Problem Description

In this problem, we are given a list of integers representing barcodes. The goal is to rearrange the barcodes so that no two consecutive barcodes are the same. Fortunately, the problem statement assures us that there is always a valid rearrangement that meets this condition. We need to find and return one such valid arrangement.

Intuition

To solve this problem, we should first think about the constraints: no two adjacent barcodes can be equal. A natural approach to prevent adjacent occurrences of the same barcode would be to arrange the barcodes in such a way that the most frequent ones are as spread out as possible. This way, we reduce the chance that they will end up next to each other.

We can do this by:

1. Counting the frequency of each barcode using Counter from the collections module, which gives us a dictionary-like object mapping each unique barcode to its count of appearances.
2. Sorting the barcodes based on their frequency. However, simply sorting them in non-ascending order of their frequency is not enough, as we need to make sure that once they are arranged, the condition is met. It's a good idea to also sort them by their value to have a deterministic result if frequencies are equal.
3. Once sorted, the array is split into two: the first half will occupy even positions, and the second half will fill the odd positions in the result array. We do this because even-indexed places followed by odd-indexed ones are naturally never adjacent, which is ideal for our requirement to space out the frequent barcodes.
4. Create a new array for the answer, filling it first with the elements at even indices and then at odd indices.

By doing this, we ensure that the most frequent elements are spread across half the array length, minimizing any chance they are adjacent to each other. The end result is a carefully structured sequence that satisfies the given condition of the problem.

Learn more about Greedy, Sorting and Heap (Priority Queue) patterns.

Solution Approach

The solution approach follows a series of logical steps that strategically utilize Python language features and its standard library functionalities. Here's a breakdown of how the solution is implemented:

1. Counting Frequencies: We use the Counter class from the collections module to count the frequency of each barcode in the given list. The Counter object, named cnt, will give us a mapping of each unique barcode value to its count of appearances.

   cnt = Counter(barcodes)

2. Sorting Based on Frequency: We then sort the original list of barcodes using the sort method, applying a custom sorting key. The sorting key is a lambda function that sorts primarily by frequency (-cnt[x], where negative is used for descending order) and secondarily by the barcode value (x) in natural ascending order in case of frequency ties.

   barcodes.sort(key=lambda x: (-cnt[x], x))

3. Structuring The Answer: With the barcodes now sorted, half of the array (rounded up in the case of odd-length arrays) can be placed at even indices and the remaining half at odd indices in a new array named ans. This is to ensure that we space out high-frequency barcodes, minimizing the risk of identical adjacent barcodes:

   • We first calculate the size of the array n and then create the answer array ans of that same size filled with zeros.

   • The clever part comes with Python's slice assignment:

     • ans[::2] is a way to refer to every second element of ans, starting from the first. We assign the first half of the barcodes to these positions.
     • ans[1::2] refers to every second element starting from the second, to which we assign the latter half of the barcodes.

   Here's how the code accomplishes that:

   n = len(barcodes)
   ans = [0] * n
   ans[::2] = barcodes[: (n + 1) // 2]
   ans[1::2] = barcodes[(n + 1) // 2 :]

4. Returning the Result: The last line of the function simply returns the ans array, which is now a rearranged list of barcodes where no two adjacent barcodes are equal.

By using this approach, the algorithm effectively distributes barcodes of high frequency throughout the arrangement to satisfy the conditions of not having identical adjacent barcodes.

class Solution:
    def rearrangeBarcodes(self, barcodes: List[int]) -> List[int]:
        cnt = Counter(barcodes)
        barcodes.sort(key=lambda x: (-cnt[x], x))
        n = len(barcodes)
        ans = [0] * n
        ans[::2] = barcodes[: (n + 1) // 2]
        ans[1::2] = barcodes[(n + 1) // 2 :]
        return ans

Example Walkthrough

Let's consider a small example to illustrate the solution approach. Suppose we're given the following list of barcodes:

barcodes = [1, 1, 1, 2, 2, 3]

Following the steps of the solution approach:

1. Counting Frequencies: We use Counter to determine the number of occurrences of each barcode. In this example:

   cnt = Counter([1, 1, 1, 2, 2, 3])  # Results in Counter({1: 3, 2: 2, 3: 1})

2. Sorting Based on Frequency: Using the sorting key in sort, we arrange the barcodes first by decreasing frequency and then by their natural value:

   sorted_barcodes = [1, 1, 1, 2, 2, 3]  # 1 is most common, then 2, and 3 is least common

3. Structuring The Answer: The sorted list is now to be split across even and odd indices in a new array. The length of the barcodes list n is 6. The first half of the array (which is the first 3 elements, as (6 + 1) // 2 is 3) is placed at even indices:

   ans = [0, 0, 0, 0, 0, 0]
   ans[::2] = [1, 1, 1]

   And the latter half of the array (which is the last 3 elements) is placed at odd indices:

   ans[1::2] = [2, 2, 3]

   So, ans becomes:

   ans = [1, 2, 1, 2, 1, 3]

4. Returning the Result: The final ans array is returned as the rearranged list of barcodes:

   return [1, 2, 1, 2, 1, 3]

This rearranged array adheres to the condition that no two consecutive barcodes are the same. In this example, barcode 1, which occurred most frequently, is well spread out, followed by barcode 2, and barcode 3 is placed at the end to satisfy the condition.

Solution Implementation

Time and Space Complexity

Time Complexity

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

1. Counting elements with Counter(barcodes) takes O(n) time, where n is the number of elements in the barcodes list.

2. Sorting the barcodes list with the custom sort key based on their count and value takes O(n log n) time.

3. Slicing the list into two parts (for odd and even positions) is done in O(n) time.

4. Assigning the sliced lists to ans using list slicing operations ans[::2] and ans[1::2] is done in O(n) time.

So, combining these operations, the overall time complexity of the code is O(n log n) due to the sort operation.

Space Complexity

The space complexity of the code is influenced by:

1. The cnt object (counter of the elements), which takes O(unique) space, where unique is the number of unique elements in barcodes.

2. The ans list, which is a new list of the same length as the input list, taking O(n) space.

Therefore, the total space complexity is O(n + unique), which simplifies to O(n) if we consider that the number of unique items is less than or equal to n.

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