oj.uz

Submission #623433

# Submission time Handle Problem Language Result Execution time Memory
623433 2022-08-05T15:26:02 Z jophyyjh Carnival Tickets (IOI20_tickets) C++14
100 / 100
 713 ms 84984 KB 
/**
 * Another problem from tinjyu's training, though we didn't directly go through the
 * full solution. The "minimum sum of dist. to a point" is classic; we know that
 * point b is the median, i.e. any val in [(n/2)-th_num, #(n/2+1)-th_num]. The obs.
 * here is that the actual val we take is irrelevant. Since n is even, the val would
 * be cancelled out so the 
 * prize = #sum_of_the_(n/2)_larger_num - #sum_of_the_(n/2)_smaller_num.
 * 
 * This means in each round, we maximize #sum_of_(n/2)_num - #sum_of_remaining
 * because this naturally finds the larger n numbers to minus the n smaller ones
 * In k=1, we know that if we wanna find a num to be part of the smaller numbers,
 * this num has to be the smallest in its row. Similarly, the largest num in the row
 * shall be picked if it's a "larger" element. With some greedy arguments, the n rows
 * with a larger #min + #max shall pick its maximum, while the remaining rows/colors
 * should pick its minimum.
 * 
 * Sadly, this greedy approach isn't correct for other values of k, but the thing we
 * know (and can prove) using our experience with k=1 is:
 *      Each time a "plus" element is picked, it must be largest value of that
 *      color. Each time when a "minus" element is picked, it must be the
 *      smallest value in the row.
 * Yep, this is valid, though we don't know what colors should be picked as "plus" or
 * "minus" (yet).
 * 
 * I then looked at [S3], which i didn't finish. After this, I went to [S4], i.e.
 * k=n. In this scenario, every num in x[][] is either added to or subtracted from
 * the final ans. Naturally, we want the largest nm/2 values to be "plus", while the
 * nm/2 smallest ones to be "minus". It turns out that we can always give a
 * construction that achieves this. Imagine that we've decided which elements are
 * "plus" & which ones are "minus"; as long as the num of rows with all "plus" <= n/2
 * and num of rows with all "minus" <= n/2, we can always find a round of tickets
 * with n/2 "plus" & n/2 "minus". If the contrary happens, either "plus" or "minus"
 * elements have a num >nm/2, which is impossible. This means we can always induct
 * downwards.
 * 
 * Wow, [S4] is really inspiring. Now, for each row of values a prefix is chosen for
 * "minus" and a suffix is chosen for "plus". We can use dp to find exactly nk/2
 * "minus" and exactly nk/2 "plus" with maximum profit, and give a construction since
 * this is just [S4] with m -> k. Well, at least I thought so, but this results in a
 * O((nm)^2) dp (assuming k is close to m). This implies that we can only score 12
 * pts in [S5], AND we'll have to code tons more code to score previous subtasks.
 *      (I admit that this is sth I hate about IOI)
 * Define r[i][j] to be the difference in color i when we pick j "plus". In other
 * words, r[i][j] = #sum_of_suffix_with_len_j - #sum_of_prefix_with_len_(k-j). It's
 * easy to see that r[i][] is increasing, but after carefully examining this, the
 * difference between adjacent terms in r[i][] is decreasing. Therefore, we can just
 * greedily find the nk/2 "plus". Finally, I've finished this problem!
 * 
 * Time Complexity: O(mn * log(mn))
 * Implementation 1             (Full solution, greedy + sorting + construction)
*/

#include <bits/stdc++.h>
#include "tickets.h"

typedef long long   ll;
typedef std::vector<int>    vec;


struct pair_t {
    ll diff;
    int color;
    int plus;   // num of plus
};

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

    std::vector<pair_t> pairs;
    ll price = 0;
    for (int i = 0; i < n; i++) {
        std::vector<ll> prefix(m + 1);
        prefix[0] = 0;
        for (int j = 0; j < m; j++)
            prefix[j + 1] = prefix[j] + x[i][j];
        std::vector<ll> r(k + 1);
        for (int j = 0; j <= k; j++)
            r[j] = -prefix[k - j] + (prefix[m] - prefix[m - j]);
        price += r[0];
        for (int j = 1; j <= k; j++) {
            assert(r[j] >= r[j - 1]);
            pairs.push_back(pair_t{r[j] - r[j - 1], i, j});
        }
        for (int j = 1; j < k; j++)
            assert(r[j] - r[j - 1] >= r[j + 1] - r[j]);
    }
    std::sort(pairs.begin(), pairs.end(),
              [](const pair_t& p1, const pair_t& p2) {
                  return p1.diff > p2.diff || (p1.diff == p2.diff && p1.plus < p2.plus);
              });
    vec plus(n, 0);
    for (int l = 0; l < n * k / 2; l++)
        price += pairs[l].diff, plus[pairs[l].color] = pairs[l].plus;
    
