#1526

Hard

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Minimum Number of Increments on Subarrays to Form a Target Array

The problem asks for the minimum number of operations to transform an initial array of zeros into a target array using subarray increments.

array dynamic programming stack greedy monotonic stack

Problem Statement

You are given an integer array target, and an integer array initial of the same size as target, which initially consists of zeros. You can increment the values of any subarray in initial by one in one operation. Your task is to return the minimum number of operations required to transform initial into target.

For each operation, choose a subarray and increment every element within that subarray by one. The goal is to minimize the total number of operations needed. Consider the transitions and how the operations can be optimized using dynamic programming and greedy strategies.

Examples

Example 1

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

Output: 3

We need at least 3 operations to form the target array from the initial array. [0,0,0,0,0] increment 1 from index 0 to 4 (inclusive). [1,1,1,1,1] increment 1 from index 1 to 3 (inclusive). [1,2,2,2,1] increment 1 at index 2. [1,2,3,2,1] target array is formed.

Example 2

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

Output: 4

[0,0,0,0] -> [1,1,1,1] -> [1,1,1,2] -> [2,1,1,2] -> [3,1,1,2]

Example 3

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

Output: 7

[0,0,0,0,0] -> [1,1,1,1,1] -> [2,1,1,1,1] -> [3,1,1,1,1] -> [3,1,2,2,2] -> [3,1,3,3,2] -> [3,1,4,4,2] -> [3,1,5,4,2].

Constraints

1 <= target.length <= 105
1 <= target[i] <= 105

Solution Approach

State Transition Dynamic Programming

The problem can be solved using state transition dynamic programming by calculating the difference between consecutive elements in the target array. For each element in target, compute the minimum number of operations needed to increment it to the desired value by applying increments to the subarrays where applicable.

Greedy Strategy with Minimum Range Increments

In the greedy approach, the key observation is that for each range in the target array, we increment the entire range by the smallest value in that range to minimize the number of operations.

Monotonic Stack for Range Management

A monotonic stack can help track the required subarrays efficiently. By iterating through the target array and managing ranges with a stack, the solution ensures that the increments are applied in an optimal order.

Complexity Analysis

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

The time and space complexity depend on the final approach. The state transition dynamic programming approach typically results in a time complexity of O(n), where n is the length of the target array, and a space complexity of O(n). The greedy and monotonic stack-based solutions may also run in O(n) time with similar space complexity.

What Interviewers Usually Probe

Does the candidate effectively use state transition dynamic programming for the problem?
Is the candidate comfortable applying greedy strategies for range-based optimization?
Does the candidate demonstrate proficiency in managing subarray operations with a monotonic stack?

Common Pitfalls or Variants

Common pitfalls

Incorrectly handling the subarray increments could lead to more operations than necessary.
Failing to optimize range increments might result in a solution that exceeds the required number of operations.
Overcomplicating the solution with unnecessary nested loops or complex data structures can increase time complexity.

Follow-up variants

Modifying the problem to work with a non-zero initial array, requiring adjustments to the dynamic programming approach.
Allowing multiple increments on the same subarray, testing the candidate's ability to handle more complex operations.
Changing the target array values to be random or non-sequential, requiring more adaptive range-based strategies.

FAQ

What is the primary strategy to solve the 'Minimum Number of Increments on Subarrays to Form a Target Array' problem?

The primary strategy is to use state transition dynamic programming, focusing on minimizing the number of operations by calculating the differences between adjacent target values.

How does the greedy approach help in solving this problem?

The greedy approach helps by incrementing the entire subarray by the minimum value in that range, which ensures that the total number of operations is minimized.

What role does the monotonic stack play in this problem?

The monotonic stack helps manage the subarrays efficiently, ensuring that increments are applied in the most optimal order.

What are the time and space complexities of the 'Minimum Number of Increments on Subarrays to Form a Target Array' problem?

The time complexity is O(n), where n is the length of the target array, and the space complexity is O(n), assuming dynamic programming or greedy approaches are used.

Can this problem be solved using a brute force approach?

While a brute force approach is possible, it would be inefficient, as it would involve checking all possible subarray increments, leading to high time complexity.

terminal

Solution

Solution 1: Dynamic Programming

We define $f[i]$ as the minimum number of operations required to obtain $target[0,..i]$, initially setting $f[0] = target[0]$.

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class Solution:
    def minNumberOperations(self, target: List[int]) -> int:
        return target[0] + sum(max(0, b - a) for a, b in pairwise(target))

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