Zero Array Transformation III

This highly efficient approach reframes the problem using linear programming duality. The original problem of maximizing removed queries is transformed into a dual problem of minimizing a certain cost. This dual problem has a special structure that allows it to be solved optimally with a greedy algorithm, which is much faster than general-purpose max-flow or LP solvers.

• Surplus Calculation: As in other approaches, compute d[i] = total_coverage[i] - nums[i]. If any d[i] < 0, return -1.
• LP Duality: The problem can be formulated as a 0-1 integer program. By the principles of LP duality, this primal problem is equivalent to a dual LP: minimize sum(d[i] * y_i) subject to sum_{k=l_j to r_j} y_k >= 1 for each query j, and y_i >= 0. The optimal value of this dual problem is our answer.
• Greedy Algorithm: This specific type of dual LP can be solved optimally using a greedy algorithm. The strategy is to process queries in a specific order and satisfy their constraints using the 'cheapest' resources available.
  1. Sort all queries by their right endpoint r_j in ascending order.
  2. Initialize an array y of size n to all zeros. We use a Fenwick tree (BIT) to maintain y to support fast range sums and point updates.
  3. Build a segment tree over the d array to efficiently find the index i with the minimum d[i] within any range [l, r]. To handle ties, we store pairs (d[i], i) and prioritize the one with the larger index i.
  4. Iterate through the sorted queries [l, r]. For each query: a. Use the Fenwick tree to find the current sum S = sum_{k=l to r} y_k. b. If S < 1, the constraint for this query is not met. We need to increase the sum by needed = 1 - S. c. To do this with minimum cost, use the d-segment tree to find the index i* in [l, r] with the minimum d[i*] (tie-breaking with largest i*). d. Increase y[i*] by needed. This is done by updating the Fenwick tree at i*. The total cost increases by needed * d[i*].
• Result: The accumulated total cost is the answer. Since y_i can be fractional, the cost is a double. The final answer is this cost rounded to the nearest integer.

This method is the most optimal. After calculating the surplus array d, we solve the dual of the problem's LP formulation. The greedy strategy works as follows:

We sort the queries by their right endpoint. This order is crucial. We iterate through the sorted queries. For each query [l, r], we check if its corresponding dual constraint (sum of y_k >= 1) is satisfied. The y_k values represent a sort of 'dual price' we assign to each index. We can efficiently check the sum using a Fenwick tree.

If the constraint is not met, we need to increase the y values for indices in [l, r]. To minimize the total cost (sum d[i]y[i]), we should put the required increase onto the y_k where d[k] is smallest. To find this cheapest index k in the range [l, r] efficiently, we use a segment tree built on the d array. In case of ties in d values, we pick the rightmost index, as this has the best chance of helping satisfy constraints for subsequent queries (which are sorted by right endpoint).

We update the chosen y[k] in our Fenwick tree and add the cost incurred to our total. After processing all queries, the total accumulated cost gives the maximum number of queries we can remove.

class Solution {
    // Requires a Segment Tree for Range Minimum Query on d, returning a pair {value, index}.
    // Requires a Fenwick Tree for range sum/point update on y (using doubles).

    public int solve(int[] nums, int[][] queries) {
        int n = nums.length;
        long[] coverage = new long[n + 1];
        for (int[] query : queries) {
            coverage[query[0]]++;
            if (query[1] + 1 <= n) {
                coverage[query[1] + 1]--;
            }
        }

        long[] d = new long[n];
        long currentCoverage = 0;
        for (int i = 0; i < n; i++) {
            currentCoverage += coverage[i];
            d[i] = currentCoverage - nums[i];
            if (d[i] < 0) return -1;
        }

        Arrays.sort(queries, (a, b) -> Integer.compare(a[1], b[1]));

        SegTree dSegTree = new SegTree(d); // For finding min d[i] in a range
        FenwickTree yFenwickTree = new FenwickTree(n); // For y_i values
        
        double totalCost = 0.0;

        for (int[] query : queries) {
            int l = query[0], r = query[1];
            double currentSum = yFenwickTree.query(l, r);
            
            if (currentSum < 1.0 - 1e-9) { // Use tolerance for float comparison
                double needed = 1.0 - currentSum;
                Pair<Long, Integer> best = dSegTree.query(l, r); // find {min_d, index}
                int bestIndex = best.getValue();
                long costOfIndex = best.getKey();
                
                yFenwickTree.update(bestIndex, needed);
                totalCost += needed * costOfIndex;
            }
        }
        return (int) Math.round(totalCost);
    }
}