561. Array Partition

Problem Description

The task is formulated around an optimization problem using an integer array named nums. This array has 2n elements, meaning that its length is an even number. The objective is to pair up the integers into n pairs in such a way that the sum of the minimum values from each pair is maximized. In mathematical terms, for pairs (a_1, b_1), (a_2, b_2), ..., (a_n, b_n), we need to find the maximum value of sum(min(a_i, b_i)) for all i from 1 to n.

To fulfill the task:

1. We have to create the pairs from the elements in the array.
2. Then, calculate the minimum of each pair.
3. Finally, calculate and return the sum of all these minimum values.

The objective is to construct these pairs wisely so that we get the maximum possible sum of the minimums.

Intuition

The intuition behind the solution relies on the idea of minimizing the loss of the larger numbers and ensuring that we can maximize the sum of the smaller numbers.

A crucial observation is that in each pair (a, b), the minimum value is the one that will be included in our final sum, whereas the larger one is effectively 'wasted' in terms of contributing to the sum. Therefore, we should minimize the waste by making sure that the difference between the paired numbers is as small as possible.

To meet this goal, the following steps are considered:

1. Sort the array in ascending order. Sorting allows us to easily group the elements into pairs in a way that minimizes the differences between the numbers within each pair.
2. Once we have the sorted array, we can pair each element with its adjacent element without worrying about missing out on a possibly smaller or equal pair.
3. Now that we have the pairs as adjacent elements, we know that within each pair, the first element is the smaller one (due to the array being sorted).
4. To find the sum of all minimum elements from each pair, we can add up every second element starting from the first element in the sorted array.

The provided code does exactly that: sum(sorted(nums)[::2]) sorts the array and then uses slicing to get every second element starting from the first one (which are all the minimums in each pair), and then sums them up to get the result.

Learn more about Greedy and Sorting patterns.

Solution Approach

The implementation of the solution is straightforward and leverages Python's built-in functions to achieve the goal efficiently. Here's a step-by-step guide through the algorithm and the patterns used in the solution:

1. Sorting the Array: The initial step is to use Python's built-in sorting function with sorted(nums). Sorting is a very common technique in such problems as it often simplifies and provides a structure to the data which can be exploited in a subsequent step. In this case, sorting puts the array in ascending order so that each pair (a, b) will have a as the minimum.

2. Slicing the Array: With the sorted array, we apply slicing sorted(nums)[::2]. In Python, slicing is a way to obtain a subsequence of a list or array. The slice [::2] specifically means "start from the first element and pick every second element." This slice operation is both compact and efficient, avoiding the need for an explicit loop to go through the array and pick the elements.

3. Summing the Minimums: Lastly, by passing the sliced list to sum(), Python's built-in function that computes the sum of the numbers in an iterable, we add up the selected numbers. The minimum of each pair is included due to the slicing, and thus the sum of these minimums is maximized.

The algorithm’s time complexity is dominated by the sorting operation, which is O(n log n) in the average and worst case scenarios. The slicing and summing are linear operations, O(n), but since n log n grows faster than n, the overall time complexity remains O(n log n).

No additional data structures are used or needed since the problem can be resolved with operations on the provided array. In terms of patterns, it's worth noting that this approach elegantly combines sorting with the properties of a sorted array to avoid a more complex and potentially less efficient approach that might explicitly build and then process pairs of numbers.

The simplicity of the approach is its strength, as it doesn’t rely on complex data structures or algorithms beyond what Python provides out of the box. The use of slicing to select every other element of an already sorted array makes the implementation both elegant and intuitive.

Example Walkthrough

Consider the array nums = [3, 1, 5, 4, 2, 6] with a length of 2n = 6, indicating that we need to create n = 3 pairs. Apply the solution approach to this example:

1. Sorting the Array: First, sort the array in ascending order which gives us the sorted array [1, 2, 3, 4, 5, 6]. By sorting, the smallest numbers are positioned at the beginning, which will help maximize the sum of the minimums of each pair.

2. Slicing the Array: Now, we retrieve every second element starting from the first element using slicing. From our sorted array, this yields [1, 3, 5] which represents the minimum values from each potential pair ((1,2), (3,4), (5,6)).

3. Summing the Minimums: Finally, sum up the elements in the sliced array [1, 3, 5]. The sum is 1 + 3 + 5 = 9. This total is the maximum sum of the minimums that can be obtained by pairing up numbers in the given array.

Following this method, the code sum(sorted(nums)[::2]) provides a quick and efficient way to reach the solution, which is the sum 9 in this case. By ensuring that we pair each number with its nearest neighbor in the sorted list, we maximize the contribution of the lower number to the sum while minimizing the "waste" due to the higher number being paired with it but not counted in the sum.

Solution Implementation

Time and Space Complexity

The given Python function arrayPairSum finds the sum of min(a[i], a[i+1]) for every pair of elements in the array when the array is sorted. It sorts the array and then sums up elements at even indices (i.e., considering 0-based indexing, indices 0, 2, 4, etc).

Time Complexity:

The time complexity is primarily determined by the sorting function. Python uses Timsort for sorting, which has a time complexity of O(n log n) for an average and worst-case scenario, where n is the length of the nums array.

There is also the summation operation, which iterates over every other element of the sorted array, contributing an additional O(n/2), which simplifies to O(n) time. However, since O(n log n) dominates O(n), the overall time complexity is O(n log n).

Space Complexity:

The space complexity of the sorting operation depends on the implementation. For Timsort, the worst-case space complexity is O(n), because it might need temporary space to hold elements while merging. Timsort is a hybrid sorting algorithm that requires temporary storage for the merge operations.

However, since the input array itself is sorted in-place and the result is computed using that without requiring extra space, other than what is needed for the sorted array and the sum variable, the space complexity may be considered O(1) or constant if the space used by the sorting algorithm is not taken into account, which is typically the case for space complexity analysis in Python where sorting is considered to be in-place.

In summary:

• Time Complexity: O(n log n)
• Space Complexity: O(1) (disregarding the space used by the sorting algorithm)

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