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Zero Array Transformation II

Difficulty: Medium

Problem Description

You are given an integer array nums of length n and a 2D array queries where queries[i] = [l_i, r_i, val_i]. Each queries[i] represents the following action on nums:

  • Decrement the value at each index in the range [l_i, r_i] in nums by at most val_i.
  • The amount by which each value is decremented can be chosen independently for each index.

A Zero Array is an array with all its elements equal to 0. Return the minimum possible non-negative value of k, such that after processing the first k queries in sequence, nums becomes a Zero Array. If no such k exists, return -1.

Key Insights

  • The problem involves decrementing values in a specific range of an array multiple times and determining the minimum number of operations required to zero out the array.
  • Each query allows independent decrementing of values within a specified range, which suggests the use of a greedy approach to minimize the number of queries needed.
  • The constraints on the values and lengths of the arrays imply the need for an efficient algorithm, potentially utilizing binary search or a prefix sum technique.

Space and Time Complexity

Time Complexity: O(n log q), where n is the length of the nums array and q is the number of queries. This accounts for iterating through queries and potentially searching through nums. Space Complexity: O(n), primarily for storing the modified state of the nums array.

Solution

To solve the problem, we can use a binary search approach combined with a greedy strategy. The idea is to determine the maximum number of queries we can process such that the array can still be transformed into a Zero Array. We'll maintain a difference array to efficiently apply the decrements in the specified ranges. The binary search will help us find the smallest k such that the transformation is possible.

  1. For a given k, simulate applying the first k queries on nums using a difference array to track the decrements.
  2. After applying the decrements, check if all values in nums are zero or can be made zero with the allowed decrements.
  3. Use binary search over the range of possible query counts to find the minimum k.

Code Solutions

def min_queries_to_zero_array(nums, queries):
    def can_zero_with_k(k):
        # Initialize the difference array
        n = len(nums)
        diff = [0] * (n + 1)
        
        # Apply the first k queries using the difference array
        for i in range(k):
            l, r, val = queries[i]
            diff[l] -= val
            if r + 1 < n:
                diff[r + 1] += val
        
        # Compute the current state of nums after applying the differences
        current = 0
        for i in range(n):
            current += diff[i]
            nums[i] -= current
        
        # Check if all values are zero or can be made zero
        return all(x <= 0 for x in nums)
    
    left, right = 0, len(queries)
    result = -1
    
    while left <= right:
        mid = (left + right) // 2
        if can_zero_with_k(mid):
            result = mid
            right = mid - 1
        else:
            left = mid + 1
    
    return result
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