#3229

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Operations to Make Array Equal to Target

This problem requires calculating the minimum number of operations to transform one array into another using state transition dynamic programming.

array dynamic programming stack greedy monotonic stack

Problem Statement

You are given two arrays, nums and target, both of the same length. In a single operation, you can choose any subarray of nums and increment or decrement each element within that subarray by 1. Your task is to find the minimum number of operations required to make nums equal to target.

In this problem, you need to utilize state transition dynamic programming. The key observation is to compute the difference nums'[i] = nums[i] - target[i]. By transforming this difference into an array of all zeroes, you minimize the number of operations needed. The goal is to make nums' into an array of zeroes using the fewest steps possible.

Examples

Example 1

Input: nums = [3,5,1,2], target = [4,6,2,4]

Output: 2

We will perform the following operations to make nums equal to target : - Increment nums[0..3] by 1, nums = [4,6,2,3] . - Increment nums[3..3] by 1, nums = [4,6,2,4] .

Example 2

Input: nums = [1,3,2], target = [2,1,4]

Output: 5

We will perform the following operations to make nums equal to target : - Increment nums[0..0] by 1, nums = [2,3,2] . - Decrement nums[1..1] by 1, nums = [2,2,2] . - Decrement nums[1..1] by 1, nums = [2,1,2] . - Increment nums[2..2] by 1, nums = [2,1,3] . - Increment nums[2..2] by 1, nums = [2,1,4] .

Constraints

1 <= nums.length == target.length <= 105
1 <= nums[i], target[i] <= 108

Solution Approach

State Transition Approach

First, calculate the difference nums'[i] = nums[i] - target[i] for each index. Then, the objective becomes to make all elements of nums' equal to zero using the minimum number of operations. This is done by selecting subarrays and adjusting them through increments and decrements.

Greedy and Dynamic Programming Combination

The key challenge here is to identify the right subarrays to apply the increments or decrements. A greedy strategy combined with dynamic programming helps minimize the number of operations by tracking the current state and efficiently computing the next steps.

Monotonic Stack Optimization

A monotonic stack can be used to optimize the transitions between states. By keeping track of the changes in nums' efficiently, the number of operations required to bring the entire array to zero can be minimized.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time and space complexity depend on the chosen approach. A direct dynamic programming solution could lead to O(n) time complexity, where n is the length of the array. However, optimizations like monotonic stacks could reduce the time complexity further by improving the handling of subarray transitions. Space complexity will also depend on the implementation, with an O(n) space complexity for storing the difference array in the basic case.

What Interviewers Usually Probe

Looking for familiarity with dynamic programming and greedy algorithms.
Candidate should demonstrate an understanding of how to minimize operations via subarray manipulation.
Interviewer may want to see an efficient solution involving space-time trade-offs.

Common Pitfalls or Variants

Common pitfalls

Misunderstanding the problem and trying to apply simple operations without considering subarray manipulations.
Failing to optimize the solution for large arrays, resulting in excessive time or space complexity.
Ignoring edge cases such as arrays where nums and target are already equal, which could result in an unnecessary increase in operations.

Follow-up variants

Changing the operation constraint from only subarrays to any segment.
Introducing more complex operations such as multiplication or division within the transformation.
Allowing partial matching or approximations of nums to target.

FAQ

What is the minimum operations to make an array equal to target?

The problem requires transforming one array into another using the least number of operations, where each operation consists of incrementing or decrementing a subarray.

How do I solve the problem using dynamic programming?

You solve this by calculating the difference between the two arrays at each index and applying state transition dynamic programming to minimize the required operations.

What are some common mistakes when solving this problem?

A common mistake is failing to consider the subarray manipulation efficiently or not optimizing the solution for large arrays.

What is the time complexity of this problem?

The time complexity can be O(n) in the basic case, but it may depend on the implementation details, especially if optimization techniques are applied.

What are some optimization techniques for this problem?

Techniques like using monotonic stacks or further optimizing the dynamic programming approach can reduce both time and space complexity.

terminal

Solution

Solution 1: Dynamic Programming

We can first calculate the difference between the arrays $\textit{nums}$ and $\textit{target}$. For a difference array, we find continuous intervals where the signs of the differences are the same. For each interval, we add the absolute value of the first element to the result. For the subsequent elements, if the absolute value of the difference is greater than the absolute value of the previous difference, we add the difference of the absolute values to the result.

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class Solution:
    def minimumOperations(self, nums: List[int], target: List[int]) -> int:
        n = len(nums)
        f = abs(target[0] - nums[0])
        for i in range(1, n):
            x = target[i] - nums[i]
            y = target[i - 1] - nums[i - 1]
            if x * y > 0:
                d = abs(x) - abs(y)
                if d > 0:
                    f += d
            else:
                f += abs(x)
        return f

Continue Practicing

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