Sum of Absolute Differences in a Sorted Array

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class Solution:
    def getSumAbsoluteDifferences(self, nums: List[int]) -> List[int]:
        ans = [0] * len(nums)

        for i in range(1, len(nums)):
            ans[0] += (nums[i] - nums[0])

        for j in range(1, len(nums)):
            ans[j] = ans[j - 1] + (nums[j] - nums[j - 1]) * j - (nums[j] - nums[j - 1]) * (len(nums) - j)

        return ans

10 Prerequisite LeetCode Problems

For this, the following are a good preparation:

  1. “1. Two Sum” - This problem teaches basic array manipulation and how to deal with sums, which is a key concept in the main problem.

  2. “561. Array Partition I” - This problem introduces the concept of manipulating sorted arrays to get certain sums, which is beneficial for understanding “Sum of Absolute Differences in a Sorted Array”.

  3. “238. Product of Array Except Self” - Provides a foundation on how to deal with sums or products where each element interacts with all others, a core part of the main problem.

  4. “724. Find Pivot Index” - This problem involves calculating running sums from both ends, which is helpful for the concept of maintaining running sums of differences in the main problem.

  5. “53. Maximum Subarray” - This problem involves summing elements of an array, similar to the main problem where we need to sum the differences.

  6. “283. Move Zeroes” - Teaches how to handle and manipulate an array, which is necessary for understanding how to handle differences in the main problem.

  7. “1695. Maximum Erasure Value” - This problem introduces the sliding window concept, which can be beneficial when considering ranges of differences.

  8. “41. First Missing Positive” - This problem forces the user to think about array manipulation in a different way, and can help with the numerical manipulation seen in the main problem.

  9. “581. Shortest Unsorted Continuous Subarray” - This problem is about finding subarrays with certain properties, which might help in understanding how to segment the main problem’s array for sums.

  10. “665. Non-decreasing Array” - This problem has a similar context of dealing with non-decreasing arrays, which is also the case in the main problem.

These cover array manipulation, summing elements in an array, maintaining running sums, and handling sorted and non-decreasing arrays, which are crucial in solving the “Sum of Absolute Differences in a Sorted Array” problem.

Problem Classification

The problem falls under the domain of “Arrays” and “Mathematics.”

What

  1. Input: An integer array nums sorted in non-decreasing order.
  2. Output: An integer array result of the same length as nums.
  3. Operation: For each i in nums, calculate the sum of absolute differences between nums[i] and every other element in the array, and store it in result[i].
  4. Constraints:
    • The length of nums is between 2 and (10^5).
    • Each element in nums is an integer between 1 and (10^4), and the array is sorted in non-decreasing order.

This problem can be classified as a “Computation” problem where you have to perform mathematical operations over arrays. It also has elements of “Traversal,” as you need to go through each element in the array to perform the computations. Finally, it’s a “Summation” problem since the objective is to find the sum of certain elements.

Clarification Questions

  1. Are negative numbers allowed in the array? (The problem statement specifies they are not, but good to confirm if not clear).
  2. Is it guaranteed that the input array will always be sorted in non-decreasing order?
  3. What should be returned if the input array is empty?
  4. Will the array contain duplicate numbers? (The problem doesn’t prohibit this, but asking can confirm your understanding).
  5. Are there any space complexity constraints for solving this problem?
  6. Is there a preference for optimizing time complexity or space complexity?
  7. What should be the data type of the returned array? Should it match the data type of the input array?
  8. Can we assume that the input array fits in memory?
  9. Are we allowed to use built-in functions for calculating absolute values?
  10. Is it acceptable to return a new array, or should the result replace the existing array?

These questions aim to clarify any ambiguities and ensure a complete understanding of the problem before proceeding to the solution phase.

Problem Analysis and Key Insights

  1. Sorted Array: The input array is sorted in non-decreasing order. This property can be utilized for optimization.

  2. Absolute Differences: The focus on absolute differences indicates that negative numbers and their positions won’t affect the output.

  3. Same Length: The output array will have the same length as the input array, giving us initial constraints on space complexity.

  4. Result Calculation: Each element in the result array is the sum of the absolute differences between a specific element in the input array and all other elements in the input array. This suggests that a naive approach might involve nested loops, leading to a high time complexity.

  5. Constraints: The constraints provided on the length of the array and the values give us an idea of the upper limits we have to consider, especially when thinking about time and space complexity.

  6. No Negative Numbers: All numbers in the array are positive integers or zero, meaning we don’t have to account for negative numbers in the absolute difference calculations.

These insights guide the problem-solving approach, indicating opportunities for optimization and pointing out important characteristics that can influence the choice of algorithm.

Problem Boundary

The scope of the problem is focused on array manipulation and mathematical computation. Specifically, it involves:

  1. Reading an integer array sorted in non-decreasing order.
  2. Calculating the absolute differences between each element and every other element in the array.
  3. Summing up these absolute differences for each element.
  4. Constructing a new integer array containing these summations, with the same length as the input array.

The problem does not involve any external data sources, user interactions, or system-level considerations. It’s a self-contained computational task with well-defined inputs and outputs.

The boundaries of this problem are well-defined by the constraints and the problem statement. Specifically:

  1. Input:

    • The length of the array nums is between 2 and 10^5.
    • Each integer in the array is between 1 and 10^4, and the array is sorted in non-decreasing order.

  2. Output:

    • An integer array result of the same length as nums.

