the process, sum(ok) will yield the total number of elements in the nums array that are guaranteed to be found, which is the solution to the problem.

Learn more about Binary Search patterns.

Solution Approach

The solution provided leverages the concept of maintaining two boundary conditions as we iterate over the input array, which is a common technique in problems involving subarrays or sequence behavior modifications based on certain conditions.

Here is a closer look at the step-by-step approach:

1. Initialize an array ok of the same length as nums, with all elements set to 1. This array will track whether each element in nums could be binary search-able (1 for yes, 0 for no).

2. Set mx (maximum boundary) to a very small number (the problem uses -1000000 as an initially small value) and mi (minimum boundary) to a very large number (the problem uses 1000000), ensuring these boundaries will not affect the first comparison.

3. Iterate over nums from left to right, updating mx to be the maximum value seen so far. If the current element x is less than mx, it means x had a larger number before it, and thus, x can't be guaranteed to always be searchable. Set ok[i] to 0.

4. Iterate over nums from right to left, updating mi to be the minimum value seen so far. If the current element nums[i] is greater than mi, it implies that there's a smaller element to its right, so if nums[i] were chosen as a pivot, it would mistakenly discard the smaller number. Set ok[i] to 0 for such cases.

5. Finally, sum up all elements in ok. This sum represents the count of elements that are guaranteed to be binary search-able.

Let's relate the steps with the actual code segments:

• The ok array initialization is directly represented by ok = [1] * n.
• Setting the mx and mi boundaries uses mx, mi = -1000000, 1000000.
• The two for loops represent the left-to-right and right-to-left iterations with corresponding logic to update ok.
• The sum of elements in ok is obtained by return sum(ok).

By combining the edge constraints from both ends of the array, we ensure that only those elements that satisfy the conditions for both leftward and rightward iterations are considered binary search-able. The nature of the algorithm is such that it has O(n) time complexity since it passes over the array twice—once from each end—and performs constant-time operations within each iteration.

In terms of algorithmic patterns, this approach can be recognized as a variant of the two-pointer technique where instead of moving two pointers towards each other, we move boundaries in a way that encodes the validity of each element based on the search-ability condition described in the problem.

Example Walkthrough

Let's use a small example to illustrate the solution approach. We will walk through both the left-to-right and right-to-left iterations steps to clarify how the ok array gets updated.

Consider the array nums = [4, 2, 5, 7, 6, 3, 1] and the target value as 3. We will go through the algorithm to demonstrate how it ensures whether 3 or any other value can be found regardless of the random pivot selections.

1. Initialize an array ok to [1, 1, 1, 1, 1, 1, 1], i.e., initially assuming all elements are search-able.
2. Set mx to -1000000 and mi to 1000000.

Left-to-Right Check:

• Start with the first element (4). Since mx is -1000000, 4 is greater than mx, so we set mx to 4.
• Next element is 2. It is less than the current mx (which is 4), so we update ok at index 1 to 0 [1, 0, 1, 1, 1, 1, 1].
• Move to 5, which is greater than mx (4), so we update mx to 5.
• Continue to 7, which is greater than mx (5), so we update mx to 7.
• Process 6, it is less than mx (7), so we update ok at index 4 to 0 [1, 0, 1, 1, 0, 1, 1].
• Move to 3, it is less than mx (7), so update ok at index 5 to 0 [1, 0, 1, 1, 0, 0, 1].
• Finally, 1 is less than mx (7), so update ok at index 6 to 0 [1, 0, 1, 1, 0, 0, 0].

Right-to-Left Check:

• Start with the last element of the current ok state [1, 0, 1, 1, 0, 0, 0] which is 1. Since mi is 1000000, 1 is less than mi, so we set mi to 1.
• Next element is 3. It's greater than the current mi (1), so we update ok at index 5 to 0 [1, 0, 1, 1, 0, 0, 0].
• Since ok[5] was already 0, this step does not change the state of ok.
• Continue to 6, which is greater than the current mi (1), so ok at index 4 could potentially be set to 0, but it's already 0.
• Process 7, it is greater than mi (1), but ok at index 3 is still 1, and it remains 1 as 7 has no smaller element to its right.
• Move to 5, which is greater than mi (1), ok at index 2 remains 1.
• Next is 2, which is greater than the current mi (1), and since ok at index 1 is 0, it remains unchanged.
• Finally, 4 is greater than mi (1), so ok remains as [1, 0, 1, 1, 0, 0, 0].

Final Step:

• Sum up the values in ok to find the total count of elements guaranteed to be searchable. In this case, sum(ok) equals to 3.

Therefore, 3 elements in the nums array can be guaranteed to be found using our random-pivot-based binary search algorithm. These elements are 4, 5, and 7. These elements are guaranteed searchable because there are no other values to their left that would cause them to be discarded if they were not chosen first, and similarly, no values to their right that would be incorrectly discarded if these elements were chosen as pivots.

Solution Implementation

Time and Space Complexity

Time Complexity

The given Python code performs a single forward pass through the array to identify elements that are not greater than any element before them, and then performs a single backward pass to identify elements that are not less than any element after them. Each pass involves a simple comparison operation for each element of the list.

• The forward pass processes each element once, marking those elements that are smaller than the maximum element observed so far with a 0. This constitutes O(n) operations where n is the size of the nums list.

• Similarly, the backward pass also processes each element once, marking those elements that are greater than the minimum element observed so far with a 0. This also constitutes O(n) operations.

• Finally, the summing operation to count the number of 1s in the ok list also takes O(n).

Hence the overall time complexity of the code is O(n) + O(n) + O(n) which simplifies to O(n).

Space Complexity

The space complexity is determined by the additional space used by the program excluding the input.

• The code defines a list ok which has the same size as the input list nums. Therefore ok takes O(n) space.

• Variables mx and mi use constant space, so their space complexity contribution is O(1).

• The loop indices and temporary variables used for logic operations also occupy constant space.

Thus, the overall space complexity of the code is O(n) due to the ok array. The rest of the variables do not contribute significantly as they are constant space.

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