闲话

调试约一个下午后发现极大值设小了。

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考虑开两个堆 \(q_1\) 和 \(a_2\)，一个存入所有一颗星星的取法，另一个存入所有两颗星星的取法。
每次两颗两颗比较，然后如果某一次取了一颗星星，那么（设这颗星星对应关卡编号为 \(i\)）把 \(b_i - a_i\) 压入堆中。
还有一些别的细节，见代码。
代码：

#include <bits/stdc++.h>
#define int long long
#define rep(i, l, r) for (int i = l; i <= r; i++)
using namespace std;
int a[300010], b[300010], del[300010][3], ch[300010], w, n, ans, cnt, res;
map<int, int> mp;
priority_queue<pair<int, int>, vector<pair<int, int> >, greater<pair<int, int> > > q1, q2;
void debug(priority_queue<pair<int, int>, vector<pair<int, int> >, greater<pair<int, int> > > q) {
    cout << "[ ";
    while (!q.empty()) {
        cout << "{ " << q.top().first << " " << q.top().second << " }" << " ";
        q.pop();
    }
    cout << "]";
}
signed main() {
    cin.tie(0) -> ios :: sync_with_stdio(false);
    cin >> n >> w;
    rep (i, 1, n) {
        ch[i] = 0;
        cin >> a[i] >> b[i];
        q1.push({a[i], i});
        q2.push({b[i], i});
    }
    if (w == 2 * n) {
        rep (i, 1, n) {
            ans += b[i];
        }
        cout << ans << "\n";
        rep (i, 1, n) {
            cout << 2;
        }
        cout << "\n";
        return 0;
    }
    if (w % 2 == 1) {
        q1.push({0, 0});
        ch[0] = 1;
        w++;
    }
    while (cnt < w) {
        while (!q1.empty() && del[q1.top().second][1]) {
            q1.pop();
        }
        while (!q2.empty() && del[q2.top().second][2]) {
            q2.pop();
        }
        int a1 = 0x3f3f3f3f3f3f3f3f, id1 = 0, a2 = 0x3f3f3f3f3f3f3f3f, id2 = 0;
        if (!q1.empty()) {
            a1 = q1.top().first; id1 = q1.top().second; q1.pop();
            while (!q1.empty() && del[q1.top().second][1]) {
                q1.pop();
            }
            a2 = q1.top().first; id2 = q1.top().second; q1.pop();
        }
        int b1 = 0x3f3f3f3f3f3f3f3f, idb = 0;
        if (!q2.empty()) {
            b1 = q2.top().first; idb = q2.top().second; q2.pop();
        }
        if ((mp[id1] || mp[id2] || mp[idb]) && (id1 && id2 && idb)) {
            assert(0);
        }
        if (a1 + a2 <= b1) {
            // cout << a1 << " " << a2 << "\n";
            cnt -= ch[id1] + ch[id2];
            ch[id1]++; ch[id2]++;
            // mp[id1] = mp[id2] = 1;
            if (ch[id1] == 1) {
                q1.push({b[id1] - a[id1], id1});
            }
            if (ch[id2] == 1) {
                q1.push({b[id2] - a[id2], id2});
            }
            if (ch[id1] == 2) {
                del[id1][1] = 1;
                mp[id1] = 1;
            }
            if (ch[id2] == 2) {
                del[id2][1] = 1;
                mp[id2] = 1;
            }
            del[id1][2] = del[id2][2] = 1;
            q2.push({b1, idb});
            // cout << "q1: "; debug(q1); cout << "\n";
            // cout << "q2: "; debug(q2); cout << "\n";
            // ans += a1 + a2;
            cnt += ch[id1] + ch[id2];
        } else {
            // cout << b1 << "\n";
            cnt -= ch[idb];
            ch[idb] += 2;
            del[idb][1] = 1;
            del[idb][2] = 1;
            mp[idb] = 1;
            q1.push({a1, id1});
            q1.push({a2, id2});
            // cout << "q1: "; debug(q1); cout << "\n";
            // cout << "q2: "; debug(q2); cout << "\n";
            // ans += b1;
            cnt += ch[idb];
        }
    }
    assert(cnt == w);
    rep (i, 1, n) {
        ans += (ch[i] == 1 ? a[i] : (ch[i] == 2 ? b[i] : 0));
        assert(ch[i] <= 2);
        res += ch[i];
    }
    // assert(res == w || res == w - 1);
    cout << ans << "\n";
    rep (i, 1, n) {
        cout << ch[i];
    }
    cout << "\n";
}