  3. Functionality:

    • The task is to populate the result array such that each element result[i] is the sum of the absolute differences between nums[i] and all other elements in nums.

  4. Time Complexity:

    • While not explicitly stated, the constraints suggest that a highly efficient algorithm is likely needed. An algorithm with time complexity worse than O(n log n) might not be acceptable.

  5. External Factors:

    • None. There are no external databases, user inputs, or other systems interacting with this problem. It’s an isolated computational task.

By paying attention to these factors, you establish the boundaries within which the problem must be solved.

Distilling the Problem to Its Core Elements

  1. Fundamental Concept:

    • This problem is based on the concept of array manipulation and mathematical summation. It requires you to understand how to calculate differences between elements in an array and sum them up for each index.

  2. Simplest Description:

    • Imagine you have a list of numbers. For each number, you need to find how far it is from every other number in the list. Add up those distances for each number.

  3. Core Problem:

    • The core problem is to calculate, for each element in the array, the sum of its absolute differences with every other element in the array.

  4. Key Components:

    • Input array nums
    • Output array result
    • Absolute difference calculation
    • Summation

  5. Minimal Set of Operations:

    • Loop through each element in the input array.
    • Inside this loop, perform another loop to calculate the absolute difference between the current element and all other elements.
    • Sum up these absolute differences.
    • Store the sum in the corresponding position in the output array.

By breaking down the problem into these key components and operations, you can develop a clearer picture of what needs to be done to arrive at a solution.

Visual Model of the Problem

To visualize the problem, consider using a grid or table format. You can set it up as follows:

  1. Columns: Represent each element in the array nums.
  2. Rows: For each element in nums, list the absolute differences between that element and all other elements in the array.

Here’s how it would look for the first example nums = [2,3,5]:

235
2013
3102
5320
  • The first row indicates that the sum for index 0 (which corresponds to the number 2) would be 0+1+3 = 4.
  • The second row indicates that the sum for index 1 (which corresponds to the number 3) would be 1+0+2 = 3.
  • The third row indicates that the sum for index 2 (which corresponds to the number 5) would be 3+2+0 = 5.

By visualizing the problem this way, you can see more clearly what calculations need to be made for each index in the output array result. It provides a structured way to approach solving the problem.

Problem Restatement

You have an array of integers sorted in non-decreasing order. Your task is to create a new array of the same length. Each element in this new array should be the sum of the absolute differences between that corresponding element in the original array and every other element in the original array.

For example, if the original array is [2,3,5], the new array should be [4,3,5].

Constraints:

  • The length of the array will be at least 2 and can go up to 105.
  • The integers in the array will be between 1 and 10^4, and sorted in non-decreasing order.

Abstract Representation of the Problem

Certainly. In abstract terms, you are given a sorted sequence ( S ) of ( n ) integers. Your goal is to construct a new sequence ( R ) of the same length ( n ), where each element ( R[i] ) is calculated as the sum of absolute differences between ( S[i] ) and all other elements in ( S ).

Formally, ( R[i] = \sum_{{j=0, j \neq i}}^{n-1} |S[i] - S[j]| ).

The constraints specify boundaries for ( n ) and the elements in ( S ), ensuring computational feasibility.

Terminology

Here are the specialized terms and their roles in the context of this problem:

  1. Non-Decreasing Order: This means the array is sorted in an order where each element is greater than or equal to the preceding one. It is crucial for understanding that the input array will always be sorted in such a way.

  2. Absolute Difference: This is the non-negative difference between two numbers. It is the core operation being applied between elements in the array to compute each result value.

  3. Array Indexing: The problem uses 0-based indexing. This refers to the way elements in the array are accessed, starting from index 0. Understanding this is vital for accurate calculations.

  4. Summation (( \sum )): This is a mathematical symbol used to represent the sum of a sequence of terms. In this context, it refers to the summing of all the absolute differences calculated.

  5. Constraints: These define the boundaries of the problem, specifying the minimum and maximum lengths for the array and the range of integer values it can contain. These constraints are important for solution optimization and validation.

Understanding these terms helps in accurately interpreting the problem statement and developing a correct and efficient solution.

Problem Simplification and Explanation

The problem asks you to look at a sorted list of numbers and for each number, find out how “different” it is from every other number in the list. You quantify this “difference” using absolute difference, which is just the positive difference between two numbers. You sum up all these differences for each number. That sum will be that number’s “uniqueness score” in the list.

Key Concepts:

  1. Sorted List: The numbers are arranged in ascending order, which should give us some clues for optimization.
  2. Absolute Difference: The distance between two numbers on a number line, disregarding which one is larger.
  3. Summation: Adding up the individual absolute differences to get a single value for each number in the list.

Interactions:

  • You take each number and compute its absolute difference with every other number.
  • You sum these differences to create a “score” for each number.
  • You repeat this process for every number in the list.

Metaphor: Imagine each number in the list is a person standing in a line according to their height, shortest to tallest. For each person, we want to find out how different their height is from everyone else in the line. We do this by measuring the height difference between that person and every other person, then adding up all those differences. The total sum will be that person’s “uniqueness score” based on their height.

By breaking it down this way, the problem becomes a series of simple steps involving sorting, absolute differences, and summing those differences.

Constraints

Here are some characteristics that can be exploited for an efficient solution:

  1. Sorted Array: The input array is sorted in non-decreasing order. This is a big hint for optimization because it means adjacent elements are either equal or increase as we move along the array. This property can save us time in calculating differences.

