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Submission #212651

# Submission time Handle Problem Language Result Execution time Memory
212651 2020-03-23T23:20:46 Z rama_pang Constellation 3 (JOI20_constellation3) C++14
100 / 100
 616 ms 45448 KB 
#include "bits/stdc++.h"
using namespace std;

class Dyanmic_Programming { // O(N^2) DP solution
 
 private:

  int N, M;
  int A[2005], X[2005], Y[2005], C[2005];

  long long DP[2005][2005]; // DP[height][column] = DP with domain(height, column) if we are forced to pick cell (heigth, column).
  int Star[2005][2005];

  void init() {
    for (int i = 0; i < M; i++) {
      for (int height = Y[i]; height <= N + 1; height++) {
        Star[height][X[i]] = max(Star[height][X[i]], C[i]);
      }
    }
  }
 
 public:

  long long solve() {
    init();

    for (int height = 1; height <= N + 1; height++) { // height N + 1 to merge all components into one
      for (int left = 1, right = left; left <= N; left = (++right)) {
        if (A[left] >= height) continue; // this cell is still occupied by a building (white cell).
        while (right + 1 <= N && A[right + 1] < height) {
          right++;
        }

        // column = left...right is connected to each other, now we find the transition
        vector<pair<pair<int, int>, long long>> maximum_values; // maximum values of disjoint domains for cells [height - 1][left...right]
        
        for (int l = left, r = l; l <= right; l = (++r)) {
          if (A[l] >= height - 1) continue;
          while (r + 1 <= right && A[r + 1] < height - 1) {
            r++;
          }

          { // with star
            long long mx = 0;
            for (int k = l; k <= r; k++) {
              mx = max(mx, DP[height - 1][k]);
            }
            maximum_values.push_back({{l, r}, mx});
          }
        }

        long long sum_max_values = 0;
        for (auto &values : maximum_values) {
          sum_max_values += values.second;
        }

        for (int column = left; column <= right; column++) {
          DP[height][column] = sum_max_values + Star[height][column];
        }

        for (auto &values : maximum_values) {
          for (int column = values.first.first; column <= values.first.second; column++) {
            DP[height][column] -= values.second; // This is not necessarily the correct domain and forbidden, so we subtract it 
            DP[height][column] += DP[height - 1][column] - Star[height - 1][column]; // This is the real domain and forbidden, minus the Star in (height - 1, column).
          }
        } 
      }
    }

    long long sum_all = 0;
    long long maximum = 0;
    
    for (int i = 0; i < M; i++) {
      sum_all += C[i];
    }

    for (int i = N + 1; i >= 1; i--) {
      for (int j = 1; j <= N; j++) {
        if (i == N + 1 ) {
          maximum = max(maximum, DP[i][j]);
        }
      }
    }

    return sum_all - maximum; 
  }

  void read() {
    ios::sync_with_stdio(0);
    cin.tie(0), cout.tie(0);

    cin >> N;
    for (int i = 1; i <= N; i++) {
      cin >> A[i];
    }
    cin >> M;
    for (int i = 0; i < M; i++) {
      cin >> X[i] >> Y[i] >> C[i];
    }
  }
};

class Full_Solution {

 private:

  int N, M;
  vector<int> A, X, Y, C;

  class DisjointSet {

   public:
   
    vector<int> p;

    DisjointSet(int n) {
      p.resize(n);
      iota(begin(p), end(p), 0);
    }
    
    int Find(int x) {
      return p[x] == x ? x : p[x] = Find(p[x]);
    }
  };

  class SegmentTree {

   private:

    vector<long long> tree;
    vector<long long> lazy;
    int sz;

    void push(int n) {
      for (int c = 0; c < 2; c++) {
        tree[n * 2 + c] += lazy[n];
        lazy[n * 2 + c] += lazy[n];
      }
      lazy[n] = 0;
    }

    void pull(int n) {
      tree[n] = max(tree[n * 2], tree[n * 2 + 1]);
    }

    void RangeSumUpdate(int n, int tl, int tr, int l, int r, long long x) {
      if (tr < l || r < tl || tl > tr || l > r) return;
      if (l <= tl && tr <= r) {
        tree[n] += x;
        lazy[n] += x;
        return;
      }
      push(n);
      int mid = (tl + tr) / 2;
      RangeSumUpdate(n * 2, tl, mid, l, r, x);
      RangeSumUpdate(n * 2 + 1, mid + 1, tr, l, r, x);
      pull(n);
    }

