Problem Description

In this problem, we are provided with an integer array arr where a dominant element is defined as an element x that appears more than half of the time in arr (freq(x) * 2 > m, where m is the length of arr). The array nums we're working with is guaranteed to contain one such dominant element.

The task is to find a valid split for the array nums into two subarrays where both subarrays contain the same dominant element. A split at index i is valid if 0 <= i < n - 1 (where n is the length of nums), and both resulting subarrays (nums[0, ..., i] and nums[i + 1, ..., n - 1]) have the same dominant element. The goal is to find the minimum index i at which such a valid split can occur. If there is no possible valid split, we should return -1.

To give a simple example, if nums is [2,2,1,2,2], splitting after the second 2 results in [2,2] and [1,2,2], both of which have 2 as the dominant element. So the index before which we split would be 2-1 (because it's 0-indexed), hence the minimum valid split index would be 1.

Intuition

The provided solution approach relies on the definition of a dominant element and some properties of an array with exactly one dominant element:

• Because there is only one dominant element, once we find it, we know it will be the dominant element in any valid split.
• Since the dominant element appears more than half the time, the first candidate for a possible split can only occur after we first encounter this condition in the running count of the dominant element.

Given these principles, the steps to arrive at the solution involve:

1. Finding the dominant element using a counter to determine the most common element in the array.
2. Iterating through the array, keeping track of the count of occurrences of the dominant element (cur).
3. As we iterate, we check two conditions at each index i:
   • If cur * 2 > i, it means the dominant element is currently more than half of the elements from 0 to i-1.
   • Simultaneously, we must check if the dominant element will remain dominant in the second subarray. This is done by checking if (cnt - cur) * 2 > len(nums) - i, where cnt is the total occurrences of the dominant element, and len(nums) - i is the length of the second subarray.

As soon as both conditions are satisfied, it indicates we've found a valid split. The solution outputs the index i - 1, accounting for the 0-indexed array.

By iterating over the array just once and checking the conditions, the solution effectively finds the minimum index for a valid split, thereby solving the problem in linear time with respect to the length of the array.

Learn more about Sorting patterns.

Solution Approach

The solution begins by using a Counter from the collections module to find the most common element in the array nums, which is the dominant element by the problem's definition. The Counter is a dictionary subclass designed for counting hashable objects, and the most_common(1) method retrieves the most common element along with its count.

x, cnt = Counter(nums).most_common(1)[0]

In this line of the code, x stands for the dominant element, while cnt is the number of times x appears in nums.

Next, the code uses a single-pass for loop to iterate over all the elements of the array while keeping track of two things:

• The count of the dominant element seen so far (cur).
• The index at which this count is processed (i), which is used for checking whether a split is valid.

for i, v in enumerate(nums, 1):

Here, enumerate is used with a start index of 1 to keep the index i synchronized with the length of the subarray being considered, since we're interested in the possibility of a split before the i-th element.

The iteration follows this logic:

1. Increment cur each time the dominant element is encountered.

if v == x:
    cur += 1

2. Check if we can make a valid split at index i-1. This check has two parts:

• To ensure that the dominant element is indeed dominant in the first subarray, we check if cur * 2 > i. This guarantees that there are more instances of x than all other elements combined until the current index.

• To ensure that the dominating element is also dominant in the second subarray after the split, we check if (cnt - cur) * 2 > len(nums) - i. This ensures that even after removing cur instances of x, we have enough left for x to remain dominant.

if cur * 2 > i and (cnt - cur) * 2 > len(nums) - i:
    return i - 1

If both conditions are satisfied, the current i represents a 1-indexed position at which an array can be split. Since the requested output should be 0-indexed, i - 1 is returned.

If the for loop completes without returning, it means no valid split exists that satisfies both conditions, and in this case, the function returns -1.

return -1

By employing a count tracking approach, the conditionals are checked in O(1) time per element, leading to an overall time complexity of O(n), where n is the number of elements in the array. The space complexity is also O(n) since we use a Counter to store the frequency of each element in the array.

Example Walkthrough

Let's consider a small example with the following array of numbers:

nums = [3, 3, 4, 2, 3]

Using this array, we'll walk through the solution approach described above.

1. First, we need to determine the dominant element in the array. We do this by utilizing the Counter class from the collections module:

from collections import Counter
x, cnt = Counter(nums).most_common(1)[0]

In our example, x will be 3 since it's the most common element, and cnt will be 3 because 3 appears three times in the array.

2. Then, we proceed to iterate over the array and count the occurrences of the dominant element while simultaneously checking if a valid split is possible:

cur = 0
for i, v in enumerate(nums, 1):
    if v == x:
        cur += 1
    if cur * 2 > i and (cnt - cur) * 2 > len(nums) - i:
        return i - 1

3. We examine each element in the loop:

   • At i = 1 (3): cur becomes 1. We cannot split yet because cur * 2 is not greater than cnt.
   • At i = 2 (3): cur becomes 2. We cannot split yet because, although cur * 2 is now 4 and greater than 2, the second condition (cnt - cur) * 2 > len(nums) - i is not met (as 1 * 2 is not greater than 3).
   • At i = 3 (4): cur remains 2. We still can't split because the first condition is no longer fulfilled.
   • At i = 4 (2): cur remains 2. Once again, we can't split due to the first condition not being met.
   • At i = 5 (3): cur becomes 3. This is the first time a split is possible as cur * 2 is 6 and greater than 5, and (cnt - cur) * 2 is 0, which is less than len(nums) - i which is 0.

# Since we have reached the end of the array,
# and there is no point where both conditions were true,
# the result of the function would be:
return -1

In this example, there is no valid split because at no point do both conditions satisfy during the iteration. Thus, according to our solution, the function would return -1, meaning no valid minimum index split exists.

Solution Implementation

Time and Space Complexity

Time Complexity

The given Python function minimumIndex first determines how many times the most common number x occurs in the array nums by using the Counter class from the collections module and retrieves the count cnt. This operation is O(n) where n is the length of the list nums, as it involves iterating through the entire list to compute the frequency of each element.

Following that, the function proceeds to iterate through the list nums while maintaining a cumulative count cur of how often it has encountered x so far. During iteration, it performs a constant time check in each iteration to determine if cur and cnt - cur are both more than half of the numbers seen so far and the numbers remaining, respectively.

The loop also iterates n times in the worst case (if it does not return early), so the total time complexity of the entire function thus remains O(n), where n is the length of the input list.

The exact time complexity is therefore O(n).

Space Complexity

In terms of space, the function uses additional memory for the Counter object to store the frequency of each element in nums. The space complexity for this part is O(m), where m denotes the number of unique elements in nums. In the worst case where all elements are unique, m would be equivalent to n, resulting in O(n) space complexity.

Additionally, the space used for the index counter i, the most common element and its count (x, cnt) and the running count cur is O(1), as they do not depend on the size of the input list but are merely constant size variables.

Thus, the overall space complexity of the function is O(m), which is O(n) in the worst case when all elements in nums are unique.

The exact space complexity is O(m) with a note that it simplifies to O(n) in the worst-case scenario.

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