  2. Non-decreasing Order: The constraint that elements are in non-decreasing order could be helpful in accounting for repeated numbers. If an element is repeated, the absolute difference will be zero for those repeated numbers, and we can quickly account for them without additional calculations.

  3. Constraints on Element Size: Elements range from 1 to 10^4. Since the numbers aren’t extraordinarily large, it simplifies potential worries about overflow or underflow. However, the range itself doesn’t provide a direct avenue for optimization.

  4. Array Length Constraints: The length of the array is between 2 and 10^5. The upper limit on the length means we should aim for a solution better than O(n^2) to be efficient.

  5. Same Length Output: The output array has the same length as the input, which means we don’t need additional space to store variables that track the length, and we can allocate the output array right at the start.

By taking advantage of these characteristics, we should be able to come up with an efficient algorithm for solving this problem.

The key insights from analyzing the constraints are:

  1. Sorted Input: Knowing the array is sorted in non-decreasing order gives us an avenue for optimization. This allows us to make educated decisions while traversing the array.

  2. Complexity Target: With the maximum array length set at 10^5, a naive O(n^2) solution would be inefficient. We need to aim for a more efficient algorithm, ideally O(n) or O(n log n).

  3. Element Range: The element values are between 1 and 10^4. This range isn’t too large, which rules out issues like integer overflow in calculations but doesn’t provide a direct path to optimization.

  4. Output Length: The output array must be of the same length as the input array. This simplifies space allocation and could potentially allow us to perform the calculations in-place to save memory.

These constraints guide us in forming an efficient algorithm. For example, the sorted nature of the array should help us in calculating the absolute differences in a more efficient manner than if the array were unsorted.

Case Analysis

Certainly, let’s consider a variety of test cases for the problem.

Single Digit Elements

Example:
Input: [1, 2]
Output: [1, 1]
Reasoning:
For the first element, 1, the absolute difference with 2 is 1. For the second element, 2, the absolute difference with 1 is also 1.

Duplicate Elements

Example:
Input: [3, 3, 3]
Output: [0, 0, 0]
Reasoning:
All elements are the same, so the absolute differences will be zero for all elements.

Larger Gaps

Example:
Input: [1, 100]
Output: [99, 99]
Reasoning:
Here, the array elements have a large gap. The absolute difference will be the same for both elements: 99.

Multiple Large Gaps

Example:
Input: [1, 50, 100]
Output: [149, 99, 149]
Reasoning:
The gaps between elements are significant, resulting in higher absolute differences for the elements at the edges of the array.

Edge Cases

  1. Minimum Array Length
    Input: [1, 2]
    Output: [1, 1]
    The array length is the smallest possible based on the problem constraints. It helps us verify if the code can handle the lower limit of the array size.

  2. All Elements are the Same
    Input: [5, 5, 5]
    Output: [0, 0, 0]
    This tests if the code can handle an array where all elements are the same, and therefore, all differences are zero.

  3. Descending Order (violates constraint)
    Input: [5, 4, 3]
    Although this is outside of the problem constraints (array should be sorted in non-decreasing order), it’s worth noting that the solution should not work for such cases, or it needs to first sort the array.

By examining these examples, we not only cover a broad input space but also understand how different characteristics, like element duplication or large gaps, affect the problem. It’s crucial that the solution handles all these scenarios well.

Visualizing these cases can provide a better understanding of the problem and solution dynamics. One effective way to do this is to use graphs or plots. Below are some visualization ideas for the mentioned cases:

Single Digit Elements

Plot a simple bar graph with 1 and 2 on the x-axis and their respective output (1) on the y-axis. It shows a 1-to-1 relationship.

Duplicate Elements

Again, a bar graph with all 3s on the x-axis and zeroes on the y-axis. This visually demonstrates that when all elements are equal, the output is an array of zeros.

Larger Gaps

Plot a line graph with points at 1 and 100 on the x-axis, showing a massive gap between them. Place markers at these points with y-values 99, showing that the output for both is the same and significantly large.

Multiple Large Gaps

You could use a line graph with points at 1, 50, and 100. Draw lines to represent the gaps between these points, and place markers for the resulting output. This shows how the edges have the same large output and the middle point has a lower value.

Edge Cases

  1. Minimum Array Length: A simple bar graph for the two elements would suffice.
  2. All Elements are the Same: A bar graph where all bars have zero height will demonstrate the zero differences clearly.
  3. Descending Order: A plot showing a descending order of elements with markers or annotations indicating that this is outside of the problem constraints.

Visualizations like these can offer a deeper insight into how the problem behaves under various circumstances, ultimately aiding in designing a more robust solution.

From analyzing the different cases, key insights emerge that can guide the problem-solving approach:

  1. Single Digit Elements: The difference is minimal, and the results mirror these minimal differences. It indicates that the magnitude of numbers plays a role in the outcome.

  2. Duplicate Elements: When all elements are the same, the output will be an array of zeros. This can be a shortcut in the solution if we encounter such a case, avoiding unnecessary calculations.

  3. Larger Gaps: A single large gap between two elements results in a significant output value. This suggests that the distance between numbers is critical, and a more significant distance results in higher output values.

  4. Multiple Large Gaps: Having more than one large gap shows that the numbers in the middle can have smaller output values compared to the edges. It highlights that the position of a number in the sorted array affects its resulting sum of differences.

