Zero array Transformation I, II

XXX. Zero Array Difference

Difference Array

Range Update

Binary Search

Problem Statement:

You are given an array nums of integers and a list of queries queries. Each query consists of one of the following:

• For Zero Array Difference, each query is a range [start, end] that decrements all elements in the range by 1.
• For Zero Array Transformation II, each query is a range [start, end, val] that increments the range by val.

The goal is to determine if it is possible to make all elements in nums non-negative using the provided queries.

Algorithm:

Zero Array Difference:

1. Initialize a Difference Array:

   Create a difference array diff to represent incremental changes for range updates.

2. Process Each Query:

   Decrement diff[start] and increment diff[end + 1] to reflect the range decrement operation.

3. Apply the Difference Array:

   Use cumulative sums to apply the effects of the difference array to nums.

4. Check Validity:

   If any element in nums becomes negative, return false; otherwise, return true.

Zero Array Transformation II:

1. Binary Search:

   Use binary search to determine the minimum number of queries required to make all elements non-negative.

2. Range Updates with Difference Array:

   Increment diff[start] and decrement diff[end + 1] for the first m queries.

3. Validation:

   Check if applying the cumulative sums of the difference array satisfies the non-negativity condition for nums.

Complexity:

Zero Array Difference: O(n + q), where n is the size of the array and q is the number of queries.
Zero Array Transformation II: O((n + q) log(q)), due to the binary search over the queries.

Java Implementations:

Zero Array Difference:

class Solution {
    // Difference Array Solution
    public boolean isZeroArrayDiff(int[] nums, int[][] queries) {
        int n = nums.length;
        int[] diff = new int[n + 1]; // Difference array for range updates

        // Process each query to update the difference array
        for (int[] query : queries) {
            int start = query[0]; // Start index of the range
            int end = query[1];   // End index of the range

            diff[start]--; // Decrement at the start of the range
            if (end + 1 < n) diff[end + 1]++; // Increment just after the end of the range (if within bounds)
        }

        int cumulative = 0; // Variable to track cumulative range updates
        for (int i = 0; i < n; i++) {
            cumulative += diff[i]; // Calculate the cumulative effect on each index
            nums[i] += cumulative; // Apply the cumulative effect to nums[i]
            if (nums[i] < 0) return false; // If any value becomes negative, transformation fails
        }

        return true; // If all elements can be zeroed, return true
    }
}

Zero Array Transformation II:

class Solution {
    public int minZeroArray(int[] nums, int[][] queries) {
        int n = nums.length;

        // Check if all elements can be zeroed with all queries
        if (!check(nums, queries, queries.length)) return -1;

        // Binary search to find the minimum number of queries
        int l = 0, r = queries.length;
        while (l <= r) {
            int m = l + (r - l) / 2;
            if (check(nums, queries, m)) {
                r = m - 1; // Try for fewer queries
            } else {
                l = m + 1; // Increase query range
            }
        }

        return l; // Minimum number of queries required
    }

    private boolean check(int[] nums, int[][] queries, int m) {
        int n = nums.length;
        int[] arr = new int[n + 1]; // Difference array for range updates

        // Apply only the first m queries
        for (int i = 0; i < m; i++) {
            int start = queries[i][0], end = queries[i][1], val = queries[i][2];
            arr[start] += val;
            if (end + 1 < n) arr[end + 1] -= val;
        }

        int cur = 0; // Cumulative sum to apply difference array
        for (int i = 0; i < n; i++) {
            cur += arr[i]; // Apply cumulative sum to nums[i]
            if (cur < nums[i]) return false; // If any value cannot be zeroed, return false
        }
        return true; // All elements can be zeroed
    }
}