IOI '20 - Tickets (41pts)

#include "tickets.h"
#include <vector>
#include <algorithm>
#include <iostream>

using namespace std;

const long long INF = 1e18;

long long find_maximum(int k, vector<vector<int>> x) {
    int n = x.size();
    int m = x[0].size();

    vector<vector<int>> answer(n, vector<int>(m, -1));
    long long prize = 0;

    int MAX = 0;
    for (int i = 0; i < n; ++i) {
        for (int j = 0; j < n; ++j) {
            MAX = max(MAX, x[i][j]);
        }
    }

    if (k == 1) {
        vector<vector<long long>> dp(1+n, vector<long long>(1+n/2, -INF));
        dp[0][0] = 0;
        // dp on prefix and number of positives
        vector<vector<bool>> chosePositive(1+n, vector<bool>(1+n/2));
        // chosePositive allows backtracking
        for (int i = 0; i < n; ++i) {
            for (int pos = 0; 2*pos <= n; ++pos) {
                if (dp[i][pos] == -INF) continue;

                // do not choose positive
                long long v = dp[i][pos] - x[i].front();
                if (v > dp[i+1][pos]) {
                    dp[i+1][pos] = v;
                    chosePositive[i+1][pos] = false;
                }

                // choose positive (if possible)
                if (2*pos == n) continue; // no more positives
                v = dp[i][pos] + x[i].back();
                if (v > dp[i+1][pos+1]) {
                    dp[i+1][pos+1] = v;
                    chosePositive[i+1][pos+1] = true;
                }
            }
        }

        prize = dp[n][n>>1];
        // Now reconstruct answer
        for (int i = n, pos = n>>1; i >= 1; --i) {
            if (!chosePositive[i][pos]) answer[i-1][0] = 0;
            else answer[i-1][m-1] = 0, pos--;
        }
    } else if (m == k || MAX == 1) {
        vector<int> cnt[2], remain[2];
        cnt[0].resize(n), cnt[1].resize(n);
        remain[0].resize(n), remain[1].resize(n);

        vector<pair<int, int>> v;
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < m; ++j) v.emplace_back(x[i][j], i);
        }
        sort(v.begin(), v.end());
        for (int i = 0; (i<<1) < n*m; ++i) {
            ++cnt[0][v[i].second];
        }
        for (int i = 0; i < n; ++i) {
            cnt[1][i] = m-cnt[0][i];
            remain[0][i] = cnt[0][i];
            remain[1][i] = cnt[1][i];
        }

        for (int i = 0; i < k; ++i) {
            int used0 = 0;
            for (int y = 0; y < n; ++y) {
                if (!remain[1][y]) { // zero
                    prize -= x[y][cnt[0][y]-remain[0][y]];
                    answer[y][cnt[0][y]-remain[0][y]] = i;
                    --remain[0][y];
                    ++used0;
                } else if (!remain[0][y]) { // one
                    prize += x[y][cnt[0][y]+remain[1][y]-1];
                    answer[y][cnt[0][y]+remain[1][y]-1] = i;
                    --remain[1][y];
                }
            }
            for (int y = 0; y < n; ++y) { // both
                if (!remain[0][y] || !remain[1][y]) continue;
                if (2*used0 < n) {
                    prize -= x[y][cnt[0][y]-remain[0][y]];
                    answer[y][cnt[0][y]-remain[0][y]] = i;
                    --remain[0][y];
                    ++used0;
                } else {
                    prize += x[y][cnt[0][y]+remain[1][y]-1];
                    answer[y][cnt[0][y]+remain[1][y]-1] = i;
                    --remain[1][y];
                }
            }
        }
    }
    allocate_tickets(answer);
    return prize;
}