  5. Edge Cases:

  • Minimum Array Length: With only two elements, the problem becomes trivial, and the output is the absolute difference between the two, mirrored for each element.
  • All Elements are the Same: This results in an output array of zeros, offering a shortcut in the solution.
  • Descending Order: This is outside the problem constraints but, if considered, would require the array to be sorted first.

These insights point towards the importance of element magnitude, distance between elements, and their position in the sorted array. Understanding these aspects can help in formulating an efficient approach to solve the problem.

Identification of Applicable Theoretical Concepts

  1. Sorting: The input array is already sorted in non-decreasing order, which is a major simplification. Knowing that the array is sorted means we can exploit this order for optimized calculations.

  2. Mathematical Symmetry: The absolute difference operation has symmetrical properties. That is, |a - b| = |b - a|. This property can be used to avoid redundant calculations.

  3. Prefix Sum: Since we are repeatedly calculating the summation of differences for each element, a prefix sum can help reduce the time complexity of these summations. This concept can be adapted to solve the problem more efficiently than brute-force methods.

  4. Arithmetic Progression: Being a sorted array, if elements form an arithmetic progression, you can use the formula for the sum of an arithmetic series to calculate the answer quickly.

  5. Dynamic Programming: The problem involves repetitive calculations that can be optimized using dynamic programming techniques. A bottom-up approach can store already computed sums and use them to calculate the subsequent sums.

  6. Divide and Conquer: Though not strictly necessary, in larger problem scopes, divide-and-conquer could be used to break down the problem into smaller sub-problems and solve them individually.

  7. Spatial Locality: The problem inherently benefits from considering the ’local’ nature of each number in the array—how it relates to its immediate neighbors—because of the sorted nature of the array.

Understanding these concepts and properties can guide the design of an efficient algorithm for solving the problem.

Simple Explanation

Imagine you have a row of people standing in a line based on their heights, shortest to tallest. Now, each person wants to know how different their height is from everyone else’s. They would walk down the line and measure the height difference with every other person.

To make it simpler, think of it as a row of flower pots of different heights. Each flower pot wants to know the total height difference between itself and all other flower pots in the line. So, you pick one flower pot and then measure how much taller or shorter it is compared to each of the other flower pots. You add up all these differences, and that’s the number you write down for that flower pot. You do the same for each flower pot in the line.

The challenge is to find a quick way to calculate these numbers for all the flower pots without having to measure each one against all others one by one.

Problem Breakdown and Solution Methodology

Let’s visualize this as a process of stacking and unstacking boxes of different sizes on a shelf.

The Approach

  1. Initial Setup:

    • Imagine you have a shelf for each number in the list, and you’re going to stack boxes on each shelf. The height of each box represents the number’s value.

  2. Understand the Neighbors:

    • Every shelf interacts only with its neighbors. Think of this as placing or removing boxes to make the heights match or identify the difference.

  3. Iterative Comparison:

    • Starting from the first shelf, you compare it with its neighbors one by one. Stack or unstack boxes to equalize the heights and count the boxes moved.

  4. Summation and Movement:

    • Once you’ve counted all the boxes moved for one shelf, jot that down. Then move on to the next shelf and repeat the process.

  5. Loop Completion:

    • Continue this process for all shelves. Each shelf will now have a count of boxes moved, and that forms your answer.

  6. Optimization:

    • Instead of comparing every shelf with every other one, you could use a trick. You can find the total box difference for the first shelf and then cleverly update that for each subsequent shelf.

Changes in Parameters

  • If more numbers are added to the list, that means more shelves and boxes. The complexity increases but the approach remains the same.
  • If the numbers are larger, the height of each box would be higher. This doesn’t fundamentally change our approach.

Example Case

Let’s use the example: [2, 3, 5]

  1. Initial Setup: Three shelves for numbers 2, 3, 5.

  2. Understand the Neighbors:

    • For the first shelf (2), its neighbors are 3 and 5.

  3. Iterative Comparison:

    • 2 is 1 less than 3, and 3 less than 5. So, 1 + 3 = 4 boxes moved.

  4. Summation and Movement:

    • Write down 4 for the first shelf.
    • Move to the second shelf (3), it’s 1 more than 2 and 2 less than 5. So, 1 + 2 = 3 boxes moved.
    • For the third shelf (5), it’s 3 more than 2 and 2 more than 3. So, 3 + 2 = 5 boxes moved.

  5. Loop Completion:

    • You’ve written down the box count for all shelves.

  6. Output:

    • The box counts are [4, 3, 5], which is your answer.

By breaking down the problem in this manner, each part becomes a manageable task, leading us to the solution.

Inference of Problem-Solving Approach from the Problem Statement

The key terms in this problem guide the approach for solving it. Here are the important terms:

  1. Integer Array (nums) Sorted in Non-Decreasing Order:

    • This tells us that we’re working with a list of sorted numbers, which often allows for more efficient algorithms. It guides us to use the properties of sorted arrays for quick calculations.

  2. Same Length (result):

    • This specifies that the output array must be the same length as the input, which gives us a space constraint and helps in pre-allocating memory for the result.

  3. Summation of Absolute Differences:

    • The absolute difference between numbers is non-negative and commutative. This informs our strategy to use simple loop constructs to calculate the sum without worrying about the sign.

