1200. Minimum Absolute Difference

Problem Description

The problem provides an array of distinct integers named arr. The objective is to find all pairs of elements that have the minimum absolute difference when compared to all other possible element pairs. In other words, we're looking for pairs of numbers that are closer together than any other pairs.

To satisfy the conditions of the problem, there are a few requirements:

• Each pair must be constructed from elements of arr.
• In each pair [a, b], a should be less than b.
• The difference b - a must be equal to the smallest absolute difference of any two elements in arr.

The result should be a list of these pairs in ascending order. Ascending order here applies to the order of the pairs themselves, meaning the first elements of each pair should be in ascending order, and in the case of a tie, the second elements should be in ascending order.

Intuition

To find the pairs with the minimum absolute difference, our approach involves sorting the array and comparing adjacent elements. The reason for sorting is that the smallest difference will always be between adjacent elements; no element can have a smaller difference with a non-adjacent element due to the properties of real numbers.

Here's the step-by-step intuition:

1. Sort the Array: By sorting arr, we ensure that we're only comparing differences between neighbors, which simplifies the problem significantly. We know that the minimum difference can only occur between neighboring numbers because the array consists of distinct integers.

2. Find the Minimum Difference: We iterate through the sorted array, considering each pair of adjacent elements, to find the minimum difference that exists in the array. This step is key to identifying what the "minimum absolute difference" actually is.

3. Gather All Pairs with Minimum Difference: Once we know the minimum difference, we iterate through the array again, this time collecting all pairs of adjacent elements that have this minimum difference. We construct a list of lists, where each inner list contains a pair of numbers.

The code provided utilizes a Pythonic way of dealing with the adjacent elements through the use of the pairwise() function, which when provided an iterable, yields tuples containing pairs of consecutive elements. This function is a cleaner and more readable way of accessing adjacent pairs without having to manage indexes manually.

Altogether, the solution efficiently gathers the pairs in a single pass after sorting, keeping the process succinct and the code readable.

Learn more about Sorting patterns.

Solution Approach

To implement the solution to this problem, we are taking advantage of the Python language features and some common algorithmic patterns. Here is a detailed explanation of how the solution approach works:

1. Sorting the Array:
   The very first step is to sort the array, arr.sort(). Sorting is a foundational algorithm in computer science, which organizes elements of a list in a certain order – in this case, ascending numerical order. Since the array consists of distinct integers, sorting will arrange the numbers such that if there's any pair with a minimum absolute difference, it will have neighboring elements.

2. Finding the Minimum Absolute Difference:
   With the sorted array, we employ a generator expression within the min function: min(b - a for a, b in pairwise(arr)). The pairwise function is used to create an iterator that will return paired tuples (e.g., (arr[0], arr[1]), (arr[1], arr[2]), ...). The min function then finds the smallest difference between these successive pairs.

3. List Comprehension to Collect Pairs with Minimum Difference:
   After finding the minimum absolute difference, we step through the array again with a list comprehension [ [a, b] for a, b in pairwise(arr) if b - a == mi ]. This comprehension checks each adjacent pair again to see if their difference is equal to the previously found minimum difference (mi). When a pair matches, it is added to the output list.

4. Returning the Result:
   The list of collected pairs that satisfy the minimum absolute difference condition is returned as the final result.

The algorithms and patterns used in this solution are:

• Sorting: A fundamental operation that reorders the items of a list (or any other sequence) to make subsequent operations (like finding a minimum difference) more efficient.
• List Comprehension: A convenient and readable way to construct lists in Python, used here for collecting the required pairs.
• Generator Expression: Used to generate values on the fly; in this case, it helps to find the minimum absolute difference without creating an intermediate list of differences.
• Itertools pairwise() Utility: Although pairwise is not an inbuilt function in Python versions prior to 3.10, it can be found in the itertools module or be custom implemented. It’s a utility to abstract away the complexity of iterating over pairs of consecutive elements.

By breaking down the problem into these steps, the implementation remains clean, efficient, and easy to understand.

Example Walkthrough

Let's walk through a small example to demonstrate how the solution approach is applied. Suppose we have the following array of distinct integers:

1arr = [4, 2, 1, 3]

We are tasked with finding all pairs of elements that have the minimum absolute difference. Here's how we will apply the explained solution approach to this array:

1. Sort the Array:

   First, we will sort the array:

   1arr.sort() => arr = [1, 2, 3, 4]

   After sorting, the array's elements are arranged in ascending order, making it easier to find pairs with the minimum absolute difference, which will be between neighbors.

2. Find the Minimum Absolute Difference:

   Next, we examine adjacent pairs to find the smallest difference. Using the pairwise() function (or iterating through neighboring pairs manually), we get:

   1Pairs: (1, 2), (2, 3), (3, 4)
   2Differences: 1, 1, 1

   The minimum difference here is 1. This value is determined by iterating through the pairs and finding the smallest difference, which is achieved using:

   1min(b - a for a, b in pairwise(arr))

3. Collect Pairs with Minimum Difference:

   Now that we know the minimum difference is 1, we look for all pairs with this difference:

   1[ [a, b] for a, b in pairwise(arr) if b - a == 1 ]
   2
   3Resulting Pairs: [[1, 2], [2, 3], [3, 4]]

   Each of these pairs has the same difference of 1, which is the minimum we found in the previous step.

4. Returning the Result:

   Lastly, we return the resultant pairs:

   1[[1, 2], [2, 3], [3, 4]]

This is the final result, containing all pairs of elements in arr which have the smallest absolute difference of any two elements, formatted according to the problem requirements. The pairs are already in ascending order due to the initial sorting of arr.

Through this example, the implementation follows the steps laid out in the solution approach and demonstrates the effectiveness of sorting, generator expressions, and list comprehensions to solve the problem with minimum absolute difference pairs efficiently.

Solution Implementation

Time and Space Complexity

The given Python code snippet defines a function minimumAbsDifference that finds all the pairs of elements in a list with the smallest absolute difference. Here is the analysis of its computational complexity:

Time Complexity

1. Sorting the array: arr.sort() is used, which has a time complexity of O(n log n) where n is the number of elements in arr.
2. Finding the minimum absolute difference: The comprehension min(b - a for a, b in pairwise(arr)) iterates over the array once, which gives a time complexity of O(n-1), simplifying to O(n).
3. Generating the list of pairs with the smallest absolute difference: The list comprehension [[a, b] for a, b in pairwise(arr) if b - a == mi] also iterates over the array once, resulting in the time complexity of O(n).

Since the sorting step dominates the overall time complexity, the final time complexity of the entire function is O(n log n).

Space Complexity

1. The sorted list: In-place sorting is used so no additional space complexity is introduced for sorting beyond the variable reassignments, which is O(1).
2. The minimum absolute difference: The minimum value mi is found with a generator expression, so the space complexity for this step is O(1).
3. The output list: The space complexity is proportional to the number of pairs with the smallest absolute difference. In the worst case, all pairs have the same minimum absolute difference, thus the space complexity is O(n).

So, the space complexity for the function is O(n) with respect to the output size.

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