Zero Array Transformation IV

This approach simulates the process for every possible number of queries k, from 1 up to the total number of queries. For each k, it checks if it's possible to make the array nums all zeros. This check is done by solving a subset sum problem for each element nums[j]. The subset sum problem itself is solved using a brute-force backtracking (or recursive) method, which explores all possibilities.

• For each possible number of queries k from 1 to queries.length:
  • Call a function canMakeZero(k) to check if nums can be turned into a zero array.
  • If canMakeZero(k) returns true, then k is the minimum number of queries, so return k.
• If the loop completes without finding a solution, return -1.
• The canMakeZero(k) function checks each index j of nums:
  • It gathers all val_i from the first k queries that are applicable to index j.
  • It uses a recursive backtracking function to check if nums[j] can be formed by a sum of a subset of these values.
  • If any index j fails this check, canMakeZero(k) returns false.
  • If all indices pass, it returns true.

The core idea is to test each potential answer k sequentially. For a given k, we must determine if there's a way to apply the first k queries to make every nums[j] zero. This decomposes into n independent subset sum problems. For each nums[j], we need to find if it can be expressed as a sum of some of the val_i from the first k queries that cover index j.

The backtracking function canFormSum explores all subsets of the applicable query values to see if any sum up to the target nums[j]. It does this by making two recursive calls at each step: one where the current value is included in the sum, and one where it's excluded.

class Solution {
    public int zeroArray(int[] nums, int[][] queries) {
        for (int k = 1; k <= queries.length; k++) {
            if (canMakeZero(nums, queries, k)) {
                return k;
            }
        }
        return -1;
    }

    private boolean canMakeZero(int[] nums, int[][] queries, int k) {
        for (int j = 0; j < nums.length; j++) {
            if (nums[j] == 0) continue;
            java.util.List<Integer> options = new java.util.ArrayList<>();
            for (int i = 0; i < k; i++) {
                if (queries[i][0] <= j && j <= queries[i][1]) {
                    options.add(queries[i][2]);
                }
            }
            if (!canFormSum(nums[j], options, 0)) {
                return false;
            }
        }
        return true;
    }

    private boolean canFormSum(int target, java.util.List<Integer> options, int index) {
        if (target == 0) return true;
        if (target < 0 || index == options.size()) return false;
        
        // Exclude current option
        if (canFormSum(target, options, index + 1)) {
            return true;
        }
        // Include current option
        if (canFormSum(target - options.get(index), options, index + 1)) {
            return true;
        }
        return false;
    }
}