  4. Constraints on Array Length and Element Value:

    • Knowing that nums.length is between 2 and 105 and elements are between 1 and 104 helps in assessing the computational complexity we can afford.

  5. Indexing (0-indexed):

    • The fact that we are working with zero-based indexing informs how we’ll set up our loops and make comparisons between elements.

Each of these terms or concepts introduces conditions that guide the approach for solving this problem. For example, the sorted nature of the array points us towards exploiting this order for more efficient calculations. The constraints on length and values help us determine the feasibility of different algorithms based on their time complexity. The term “absolute differences” steers us toward a calculation approach that does not have to account for the sign of numbers.

Drawing tables or diagrams can be an effective way to visualize key properties and constraints of the problem. Here’s how you can do it:

  1. Sorted Integer Array (nums):

    • Create a table with the array indices as columns and the corresponding array elements as values below the indices. This table visually reinforces that the array is sorted, making it easier to spot patterns or trends.

  2. Same Length (result):

    • Next to the first table, draw another empty table with the same number of columns. This helps visualize that the output will be of the same length as the input.

  3. Summation of Absolute Differences:

    • You can draw arrows between elements in the nums array to represent the absolute difference calculations. This shows how each element relates to all other elements in the array.

  4. Constraints on Array Length and Element Value:

    • These could be noted as text next to your tables or represented by boundary lines in the table. For example, if you know the maximum length is 105, you could draw a boundary after the 105th column.

  5. Indexing (0-indexed):

    • Make sure the column indices in your tables start at 0 to reflect the 0-based indexing. This helps ensure that you account for it correctly in your calculations.

  6. Example Cases:

    • In a separate section, you could list example input arrays and their expected output arrays. For each example, you could show a step-by-step calculation for one or two result elements to demonstrate how they should be computed.

  7. Edge Cases:

    • Create additional tables to specifically represent edge cases like the smallest or largest possible arrays, or special conditions like all elements being equal.

By laying out these elements visually, you can get a clearer understanding of how each part of the problem and its constraints interact, helping you formulate a strategy for solving it.

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Stepwise Refinement

  1. Stepwise Refinement:

    1. Initialize Output Array:

      • Start by initializing an empty output array, result, with the same length as nums.

    2. Calculate Individual Sums:

      • Iterate through each element in nums. For each element, calculate the sum of the absolute differences between that element and all other elements in nums.

    3. Store in Output Array:

      • Store each calculated sum in the corresponding position in result.

    4. Return Result:

      • After completing the iterations and populating result, return it as the output.

  2. Granular, Actionable Steps:

    1. Initialize Output Array:

      • Declare an empty list called result that has the same length as nums.

    2. Outer Loop:

      • Loop i from 0 to the length of nums - 1.

    3. Inner Loop and Sum Calculation:

      • Inside the outer loop, initialize a variable sum to 0.
      • Loop j from 0 to the length of nums - 1.
        • If i is not equal to j, add the absolute difference between nums[i] and nums[j] to sum.

    4. Update Result:

      • Set result[i] equal to sum.

    5. Return Result:

      • Exit the loop and return the result array.

  3. Independent Parts:

    • The calculation for each element in the result array is independent of the others. You could, theoretically, calculate multiple elements in parallel.

  4. Repeatable Patterns:

    • The act of iterating through the array and calculating the sum of absolute differences is a repeatable pattern. For each element nums[i], the same series of calculations is performed.

By following this stepwise refinement, you can break down the high-level approach into specific, actionable steps, making it easier to implement the solution.

Solution Approach and Analysis

  1. Initialize Result Array:

    • Our first step is to initialize an empty array, result, which will store the final answers. This array will be the same length as the input array nums.

  2. Iterate Through nums:

    • Next, we iterate through each element in the nums array. For each element, we have to find how far it is from every other element in the array.

  3. Calculate Sums:

    • As we’re iterating through nums, we initialize a variable sum to zero for each element.
    • For each element, we use another loop to go through the array again and add the absolute difference between the current element and every other element to sum.

  4. Store in Result Array:

    • The sum we calculate for each element is then stored in the result array at the corresponding index.

  5. Return Result Array:

    • Finally, we return the result array as the answer.

Metaphor: Think of each element in nums as a house in a neighborhood. For each house, you want to find out how far it is from all the other houses. You then add up those distances. You do this for every house and store those total distances in a separate list.

Changes in Parameters:

  • Increasing the length of nums would exponentially increase computation time unless optimized.
  • If nums contains many repeated elements, we could optimize by performing calculations for unique numbers and then copying the results for the duplicates.

Example:

For nums = [2, 3, 5], our result array will also have 3 elements.

  • For 2:

    • The distances to the other elements are |2-3| = 1 and |2-5| = 3.
    • Sum = 1 + 3 = 4
    • result[0] = 4

  • For 3:

    • The distances to the other elements are |3-2| = 1 and |3-5| = 2.
    • Sum = 1 + 2 = 3
    • result[1] = 3

  • For 5:

    • The distances to the other elements are |5-2| = 3 and |5-3| = 2.
    • Sum = 3 + 2 = 5
    • result[2] = 5

Our final result array is [4, 3, 5], which is the output.

This step-by-step approach aligns well with the problem requirements and can be adapted based on specific constraints and conditions.

Identify Invariant

The invariant in this problem is the sorted nature of the input array nums. It’s sorted in non-decreasing order. This invariant could potentially be exploited for optimization but isn’t strictly necessary for solving the problem as stated.

