minimum-time-to-break-locks-i.cpp

// Time:  O(n^3)
// Space: O(n^2)

// hungarian algorithm, weighted bipartite matching
class Solution {
public:
    int findMinimumTime(vector<int>& strength, int K) {
        const auto& ceil_divide = [](int a, int b) {
            return (a + b - 1) / b;
        };
    
        vector<vector<int>> adj(size(strength), vector<int>(size(strength)));
        for (int i = 0; i < size(strength); ++i) {
            for (int j = 0; j < size(strength); ++j) {
                adj[i][j] = ceil_divide(strength[i], 1 + j * K);
            }
        }
        return hungarian(adj).first;
    }

private:
    // Template modified from:
    // https://github.com/kth-competitive-programming/kactl/blob/main/content/graph/WeightedMatching.h
    pair<int, vector<int>> hungarian(const vector<vector<int>> &a) {  // Time: O(n^2 * m), Space: O(n + m)
        if (a.empty()) return {0, {}};
        int n = size(a) + 1, m = size(a[0]) + 1;
        vector<int> u(n), v(m), p(m), ans(n - 1);
        for (int i = 1; i < n; ++i) {
            p[0] = i;
            int j0 = 0; // add "dummy" worker 0
            vector<int> dist(m, numeric_limits<int>::max()), pre(m, -1);
            vector<bool> done(m + 1);
            do { // dijkstra
                done[j0] = true;
                int i0 = p[j0], j1, delta = numeric_limits<int>::max();
                for (int j = 1; j < m; ++j) {
                    if (!done[j]) {
                        auto cur = a[i0 - 1][j - 1] - u[i0] - v[j];
                        if (cur < dist[j]) dist[j] = cur, pre[j] = j0;
                        if (dist[j] < delta) delta = dist[j], j1 = j;
                    }
                }
                for (int j = 0; j < m; ++j) {
                    if (done[j]) u[p[j]] += delta, v[j] -= delta;
                    else dist[j] -= delta;
                }
                j0 = j1;
            } while (p[j0]);
            while (j0) { // update alternating path
                int j1 = pre[j0];
                p[j0] = p[j1], j0 = j1;
            }
        }
        for (int j = 1; j < m; ++j) if (p[j]) ans[p[j] - 1] = j - 1;
        return {-v[0], ans}; // min cost
    }
};

// Time:  O(n * 2^n)
// Space: O(2^n)
// bitmasks, dp
class Solution2 {
public:
    int findMinimumTime(vector<int>& strength, int K) {
        const auto& ceil_divide = [](int a, int b) {
            return (a + b - 1) / b;
        };

        vector<int> dp(1 << size(strength), numeric_limits<int>::max());
        dp[0] = 0;
        for (int mask = 1; mask < size(dp); ++mask) {
            const int x = 1 + (__builtin_popcount(mask) - 1) * K;
            for (int i = 0; i < size(strength); ++i) {
                if (!(mask & (1 << i))) {
                    continue;
                }
                dp[mask] = min(dp[mask], dp[mask ^ (1 << i)] + ceil_divide(strength[i], x));
            }
        }
        return dp.back();
    }
};