oj.uz

Submission #291424

# Submission time Handle Problem Language Result Execution time Memory
291424 2020-09-05T10:27:45 Z rama_pang Arranging Tickets (JOI17_arranging_tickets) C++14
100 / 100
 2168 ms 15832 KB 
#include <bits/stdc++.h>
using namespace std;

// Solution:
// We can binary search the answer. We also break the circle to
// a line segment.
//
// If intervals [a, b] and [c, d] is reversed, and a < b < c < d,
// then we originally have interval values:
// ([1, a), 0), ([a, b], 1), ((b, c), 0), ([c, d], 1), ((d, N], 0)
// and reversed values are:
// ([1, a), 2), ([a, b], 1), ((b, c), 2), ([c, d], 1), ((d, N], 2)
// We can see that all regions increased - thus all flipped intervals
// must share a point t.
//
// For a fixed point t, let a[i] be the sum of people travelling through i,
// and b[i] be the final one once we reversed optimally. Then b[t] = max(b[i])
// or b[t] = max(b[i]) - 1. This allows us to check only a[t] - max_ans and
// a[t] - max_ans + 1 flips for a point t, where max_ans is the current
// mid in the binary search. Proof follows.
//
// Assume then b[t] < max(b[i]) - 1. Then we can find two reversed segments [a, b] and
// [c, d], such that a is the minimum coordinate of reversed segment and d is the
// maximum one of reversed segments. By definition, [a, b] and [c, d] passes through t, 
// and we can unreverse them. This will not change max(b[i]), and increment b[t] by either
// 1 or 2. Thus we can keep doing this until the condition is satisfied without breaking
// the solution.
//
// This yields an O(M^2 log^2 M) solution. We can further speed this up by noticing that
// we only need to check t where a[t] = max(a[i]) and t is the minimum and maximum i that
// is equal max(a[i]). Why? Since b[i] - a[i] is smaller the closer i is to t, let j not be
// in the common interval of all reversed intervals. Then b[t] - a[t] + 1 <= b[j] - a[j].
// If we assume a[t] + 1 <= a[j], we can sum them to get b[t] + 2 <= b[j]. But by definition
// b[t] = max(b[i]) or b[t] = max(b[i]) - 1, thus we get a contradiction. Therefore, a[t] = 
// max(a[i]) must be satisfied.
//
// Now we prove the second condition (only checking the leftmost and rightmost t such that a[t] =
// max(a[i])). First, there is no reversed interval [x, y], such that l < x < y < r. Assume otherwise.
// Then, when we reverse [x, y], let b' be the resulting arrangement. We have b'[t] = b[t] + 1
// and b'[l] = b[l] - 1, and b'[l] >= b'[t] since we use t = l. Then we get b[l] >= b[t] + 2 which
// is a contradiction since b[t] = max(b[i]) or b[t] = max(b[i]) - 1. Thus no such [x, y] exist, which
// means we only need to consider t = l or t = r.
//
// After we determine max_ans and t, we can do the reversal greedily. We can use a segment tree or a
// priority queue to support the operations: point update += x at position i and remove x items from 
// the right to simulate the greedy selection. We check after we do all flips to see if we satisfy
// everything.
//
// Time: O(N log^2 N)

int main() {
  ios::sync_with_stdio(0);
  cin.tie(0), cout.tie(0);

  int N, M;
  cin >> N >> M;

  vector<int> A(M), B(M), C(M);
  for (int i = 0; i < M; i++) {
    cin >> A[i] >> B[i] >> C[i];
    A[i]--, B[i]--;
    if (A[i] > B[i]) {
      swap(A[i], B[i]);
    }
  }

  vector<long long> sum(N);
  vector<vector<pair<int, int>>> segs(N);
  for (int i = 0; i < M; i++) {
    sum[A[i]] += C[i];
    sum[B[i]] -= C[i];
    segs[A[i]].emplace_back(B[i], C[i]);
  }
  for (int i = 1; i < N; i++) {
    sum[i] += sum[i - 1];
  }

  long long mx = *max_element(begin(sum), end(sum));
  int ll = -1, rr = -1;
  for (int i = 0; i < N; i++) {
    if (sum[i] == mx) {
      ll = i;
      break;
    }
  }
  for (int i = N - 1; i >= 0; i--) {
    if (sum[i] == mx) {
      rr = i;
      break;
    }
  }