Invariants are properties that remain unchanged throughout the execution of a piece of code or during the problem-solving process. Identifying them can often lead to insights that simplify the algorithm or make it more efficient. In this case, knowing that the array is sorted could guide us to consider algorithms or approaches that take advantage of sorted arrays. However, the problem itself doesn’t necessitate using this property for a basic solution.

Identify Loop Invariant

What is the loop invariant in this problem?

Is invariant and loop invariant the same for this problem?

Identify Recursion Invariant

Is there an invariant during recursion in this problem?

Is invariant and invariant during recursion the same for this problem?

Thought Process

  1. Identify Requirements: You are given a sorted array, and for each element in that array, you need to find the sum of the absolute differences between that element and all other elements.

  2. Generate Insights: The array is sorted. We could potentially exploit this property.

  3. Solve for a single element: Before solving for all elements in the array, let’s think about how to solve this for one element. You could iterate through all other elements, calculate the absolute difference and sum it up.

  4. Optimize for All Elements: If you apply the single-element solution naively for all elements, the solution would be (O(N^2)), which may not be efficient for large arrays. We can try to optimize it.

  5. Exploit Sorted Array: Since the array is sorted, differences between an element and elements on its left would always be negative and with elements on its right would be positive. This insight can lead to an optimized solution.

  6. Iterative Summation: You can calculate the sum iteratively rather than recalculating it each time, which would lead to an (O(N)) solution.

Code:

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def sum_of_abs_diffs(nums):
    n = len(nums)
    result = [0] * n

    # Calculate the prefix sums for easier computation later.
    prefix_sum = [0]
    for num in nums:
        prefix_sum.append(prefix_sum[-1] + num)

    # Compute the result for each index i.
    for i in range(n):
        # Sum of absolute differences with elements on the right of i
        right_sum = prefix_sum[-1] - prefix_sum[i + 1] - (n - i - 1) * nums[i]

        # Sum of absolute differences with elements on the left of i
        left_sum = i * nums[i] - prefix_sum[i]

        result[i] = left_sum + right_sum

    return result

# Test the function
nums = [2, 3, 5]
print(sum_of_abs_diffs(nums))  # Output should be [4, 3, 5]

nums = [1, 4, 6, 8, 10]
print(sum_of_abs_diffs(nums))  # Output should be [24, 15, 13, 15, 21]

Cues and Directions

  • The sorted nature of the array is a cue that we can optimize our solution.
  • We used prefix sums to make the sum calculation efficient.

This is how you can approach this type of problem by first understanding the basic requirements, generating insights, and then methodically solving it.

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from typing import List

class Solution:
    def getSumAbsoluteDifferences(self, nums: List[int]) -> List[int]:
        n = len(nums)
        result = [0] * n

        # Calculate the prefix sums for easier computation later.
        prefix_sum = [0]
        for num in nums:
            prefix_sum.append(prefix_sum[-1] + num)

        # Compute the result for each index i.
        for i in range(n):
            # Sum of absolute differences with elements on the right of i
            right_sum = prefix_sum[-1] - prefix_sum[i + 1] - (n - i - 1) * nums[i]

            # Sum of absolute differences with elements on the left of i
            left_sum = i * nums[i] - prefix_sum[i]

            result[i] = left_sum + right_sum

        return result

Here’s what the function does:

  1. It first calculates the prefix sum of the array nums.
  2. For each index i, it computes the sum of absolute differences using both the prefix sum and the property of the sorted array to do so efficiently.

This should meet the requirements of the problem statement.

Establishing Preconditions and Postconditions

  1. Parameters:

    • The method takes one input parameter nums.
    • The type of this parameter is List[int].
    • This list represents a sequence of integers sorted in non-decreasing order. The list size and its elements are within the problem’s constraints.

  2. Preconditions:

    • nums must be a sorted list with at least 2 elements, and each element should be an integer between 1 and 10^4.
    • There are no specific requirements on the state of the program itself before the method is called.

  3. Method Functionality:

    • This method calculates the sum of absolute differences between each element and every other element in the array.
    • It uses the input array nums to compute this and does not modify the input array.

  4. Postconditions:

    • The method returns a new list containing the sum of absolute differences for each index.
    • The returned list is of type List[int] and has the same length as the input list.
    • There are no side effects; the method is pure and does not modify any state.

  5. Error Handling:

    • The method assumes that the preconditions about the input are met. If they are not met, the behavior is undefined as per this implementation.
    • It does not explicitly throw any exceptions or return special values for invalid input. To handle such cases, one could add additional checks and validations.

By understanding these aspects, we gain a complete picture of what the method does and how it should behave. This also guides how we might test or debug this method during development.

Problem Decomposition

  1. Problem Understanding:

    • We are given a sorted list of integers. For each element in the list, we have to find the sum of its absolute differences with all other elements. Finally, we have to return a new list where each element corresponds to this sum for the respective index in the input list.

  2. Initial Breakdown:

    • Calculate the sum of absolute differences for each element.
    • Store these sums in a new list.

  3. Subproblem Refinement:

    • For each element, loop through the list to find its absolute difference with every other element.
    • Sum up these differences.
    • Store the sum in the new list.

  4. Task Identification:

    • Finding the absolute difference between two numbers.
    • Summing up a list of numbers.
    • These tasks are repeated for each element in the input list.

