EGOI '23 - Bikes vs Cars (69pts)

#include <bits/stdc++.h>

using namespace std;

const int INF = 1e7;

// DSU
int root(int i, vector<int> &parent) {
    return parent[i] < 0 ? i : (parent[i] = root(parent[i], parent));
}
void unite(int i, int j, vector<int> &parent) {
    i = root(i, parent), j = root(j, parent);
    if (i == j) return;
    if (parent[i] > parent[j]) swap(i, j);
    parent[i] += parent[j];
    parent[j] = i;
}

vector<tuple<int, int, int>> solve() {
    int n, W; cin >> n >> W;
    vector<vector<int>> B(n, vector<int>(n)), C(n, vector<int>(n));
    for (int i = 1; i < n; ++i) {
        for (int j = 0; j < i; ++j) cin >> C[i][j], C[j][i] = C[i][j];
    }
    int mnB = INF, mxB = 0;
    for (int i = 1; i < n; ++i) {
        for (int j = 0; j < i; ++j) {
            cin >> B[i][j], B[j][i] = B[i][j];
            mnB = min(mnB, B[i][j]);
            mxB = max(mxB, B[i][j]);
        }
    }

    vector<tuple<int, int, int>> streets;
    if (mnB == mxB) {
        for (int i = 0; i < n-1; ++i) streets.emplace_back(i, i+1, mnB);
        int smallC = W-mnB; // cannot build C lower than this

        // sort C by weight
        vector<tuple<int, int, int>> edges;
        for (int i = 0; i < n; ++i) {
            for (int j = i+1; j < n; ++j) {
                if (C[i][j] < smallC) return {};
                edges.emplace_back(C[i][j], i, j);
            }
        }
        sort(edges.begin(), edges.end());

        vector<int> parent(n, -1);
        while (!edges.empty()) {
            int curC = get<0>(edges.back());
            for (int i = (int)edges.size()-1; i >= 0 && get<0>(edges[i]) == curC; --i) {
                int a = get<1>(edges[i]), b = get<2>(edges[i]);
                if (root(a, parent) == root(b, parent)) return {}; // reachable with larger edges
            }
            for (int i = (int)edges.size()-1; i >= 0 && get<0>(edges[i]) == curC; --i) {
                int a = get<1>(edges[i]), b = get<2>(edges[i]);
                if (root(a, parent) != root(b, parent)) unite(a, b, parent), streets.emplace_back(a, b, W-curC);
                edges.pop_back();
            }
        }
        return streets;
    } else if (W == 1) {
        vector<int> parentB(n, -1), parentC(n, -1), parent(n, -1);
        for (int i = 0; i < n; ++i) {
            for (int j = i+1; j < n; ++j) {
                if (B[i][j] == 1) unite(i, j, parentB), unite(i, j, parent);
                if (C[i][j] == 1) unite(i, j, parentC), unite(i, j, parent);
            }
        }
        if (parent[root(0, parent)] != -n) return {}; // all connected
        for (int i = 0; i < n; ++i) {
            for (int j = i+1; j < n; ++j) {
                if (B[i][j] == 0 && root(i, parentB) == root(j, parentB)) return {};
                if (C[i][j] == 0 && root(i, parentC) == root(j, parentC)) return {};
            }
        }

        for (int i = 0; i < n; ++i) {
            if (parentB[i] >= 0) streets.emplace_back(i, root(i, parentB), 1);
            if (parentC[i] >= 0) streets.emplace_back(i, root(i, parentC), 0);
        }

        return streets;
    } else {
        vector<vector<int>> dB(n, vector<int>(n, -1)), dC(n, vector<int>(n, -1));
        for (int i = 0; i < n; ++i) {
            for (int j = i+1; j < n; ++j) {
                if (B[i][j] + C[i][j] >= W) {
                    streets.emplace_back(i, j, B[i][j]);
                    dB[j][i] = dB[i][j] = max(dB[i][j], B[i][j]);
                    streets.emplace_back(i, j, W-C[i][j]);
                    dC[j][i] = dC[i][j] = max(dC[i][j], C[i][j]);
                }
            }
        }

        // Floyd
        for (int k = 0; k < n; ++k) {
            for (int i = 0; i < n; ++i) {
                for (int j = 0; j < n; ++j) {
                    dB[i][j] = max(dB[i][j], min(dB[i][k], dB[k][j]));
                    dC[i][j] = max(dC[i][j], min(dC[i][k], dC[k][j]));
                }
            }
        }

        for (int i = 0; i < n; ++i) {
            for (int j = i+1; j < n; ++j) {
                if (dB[i][j] != B[i][j] || dC[i][j] != C[i][j]) return {};
            }
        }
        return streets;
    }
}

int main() {
    ios_base::sync_with_stdio(0); cin.tie(0); cout.tie(0);
    vector<tuple<int, int, int>> streets = solve();
    if (streets.empty()) {
        cout << "NO" << endl;
    } else {
        cout << streets.size() << '\n';
        for (auto [u, v, b] : streets) cout << u << ' ' << v << ' ' << b << '\n';
    }
    return 0;
}