    vec minus(n);
    for (int i = 0; i < n; i++) {
        minus[i] = k - plus[i];
        assert(plus[i] >= 0 && minus[i] >= 0);
    }
    vec left(n, 0), right(n, m - 1);
    std::vector<vec> tickets(n, vec(m, -1));
    for (int r = 0; r < k; r++) {
        int m_count = 0, p_count = 0;
        std::vector<bool> must(n, false);   // if i must be minus or must be plus
        for (int i = 0; i < n; i++) {
            if (plus[i] == 0) {
                tickets[i][left[i]] = r, left[i]++;
                minus[i]--, m_count++, must[i] = true;
            } else if (minus[i] == 0) {
                tickets[i][right[i]] = r, right[i]--;
                plus[i]--, p_count++, must[i] = true;
            }
        }
        for (int i = 0; i < n; i++) {
            if (!must[i]) {
                if (m_count < n / 2)
                    tickets[i][left[i]] = r, left[i]++, m_count++, minus[i]--;
                else if (p_count < n / 2)
                    tickets[i][right[i]] = r, right[i]--, p_count++, plus[i]--;
                else
                    std::abort();   // absurd
            }
        }
    }
    allocate_tickets(tickets);
    return price;
}

Subtask #1
11.0 / 11.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 0 ms 212 KB Output is correct
3 Correct 1 ms 212 KB Output is correct
4 Correct 1 ms 212 KB Output is correct
5 Correct 0 ms 340 KB Output is correct
6 Correct 1 ms 724 KB Output is correct

Subtask #2
16.0 / 16.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 0 ms 212 KB Output is correct
3 Correct 1 ms 212 KB Output is correct
4 Correct 2 ms 340 KB Output is correct
5 Correct 21 ms 2308 KB Output is correct
6 Correct 463 ms 51320 KB Output is correct

Subtask #3
14.0 / 14.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 0 ms 212 KB Output is correct
3 Correct 0 ms 212 KB Output is correct
4 Correct 2 ms 468 KB Output is correct
5 Correct 21 ms 2892 KB Output is correct
6 Correct 489 ms 63156 KB Output is correct
7 Correct 518 ms 69024 KB Output is correct
8 Correct 4 ms 724 KB Output is correct
9 Correct 1 ms 296 KB Output is correct
10 Correct 0 ms 212 KB Output is correct
11 Correct 1 ms 212 KB Output is correct
12 Correct 6 ms 976 KB Output is correct
13 Correct 20 ms 2676 KB Output is correct
14 Correct 17 ms 2772 KB Output is correct
15 Correct 557 ms 73828 KB Output is correct

Subtask #4
14.0 / 14.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 0 ms 212 KB Output is correct
3 Correct 0 ms 212 KB Output is correct
4 Correct 2 ms 492 KB Output is correct
5 Correct 27 ms 3476 KB Output is correct
6 Correct 5 ms 824 KB Output is correct
7 Correct 5 ms 980 KB Output is correct
8 Correct 713 ms 83752 KB Output is correct
9 Correct 685 ms 82528 KB Output is correct
10 Correct 655 ms 82628 KB Output is correct
11 Correct 704 ms 83716 KB Output is correct

Subtask #5
12.0 / 12.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 2 ms 340 KB Output is correct
3 Correct 2 ms 340 KB Output is correct
4 Correct 2 ms 468 KB Output is correct
5 Correct 2 ms 468 KB Output is correct
6 Correct 2 ms 468 KB Output is correct
7 Correct 1 ms 212 KB Output is correct
8 Correct 1 ms 340 KB Output is correct
9 Correct 2 ms 468 KB Output is correct
10 Correct 2 ms 468 KB Output is correct
11 Correct 2 ms 468 KB Output is correct
12 Correct 2 ms 508 KB Output is correct

Subtask #6
23.0 / 23.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 2 ms 340 KB Output is correct
3 Correct 2 ms 340 KB Output is correct
4 Correct 2 ms 468 KB Output is correct
5 Correct 2 ms 468 KB Output is correct
6 Correct 2 ms 468 KB Output is correct
7 Correct 1 ms 212 KB Output is correct
8 Correct 1 ms 340 KB Output is correct
9 Correct 2 ms 468 KB Output is correct
10 Correct 2 ms 468 KB Output is correct
11 Correct 2 ms 468 KB Output is correct
12 Correct 2 ms 508 KB Output is correct
13 Correct 24 ms 2284 KB Output is correct
14 Correct 19 ms 2376 KB Output is correct
15 Correct 23 ms 2764 KB Output is correct
16 Correct 29 ms 3528 KB Output is correct
17 Correct 1 ms 340 KB Output is correct
18 Correct 2 ms 468 KB Output is correct
19 Correct 2 ms 340 KB Output is correct
20 Correct 33 ms 2980 KB Output is correct
21 Correct 23 ms 3756 KB Output is correct
22 Correct 25 ms 3652 KB Output is correct
23 Correct 27 ms 4160 KB Output is correct