  5. Task Abstraction:

    • CalculateAbsoluteDifference(a, b): Returns |a - b|
    • CalculateSumOfList(lst): Returns the sum of all elements in lst.
    • These tasks are abstracted enough to be clear and reusable but still make sense in this context.

  6. Method Naming:

    • CalculateAbsoluteDifference
    • CalculateSumOfList

  7. Subproblem Interactions:

    • Firstly, the CalculateAbsoluteDifference task will be used within a loop for each element in the list.
    • The resulting differences will then be summed up using the CalculateSumOfList task.
    • Finally, this sum will be stored in a new list.
    • The tasks are dependent on the input list, but independent of each other. They need to be performed sequentially for each element in the list.

By identifying these various components and their interactions, we can have a structured approach to solving the problem.

From Brute Force to Optimal Solution

Brute Force Solution

In the brute-force approach, for each element in the list, we go through every other element to find the absolute difference and sum these up. Here’s how we can code this:

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from typing import List

def getSumAbsoluteDifferences(nums: List[int]) -> List[int]:
    result = []
    for i in range(len(nums)):
        sum_diff = 0
        for j in range(len(nums)):
            if i != j:
                sum_diff += abs(nums[i] - nums[j])
        result.append(sum_diff)
    return result

Inefficiencies:

  1. Time Complexity: O(n^2) - We are looping through the list for each element.
  2. Space Complexity: O(n) - Storing the result in a new list of size n.

Optimized Solution

The brute-force solution is not efficient due to its high time complexity. We can optimize this by observing patterns and reducing the repeated work.

Key Observation:

For an element nums[i] at index i, the sum of its absolute differences with all other elements can be calculated as follows:

  • Sum of greater elements: (n - i - 1) * nums[i]
  • Sum of smaller elements: i * nums[i]
  • result[i] = (Sum of greater elements - Sum of smaller elements)

Steps:

  1. Pre-calculate the sum of the entire list.
  2. Loop through the list only once, using the key observation to calculate the sum of absolute differences for each element.

Here’s the code:

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from typing import List

def getSumAbsoluteDifferences(nums: List[int]) -> List[int]:
    n = len(nums)
    total_sum = sum(nums)
    left_sum = 0
    result = []
    
    for i in range(n):
        total_sum -= nums[i]  # Remaining sum after excluding nums[i]
        
        sum_diff = (total_sum - (n - i - 1) * nums[i]) + (nums[i] * i - left_sum)
        result.append(sum_diff)
        
        left_sum += nums[i]  # Sum of elements before nums[i]
        
    return result

Impact on Time and Space Complexity:

  1. Time Complexity: O(n) - We loop through the list only once.
  2. Space Complexity: O(n) - We still store the result in a new list, but we have greatly reduced the time complexity.

The optimized solution is significantly more efficient than the brute-force approach.

Code Explanation and Design Decisions

  1. Initial Parameters:

    • nums is the list of integers for which we have to find the sum of absolute differences for each element.
    • n is the length of nums, used for iteration.
    • total_sum stores the sum of all elements in nums.
    • left_sum stores the sum of all elements before the current element.

  2. Primary Loop:

    • The loop iterates through each element in nums.
    • Each iteration calculates the sum of absolute differences for one element and appends it to result.

  3. Conditions or Branches:

    • total_sum -= nums[i] removes the current element from the sum, adjusting for the calculation.
    • sum_diff calculates the sum of absolute differences, adhering to the formula based on the observation.
    • There are no conditional branches (if, else) here, just straightforward calculations.

  4. Updates or Modifications:

    • total_sum is decremented by nums[i] to exclude the current element.
    • left_sum is incremented by nums[i] to include the current element.
    • These changes reflect how we’re moving through nums and considering different sets of elements for each index.

  5. Invariant:

    • total_sum always contains the sum of the elements to the right of the current index.
    • left_sum always contains the sum of the elements to the left of the current index.
    • These invariants help us efficiently calculate sum_diff.

  6. Significance of Final Output:

    • The result list contains the sum of absolute differences for each element in nums.
    • It satisfies the problem’s requirement of finding these sums for all elements in an efficient manner.

The code is designed to solve the problem by leveraging pre-calculated sums and updating them as it iterates through the list, thereby reducing the time complexity.

Coding Constructs

  1. High-Level Strategies:

    • Pre-computation of sums for efficient look-up.
    • Iteration over the input list to update sums dynamically.

  2. Purpose for Non-Programmer:

    • Imagine you have a row of numbers. This code finds, for each number, the sum of its differences with all other numbers in the row.

  3. Logical Elements:

    • Looping constructs for iteration.
    • Variables for storing intermediate sums.
    • An array to store the final results.

  4. Algorithmic Approach in Plain English:

    • First, calculate the sum of all numbers.
    • Start from the first number, and for each number, figure out its total difference from all other numbers.
    • Keep track of the sum of numbers that you’ve already looked at and the sum of numbers that you haven’t yet looked at. This helps to quickly find the next total difference.

  5. Key Steps or Operations:

    • Calculate the initial total sum of all elements.
    • Iterate through each element:
      • Find the sum of absolute differences for the current element.
      • Update the running totals (sum of elements before and sum of elements after the current element).
      • Store the result.

  6. Algorithmic Patterns:

    • Dynamic Programming: Stores intermediate sums to avoid redundant calculations.
    • Iteration: Goes through the list once, making the algorithm O(n) in time complexity.