  auto Check = [&](int t, long long mx, long long flips) -> bool {
    vector<long long> upd_sum(N);
    for (int i = 1; i < N; i++) {
      upd_sum[i] += upd_sum[i - 1];
    }
    long long flipped = 0;
    priority_queue<array<int, 3>> pq;
    for (int pos = 0; pos <= t; pos++) {
      for (auto i : segs[pos]) {
        if (i.first > t) {
          pq.push({i.first, pos, i.second});
        }
      }
      // Look at effects to upd_sum for derivation, sum[pos] += 2 * constant
      long long need = (sum[pos] - mx + flips + 1) / 2; 
      if (pos == t) need = flips;
      need = max(need - flipped, 0ll);
      flipped += need;
      while (need > 0) {
        if (pq.empty()) {
          return false;
        }
        auto arr = pq.top(); pq.pop();
        long long del = min(need, 1ll * arr[2]);
        arr[2] -= del, need -= del;
        if (arr[2] > 0) {
          pq.push(arr);
        }
        // Add [0, A) and [B, N)
        upd_sum[0] += del;
        upd_sum[arr[1]] -=  del;
        upd_sum[arr[0]] +=  del;
        // Undo [A, B)
        upd_sum[arr[1]] -= del;
        upd_sum[arr[0]] += del;
      }
    }

    for (int i = 0; i < N; i++) {
      if (i > 0) {
        upd_sum[i] += upd_sum[i - 1];
      }
      if (sum[i] + upd_sum[i] > mx) {
        return false;
      }
    }
    return true;
  };

  auto CheckAll = [&](long long m) -> bool {
    return Check(ll, m, sum[ll] - m + 0) ||
           Check(ll, m, sum[ll] - m + 1) ||
           Check(rr, m, sum[rr] - m + 0) ||
           Check(rr, m, sum[rr] - m + 1);
  };

  long long lo = 0, hi = mx;
  while (lo < hi) {
    long long md = (lo + hi) / 2;
    if (CheckAll(md)) {
      hi = md;
    } else {
      lo = md + 1;
    }
  }

  cout << lo << "\n";
  return 0;
}

Subtask #1
10.0 / 10.0

# Verdict Execution time Memory Grader output
1 Correct 1 ms 384 KB Output is correct
2 Correct 1 ms 384 KB Output is correct
3 Correct 1 ms 384 KB Output is correct
4 Correct 0 ms 384 KB Output is correct
5 Correct 1 ms 384 KB Output is correct
6 Correct 1 ms 512 KB Output is correct
7 Correct 1 ms 384 KB Output is correct
8 Correct 1 ms 384 KB Output is correct
9 Correct 1 ms 384 KB Output is correct
10 Correct 1 ms 512 KB Output is correct
11 Correct 0 ms 384 KB Output is correct
12 Correct 0 ms 384 KB Output is correct
13 Correct 1 ms 384 KB Output is correct
14 Correct 1 ms 384 KB Output is correct
15 Correct 0 ms 384 KB Output is correct

Subtask #2
35.0 / 35.0

# Verdict Execution time Memory Grader output
1 Correct 1 ms 384 KB Output is correct
2 Correct 1 ms 384 KB Output is correct
3 Correct 1 ms 384 KB Output is correct
4 Correct 0 ms 384 KB Output is correct
5 Correct 1 ms 384 KB Output is correct
6 Correct 1 ms 512 KB Output is correct
7 Correct 1 ms 384 KB Output is correct
8 Correct 1 ms 384 KB Output is correct
9 Correct 1 ms 384 KB Output is correct
10 Correct 1 ms 512 KB Output is correct
11 Correct 0 ms 384 KB Output is correct
12 Correct 0 ms 384 KB Output is correct
13 Correct 1 ms 384 KB Output is correct
14 Correct 1 ms 384 KB Output is correct
15 Correct 0 ms 384 KB Output is correct
16 Correct 1 ms 384 KB Output is correct
17 Correct 1 ms 384 KB Output is correct
18 Correct 1 ms 384 KB Output is correct
19 Correct 1 ms 384 KB Output is correct
20 Correct 1 ms 384 KB Output is correct
21 Correct 1 ms 384 KB Output is correct
22 Correct 1 ms 384 KB Output is correct
23 Correct 1 ms 384 KB Output is correct
24 Correct 1 ms 384 KB Output is correct
25 Correct 1 ms 384 KB Output is correct
26 Correct 1 ms 384 KB Output is correct
27 Correct 1 ms 384 KB Output is correct