Subtask #7
10.0 / 10.0

# Verdict Execution time Memory Grader output
1 Correct 0 ms 212 KB Output is correct
2 Correct 0 ms 212 KB Output is correct
3 Correct 1 ms 212 KB Output is correct
4 Correct 1 ms 212 KB Output is correct
5 Correct 0 ms 340 KB Output is correct
6 Correct 1 ms 724 KB Output is correct
7 Correct 0 ms 212 KB Output is correct
8 Correct 0 ms 212 KB Output is correct
9 Correct 1 ms 212 KB Output is correct
10 Correct 2 ms 340 KB Output is correct
11 Correct 21 ms 2308 KB Output is correct
12 Correct 463 ms 51320 KB Output is correct
13 Correct 0 ms 212 KB Output is correct
14 Correct 0 ms 212 KB Output is correct
15 Correct 0 ms 212 KB Output is correct
16 Correct 2 ms 468 KB Output is correct
17 Correct 21 ms 2892 KB Output is correct
18 Correct 489 ms 63156 KB Output is correct
19 Correct 518 ms 69024 KB Output is correct
20 Correct 4 ms 724 KB Output is correct
21 Correct 1 ms 296 KB Output is correct
22 Correct 0 ms 212 KB Output is correct
23 Correct 1 ms 212 KB Output is correct
24 Correct 6 ms 976 KB Output is correct
25 Correct 20 ms 2676 KB Output is correct
26 Correct 17 ms 2772 KB Output is correct
27 Correct 557 ms 73828 KB Output is correct
28 Correct 0 ms 212 KB Output is correct
29 Correct 0 ms 212 KB Output is correct
30 Correct 0 ms 212 KB Output is correct
31 Correct 2 ms 492 KB Output is correct
32 Correct 27 ms 3476 KB Output is correct
33 Correct 5 ms 824 KB Output is correct
34 Correct 5 ms 980 KB Output is correct
35 Correct 713 ms 83752 KB Output is correct
36 Correct 685 ms 82528 KB Output is correct
37 Correct 655 ms 82628 KB Output is correct
38 Correct 704 ms 83716 KB Output is correct
39 Correct 0 ms 212 KB Output is correct
40 Correct 2 ms 340 KB Output is correct
41 Correct 2 ms 340 KB Output is correct
42 Correct 2 ms 468 KB Output is correct
43 Correct 2 ms 468 KB Output is correct
44 Correct 2 ms 468 KB Output is correct
45 Correct 1 ms 212 KB Output is correct
46 Correct 1 ms 340 KB Output is correct
47 Correct 2 ms 468 KB Output is correct
48 Correct 2 ms 468 KB Output is correct
49 Correct 2 ms 468 KB Output is correct
50 Correct 2 ms 508 KB Output is correct
51 Correct 24 ms 2284 KB Output is correct
52 Correct 19 ms 2376 KB Output is correct
53 Correct 23 ms 2764 KB Output is correct
54 Correct 29 ms 3528 KB Output is correct
55 Correct 1 ms 340 KB Output is correct
56 Correct 2 ms 468 KB Output is correct
57 Correct 2 ms 340 KB Output is correct
58 Correct 33 ms 2980 KB Output is correct
59 Correct 23 ms 3756 KB Output is correct
60 Correct 25 ms 3652 KB Output is correct
61 Correct 27 ms 4160 KB Output is correct
62 Correct 56 ms 6960 KB Output is correct
63 Correct 55 ms 7048 KB Output is correct
64 Correct 77 ms 10184 KB Output is correct
65 Correct 254 ms 28860 KB Output is correct
66 Correct 322 ms 36748 KB Output is correct
67 Correct 6 ms 1108 KB Output is correct
68 Correct 5 ms 956 KB Output is correct
69 Correct 488 ms 52588 KB Output is correct
70 Correct 586 ms 63408 KB Output is correct
71 Correct 709 ms 84984 KB Output is correct
72 Correct 619 ms 84140 KB Output is correct
73 Correct 679 ms 78616 KB Output is correct
74 Correct 532 ms 62864 KB Output is correct
75 Correct 584 ms 65104 KB Output is correct