This approach effectively breaks down the complex task of finding the sum of absolute differences for each element by iteratively updating and reusing pre-calculated sums.

Language Agnostic Coding Drills

  1. Distinct Coding Concepts:

    • Variable Initialization: The act of declaring variables and setting initial values.
    • List Handling: Creating and manipulating lists.
    • Loops: Using loops for iteration.
    • Summation: Adding up elements in a list.
    • Mathematical Operations: Basic arithmetic like subtraction and absolute value.
    • Dynamic Updates: Changing the value of variables based on loop iterations.
    • Result Aggregation: Storing intermediate results in a list.

  2. Order of Increasing Difficulty:

    • Variable Initialization: Basic skill needed for almost every program.
    • List Handling: A step above basic variable handling, but fundamental in many programs.
    • Mathematical Operations: Understanding how to perform basic calculations.
    • Loops: Requires understanding of flow control.
    • Summation: Involves both looping and mathematical operations.
    • Result Aggregation: Requires list handling and an understanding of how to collect and store results.
    • Dynamic Updates: More advanced as it involves modifying variables dynamically during loop iterations.

  3. Problem-Solving Approach:

    • Understand the Problem: Break down the problem statement into key components.
    • Initialize Variables: Declare variables to store the total sum, a running sum, and an array for the final results.
    • Calculate Initial Sum: Use a loop to go through the input list and calculate the initial sum of all elements.
    • Iterate Through List: Use another loop to go through each element in the list.
      • Calculate Current Sum of Differences: Use the initial sum, the current element, and its position to calculate the sum of absolute differences for the current element.
      • Store the Result: Append the calculated value to the result list.
      • Update Running Sums: As we progress through the loop, update the running sums for use in the next iteration.
    • Return Result: The final list contains the sum of absolute differences for each element in the input list.

Each coding drill prepares you for the next. Variable initialization sets the stage. List handling allows you to manage your data. Loops and summation build the engine of your solution. Mathematical operations are the tools for your engine. Dynamic updates and result aggregation give your solution flexibility and completeness. Together, they form a cohesive strategy for solving the problem.

Targeted Drills in Python

1. Coding Drills for General Concepts:

Variable Initialization

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# Declare a variable and initialize it
x = 0

List Handling

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# Create a list and append an element to it
my_list = [1, 2, 3]
my_list.append(4)

Mathematical Operations

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# Basic arithmetic operations
addition = 5 + 3
subtraction = 5 - 3

Loops

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# Simple for loop to iterate through a list
for i in [1, 2, 3]:
    print(i)

Summation

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# Summing elements in a list
total = 0
for i in [1, 2, 3]:
    total += i

Result Aggregation

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# Aggregating results into a list
result = []
for i in [1, 2, 3]:
    result.append(i * 2)

Dynamic Updates

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# Dynamically update a variable within a loop
running_sum = 0
for i in [1, 2, 3]:
    running_sum += i

2. Problem-Specific Drills:

Calculate Absolute Difference

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# Calculate the absolute difference between two numbers
diff = abs(5 - 3)

Why Essential: The problem requires calculating the sum of absolute differences.

3. Integrating Drills to Solve the Problem:

  1. Variable Initialization: Start by initializing variables to store the running sum of elements, the total sum of elements, and an empty list for the result.

  2. List Handling: Use list handling methods to work with the list of numbers and to store results.

  3. Loops: Use a loop to iterate through the list of numbers. Another loop will calculate the sum of all elements.

  4. Mathematical Operations and Summation: Within the loop, use basic math operations to calculate the sum of absolute differences for each element. Use the running sum for efficiency.

  5. Result Aggregation: Append the calculated sum of absolute differences for each element to the result list.

  6. Dynamic Updates: After each iteration, update the running sum to account for the next element.

  7. Problem-Specific Drill: Use the abs() function to calculate the absolute difference between two numbers as needed.

By integrating these drills in the correct order, you can build a cohesive and efficient solution for the initial problem. Each drill adds a layer of complexity, gradually building up to the final solution.

Q&A

Similar Problems

Here are 10 problems that use similar underlying concepts:

  1. Two Sum: This problem also involves iterating over a list and finding complementary elements. It employs list handling and dynamic updates, much like our original problem.

  2. Reverse Integer: It involves handling individual digits of a number, which requires similar variable initialization and mathematical operations.

  3. Single Number: Requires iterating over a list to identify an element with specific properties, employing loops, and variable updates.

  4. Maximum Subarray: This problem uses dynamic programming and needs to keep track of running totals, similar to our running sum.

  5. Best Time to Buy and Sell Stock: Similarities include use of dynamic variables to keep track of minimum and maximum values while iterating over a list.

  6. Product of Array Except Self: This involves calculating products while iterating over a list, similar to how we calculated the sum of absolute differences.

  7. Contains Duplicate: Requires iterating through a list and checking for element properties. Utilizes list handling and dynamic updates.

  8. Rotate Array: Also involves handling lists and iterating through them to perform operations on each element.

  9. Maximum Product Subarray: Like our problem, this also uses dynamic programming and loop iterations to calculate maximum product possible in subarrays.

  10. Find Minimum in Rotated Sorted Array: Similar in that it involves iterating over an array and applying mathematical comparisons to find a specific value.

Each of these problems involves at least one of the coding drills or problem-solving strategies used in our original problem, such as loops, dynamic updates, mathematical operations, or result aggregation.