Subtask #3
20.0 / 20.0

# Verdict Execution time Memory Grader output
1 Correct 1 ms 384 KB Output is correct
2 Correct 1 ms 384 KB Output is correct
3 Correct 1 ms 384 KB Output is correct
4 Correct 0 ms 384 KB Output is correct
5 Correct 1 ms 384 KB Output is correct
6 Correct 1 ms 512 KB Output is correct
7 Correct 1 ms 384 KB Output is correct
8 Correct 1 ms 384 KB Output is correct
9 Correct 1 ms 384 KB Output is correct
10 Correct 1 ms 512 KB Output is correct
11 Correct 0 ms 384 KB Output is correct
12 Correct 0 ms 384 KB Output is correct
13 Correct 1 ms 384 KB Output is correct
14 Correct 1 ms 384 KB Output is correct
15 Correct 0 ms 384 KB Output is correct
16 Correct 1 ms 384 KB Output is correct
17 Correct 1 ms 384 KB Output is correct
18 Correct 1 ms 384 KB Output is correct
19 Correct 1 ms 384 KB Output is correct
20 Correct 1 ms 384 KB Output is correct
21 Correct 1 ms 384 KB Output is correct
22 Correct 1 ms 384 KB Output is correct
23 Correct 1 ms 384 KB Output is correct
24 Correct 1 ms 384 KB Output is correct
25 Correct 1 ms 384 KB Output is correct
26 Correct 1 ms 384 KB Output is correct
27 Correct 1 ms 384 KB Output is correct
28 Correct 4 ms 640 KB Output is correct
29 Correct 4 ms 640 KB Output is correct
30 Correct 4 ms 640 KB Output is correct
31 Correct 4 ms 640 KB Output is correct
32 Correct 6 ms 640 KB Output is correct
33 Correct 5 ms 640 KB Output is correct
34 Correct 3 ms 640 KB Output is correct
35 Correct 5 ms 640 KB Output is correct
36 Correct 6 ms 640 KB Output is correct
37 Correct 2 ms 656 KB Output is correct
38 Correct 2 ms 640 KB Output is correct
39 Correct 2 ms 640 KB Output is correct

Subtask #4
20.0 / 20.0

# Verdict Execution time Memory Grader output
1 Correct 1 ms 384 KB Output is correct
2 Correct 1 ms 384 KB Output is correct
3 Correct 1 ms 384 KB Output is correct
4 Correct 0 ms 384 KB Output is correct
5 Correct 1 ms 384 KB Output is correct
6 Correct 1 ms 512 KB Output is correct
7 Correct 1 ms 384 KB Output is correct
8 Correct 1 ms 384 KB Output is correct
9 Correct 1 ms 384 KB Output is correct
10 Correct 1 ms 512 KB Output is correct
11 Correct 0 ms 384 KB Output is correct
12 Correct 0 ms 384 KB Output is correct
13 Correct 1 ms 384 KB Output is correct
14 Correct 1 ms 384 KB Output is correct
15 Correct 0 ms 384 KB Output is correct
16 Correct 1 ms 384 KB Output is correct
17 Correct 1 ms 384 KB Output is correct
18 Correct 1 ms 384 KB Output is correct
19 Correct 1 ms 384 KB Output is correct
20 Correct 1 ms 384 KB Output is correct
21 Correct 1 ms 384 KB Output is correct
22 Correct 1 ms 384 KB Output is correct
23 Correct 1 ms 384 KB Output is correct
24 Correct 1 ms 384 KB Output is correct
25 Correct 1 ms 384 KB Output is correct
26 Correct 1 ms 384 KB Output is correct
27 Correct 1 ms 384 KB Output is correct
28 Correct 4 ms 640 KB Output is correct
29 Correct 4 ms 640 KB Output is correct
30 Correct 4 ms 640 KB Output is correct
31 Correct 4 ms 640 KB Output is correct
32 Correct 6 ms 640 KB Output is correct
33 Correct 5 ms 640 KB Output is correct
34 Correct 3 ms 640 KB Output is correct
35 Correct 5 ms 640 KB Output is correct
36 Correct 6 ms 640 KB Output is correct
37 Correct 2 ms 656 KB Output is correct
38 Correct 2 ms 640 KB Output is correct
39 Correct 2 ms 640 KB Output is correct
40 Correct 268 ms 14428 KB Output is correct
41 Correct 307 ms 14804 KB Output is correct
42 Correct 255 ms 14300 KB Output is correct
43 Correct 425 ms 14300 KB Output is correct
44 Correct 222 ms 14428 KB Output is correct
45 Correct 507 ms 14096 KB Output is correct
46 Correct 312 ms 14048 KB Output is correct
47 Correct 108 ms 13556 KB Output is correct
48 Correct 117 ms 13692 KB Output is correct
49 Correct 80 ms 10184 KB Output is correct
50 Correct 62 ms 9924 KB Output is correct
51 Correct 77 ms 9788 KB Output is correct

