Problem Description

You are given two positive integer arrays nums and target, both of the same length. The goal is to transform the nums array into the target array using the least number of operations. An operation consists of choosing any subarray within nums and either incrementing or decrementing each element in that subarray by 1. Your task is to determine the minimum number of operations required to achieve this transformation.

Intuition

To solve this problem, let's first consider the difference between the corresponding elements of the arrays nums and target. We can define a difference array where each element is the difference between the elements at the same position in the two arrays. To transform nums into target, the differences between the arrays must be neutralized.

The key observation is that if consecutive elements in the difference array have the same sign (both positive or both negative), these differences can be adjusted together by selecting a subarray, thus incrementing or decrementing them collectively. For an interval of differences where the signs are consistent, we should focus on adjusting the difference at the beginning. The absolute value of the first difference in such an interval should be included in the result as it denotes how much adjustment needs to be made initially. For subsequent differences, if the new difference is larger in absolute value than the previous one, the excess needs further adjustments, so the difference of the absolute values is added to the result.

This method efficiently considers the change needed for each transition between intervals of differences. Our approach iterates through the differences, updating a cumulative count of required operations to minimize the adjustments to transform nums into target.

Learn more about Stack, Greedy, Dynamic Programming and Monotonic Stack patterns.

Solution Approach

The solution to this problem uses a straightforward iterative approach to determine the minimum operations required to transform the nums array into the target array. Here's how we implement this:

1. Calculate Initial Difference:

   • Start by measuring the initial difference between the first elements of nums and target. Let's call this the initial cost, f. Calculate it as f = abs(target[0] - nums[0]).

2. Iterate through the Arrays:

   • For each subsequent element in the arrays, compute the difference x between the current elements of target and nums, and y as the difference of their previous elements.

3. Adjust Operations Based on Change in Difference:

   • If the differences x and y share the same sign, meaning they contribute in the same direction, we compute the additional required adjustment as d = abs(x) - abs(y).
   • If this calculated d is positive, we add it to f, as this indicates an increase in the absolute difference that needs to be adjusted.
   • If the signs of the differences x and y do not match, it means the trend has changed, necessitating the adjustment of the entire current difference abs(x) to the operation count f.

4. Return Total Operations:

   • Once the iteration completes, f holds the minimum number of operations required to make nums equal to target.

This algorithm efficiently computes the necessary operations by looking at the change points in the sequence of differences, leveraging the fact that adjustments can be made collectively when differences align in their directional change. The process has a linear complexity of O(n) due to its single traversal of the array, and it operates in constant space O(1), making it optimal for handling large arrays.

Example Walkthrough

Let's walk through an example to illustrate the solution approach. Suppose we have the following arrays:

• nums = [3, 1, 2]
• target = [1, 3, 2]

Our task is to transform nums into target with minimum operations. Here's how the algorithm works:

1. Calculate Initial Difference:

   • Start by looking at the first elements of both arrays: nums[0] = 3 and target[0] = 1.
   • Calculate the absolute difference: f = abs(target[0] - nums[0]) = abs(1 - 3) = 2.
   • This means we need 2 operations initially to transform the first element of nums to match target.

2. Iterate through the Arrays:

   • Move to the next pair of elements: nums[1] = 1, target[1] = 3.
   • Calculate the current difference x = target[1] - nums[1] = 3 - 1 = 2.
   • Calculate the previous difference y = target[0] - nums[0] = 1 - 3 = -2.

3. Adjust Operations Based on Change in Difference:

   • The differences x (positive) and y (negative) have opposite signs, indicating a change in the trend.
   • Because the signs are different, add the absolute difference of the current element to f: f = f + abs(x) = 2 + 2 = 4.

4. Continue Iterating:

   • For the next elements, nums[2] = 2 and target[2] = 2, the difference x = target[2] - nums[2] = 0.
   • With no difference between them, y, calculated previously as 2 (from step 3), becomes irrelevant as the difference is resolved.

5. Return Total Operations:

   • After completing the iteration, f holds the value 4, representing the minimum number of operations needed.

This example illustrates the principle of monitoring changes in difference trends throughout nums and target, applying batch operations to efficiently match the arrays. Each adjustment is calculated based on the change in difference trends, aiming to minimize the total operation count.

Solution Implementation

Time and Space Complexity

The given code calculates the minimum number of operations required to transform one list into another. The program iterates through the lists once, performing constant-time operations for each element. Hence, the time complexity is O(n), where n is the length of the lists nums and target.

The space complexity is O(1) since the algorithm uses only a fixed amount of additional space to store variables f, x, y, and d, regardless of the input size.

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