Problem Description

In this problem, we are given a 0-indexed array nums of n integers. The task is to make all elements in the array equal by performing a specific operation repeatedly. During each second, for every element at index i, you can update nums[i] to be equal to its current value, the value of the previous element (nums[(i - 1 + n) % n]), or the value of the next element (nums[(i + 1) % n]). The modulo operation ensures that you are wrapping around the array when reaching the ends, which means it actually forms a loop or ring. The goal is to find the minimum number of seconds needed to make all the elements in the array equal.

Intuition

The key to solving this problem lies in understanding that we can always make the entire array equal to any one of its current values. This is because in each operation, you are allowed to choose the previous or next element's value, which can eventually spread any number's occurrence across the array.

Since we want to minimize the number of seconds, the best strategy would be to choose the value that will take the least amount of time to spread through the array. Intuitively, if we have clusters of the same number occurring together, we would prefer to choose one such cluster's value and spread it to the rest of the array.

The solution involves the following steps:

• We first group indices of identical elements into lists, using a dictionary where the keys are the array's values and the values are lists of indices where these numbers occur.
• Then, for each group of identical elements:
  • We calculate the distance between the first and last occurrence of the value, taking into account the wrap-around using (idx[0] + n - idx[-1]).
  • We also calculate the maximum distance between any two consecutive occurrences, as this will represent the maximum time required to change the value between these two points using max(t, j - i) for every pair of consecutive indices in the group.
  • The time needed to make the entire array equal to this value is half the maximum distance found, since the value can spread from both ends of a series.
• The minimum time across all such values is the final answer.

By applying this approach, we ensure that we pick the value that will take the least amount of time to replicate across the entire array.

Learn more about Greedy patterns.

Solution Approach

The solution provided uses Python's defaultdict to categorize indices of identical elements into lists, and the inf constant from the math module as a representation of infinity, which is employed to find the minimum value as we compare different distances.

Here is the step-by-step breakdown of the implementation:

• A defaultdict of lists is instantiated. It will map each unique value in nums to a list of indices where it occurs. This is achieved by enumerating over nums and appending the index i to the list of d[x], where x is the value at index i.
• A variable ans is initialized to infinity (inf). This will hold the minimum number of seconds required to make all elements of the array equal.
• The algorithm then iterates over the values of the dictionary, which are the lists of indices for each distinct number in the array.
  • For each list of indices (idx), it calculates t, the distance considering wrap-around between the first and last occurrence: idx[0] + n - idx[-1].
  • It then proceeds to find the maximum distance between any two consecutive occurrences of the same number within the list. This is done using Python's pairwise function (Python 3.10+). If pairwise is not available, a simple zip like zip(idx, idx[1:]) can be used to achieve similar results.
  • For every pair (i, j) of consecutive indices in idx, t is updated to the maximum of its current value and the distance between the consecutive indices j - i.
  • Finally, because the value can spread from both ends towards the middle, only half this time is needed to make all elements between i and j equal, hence t // 2.
• The algorithm updates ans with the minimum between its current value and t // 2. Since ans is initialized to infinity and we are finding the minimum over all iterations, ans will hold the minimum time needed to make all elements equal after examining all distinct numbers.
• The function returns the value stored in ans.

This approach effectively breaks down a seemingly complex problem into a series of calculations based on the distribution of values across the array, using dictionary and list structures to organize data and a simple loop to compute the minimum time.

Example Walkthrough

Let's illustrate the solution approach using a simple example:

Given the array nums = [1, 2, 3, 2, 1], which is 0-indexed:

Step 1: We create a dictionary of list indices for each unique value in nums. In our case:

• The value 1 occurs at indices [0, 4].
• The value 2 occurs at indices [1, 3].
• The value 3 occurs at index [2].

Step 2: Initialize ans to infinity.

Step 3: Evaluate each group of identical elements:

• For value 1, the list of indices is [0, 4]. Since the array forms a loop:

  • The distance considering wrap-around is (0 + 5 - 4) % 5 = 1.
  • There are no consecutive occurrences to calculate the maximum distance, so the maximum distance remains 1.
  • The time needed is 1 // 2 = 0 (since we can start from both ends, it needs no time to convert values in between).

• For value 2, the indices are [1, 3]. No wrap-around is needed.

  • The maximum distance between consecutive occurrences is (3 - 1) = 2.
  • The time needed would be 2 // 2 = 1.

• For value 3, there is only one occurrence, no distance to calculate between indices.

Step 4: Find the minimum ans:

• For value 1, ans becomes the minimum of infinity and 0, which is 0.
• For value 2, ans is the minimum of 0 and 1, which remains 0.
• For value 3, no change as there is only one occurrence.

Step 5: Return the value in ans, which is 0.

This would mean that no time is needed to make all the elements equal to 1 in this example, since we theoretically start spreading the value from both ends of the occurrences.

Solution Implementation

Time and Space Complexity

Time Complexity

The time complexity of the given code is determined by several factors:

• The loop that creates the dictionary d, which has a time complexity of O(n) since it goes through all the elements of nums once.
• The loop that goes through d.values() which can potentially iterate through all elements again in the worst case. If all the elements in nums are unique, this would again take O(n) time.
• Inside the second loop, there's a nested call to pairwise(idx). The pairwise function itself has O(k) complexity, where k is the length of the list idx passed to it. In the worst case, where nums has many repeated elements, this nested loop could have O(n) complexity if all elements are the same.

Given that pairwise is called inside the loop for every key in the dictionary d, the overall complexity of these nested loops depends on the distribution of the numbers in the nums list. The worst-case scenario happens when all elements are the same, leading to a complexity of O(n) for the iterations throughout d.values(), compounded with the complexity of pairwise, leading to a worst-case time complexity of O(n^2).

Therefore, the overall worst-case time complexity of the code is O(n^2).

Space Complexity

The space complexity can be analyzed as follows:

• The dictionary d that stores the indices of each element can potentially store n keys with a list of indices as values. In the worst case where all numbers are the same, the list of indices would also contain n values. Therefore, the worst-case space complexity for d is O(n).
• The space used by variables ans, t, idx, i, and j is constant, hence O(1).

Taking the above points into consideration, the total space complexity is O(n) for the dictionary storage.

In conclusion, the code has a time complexity of O(n^2) and a space complexity of O(n).

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