Subtask #5
15.0 / 15.0

# Verdict Execution time Memory Grader output
1 Correct 1 ms 384 KB Output is correct
2 Correct 1 ms 384 KB Output is correct
3 Correct 1 ms 384 KB Output is correct
4 Correct 0 ms 384 KB Output is correct
5 Correct 1 ms 384 KB Output is correct
6 Correct 1 ms 512 KB Output is correct
7 Correct 1 ms 384 KB Output is correct
8 Correct 1 ms 384 KB Output is correct
9 Correct 1 ms 384 KB Output is correct
10 Correct 1 ms 512 KB Output is correct
11 Correct 0 ms 384 KB Output is correct
12 Correct 0 ms 384 KB Output is correct
13 Correct 1 ms 384 KB Output is correct
14 Correct 1 ms 384 KB Output is correct
15 Correct 0 ms 384 KB Output is correct
16 Correct 1 ms 384 KB Output is correct
17 Correct 1 ms 384 KB Output is correct
18 Correct 1 ms 384 KB Output is correct
19 Correct 1 ms 384 KB Output is correct
20 Correct 1 ms 384 KB Output is correct
21 Correct 1 ms 384 KB Output is correct
22 Correct 1 ms 384 KB Output is correct
23 Correct 1 ms 384 KB Output is correct
24 Correct 1 ms 384 KB Output is correct
25 Correct 1 ms 384 KB Output is correct
26 Correct 1 ms 384 KB Output is correct
27 Correct 1 ms 384 KB Output is correct
28 Correct 4 ms 640 KB Output is correct
29 Correct 4 ms 640 KB Output is correct
30 Correct 4 ms 640 KB Output is correct
31 Correct 4 ms 640 KB Output is correct
32 Correct 6 ms 640 KB Output is correct
33 Correct 5 ms 640 KB Output is correct
34 Correct 3 ms 640 KB Output is correct
35 Correct 5 ms 640 KB Output is correct
36 Correct 6 ms 640 KB Output is correct
37 Correct 2 ms 656 KB Output is correct
38 Correct 2 ms 640 KB Output is correct
39 Correct 2 ms 640 KB Output is correct
40 Correct 268 ms 14428 KB Output is correct
41 Correct 307 ms 14804 KB Output is correct
42 Correct 255 ms 14300 KB Output is correct
43 Correct 425 ms 14300 KB Output is correct
44 Correct 222 ms 14428 KB Output is correct
45 Correct 507 ms 14096 KB Output is correct
46 Correct 312 ms 14048 KB Output is correct
47 Correct 108 ms 13556 KB Output is correct
48 Correct 117 ms 13692 KB Output is correct
49 Correct 80 ms 10184 KB Output is correct
50 Correct 62 ms 9924 KB Output is correct
51 Correct 77 ms 9788 KB Output is correct
52 Correct 783 ms 15448 KB Output is correct
53 Correct 977 ms 15832 KB Output is correct
54 Correct 1905 ms 15440 KB Output is correct
55 Correct 685 ms 15324 KB Output is correct
56 Correct 1532 ms 15040 KB Output is correct
57 Correct 2168 ms 14896 KB Output is correct
58 Correct 831 ms 14940 KB Output is correct
59 Correct 771 ms 11000 KB Output is correct