    long long RangeMaximumQuery(int n, int tl, int tr, int l, int r) {
      if (tr < l || r < tl || tl > tr || l > r) return 0;
      if (l <= tl && tr <= r) return tree[n];
      push(n);
      int mid = (tl + tr) / 2;
      return max(RangeMaximumQuery(n * 2, tl, mid, l, r), 
                  RangeMaximumQuery(n * 2 + 1, mid + 1, tr, l, r));
    }
  
   public:

    SegmentTree(int n) {
      sz = n;
      tree.assign(4 * n, 0);
      lazy.assign(4 * n, 0);
    }

    void RangeSumUpdate(int l, int r, long long x) {
      return RangeSumUpdate(1, 0, sz - 1, l, r, x);
    }

    long long RangeMaximumQuery(int l, int r) {
      return RangeMaximumQuery(1, 0, sz - 1, l, r);
    }

  };

  vector<vector<pair<int, int>>> CalculateStarDifference() {
    vector<vector<pair<int, int>>> StarDifference(N + 2); // SD[height] = Points where Star[height][column] - Star[height - 1][column] > 0, with (column, value)
    
    vector<vector<int>> YCoordinates(N + 1);
    vector<vector<int>> stars_per_column(N + 1);

    for (int i = 0; i < M; i++) {
      YCoordinates[X[i]].emplace_back(Y[i]);
    }

    for (int i = 1; i <= N; i++) {
      sort(begin(YCoordinates[i]), end(YCoordinates[i]));
      YCoordinates[i].resize(unique(begin(YCoordinates[i]), end(YCoordinates[i])) - begin(YCoordinates[i]));
    }

    for (int i = 1; i <= N; i++) {
      stars_per_column[i].resize(YCoordinates[i].size());
    }

    for (int i = 0; i < M; i++) {
      int where = lower_bound(begin(YCoordinates[X[i]]), end(YCoordinates[X[i]]), Y[i]) - begin(YCoordinates[X[i]]);
      stars_per_column[X[i]][where] = max(stars_per_column[X[i]][where], C[i]);
    }

    for (int i = 1; i <= N; i++) {
      for (int j = 1; j < stars_per_column[i].size(); j++) {
        stars_per_column[i][j] = max(stars_per_column[i][j], stars_per_column[i][j - 1]);
      }
    }

    for (int i = 1; i <= N; i++) {
      for (int j = 0; j < stars_per_column[i].size(); j++) {
        int value = stars_per_column[i][j] - (j > 0 ? stars_per_column[i][j - 1] : 0);
        StarDifference[YCoordinates[i][j]].emplace_back(i, value);
      }
    }

    return StarDifference;
  }

  vector<vector<int>> CalculateBuildingsWithHeight() {
    vector<vector<int>> BuildingsWithHeight(N + 2); // bwh[height] = A[i] such that A[i] = height - 1
    
    for (int i = 1; i <= N; i++) {
      BuildingsWithHeight[A[i] + 1].emplace_back(i);
    }
    
    return BuildingsWithHeight;
  }

  long long SumAll() { // sum of all C[i]
    long long res = 0;
    for (int i = 0; i < M; i++) {
      res += C[i];
    }
    return res;
  }

 public:

  void read() {
    ios::sync_with_stdio(0);
    cin.tie(0), cout.tie(0);

    cin >> N;

    A.resize(N + 1);

    for (int i = 1; i <= N; i++) {
      cin >> A[i];
    }

    cin >> M;

    X.resize(M);
    Y.resize(M);
    C.resize(M);

    for (int i = 0; i < M; i++) {
      cin >> X[i] >> Y[i] >> C[i];
    }
  }

  long long solve() {
    // initialization

    vector<vector<pair<int, int>>> StarDifference; // Points where Star[height][column] - Star[height - 1][column] > 0, with (height, value)
    StarDifference = CalculateStarDifference();

    vector<vector<int>> BuildingsWithHeight(N + 2); // BWH[height] = A[i] such that A[i] = height - 1
    BuildingsWithHeight = CalculateBuildingsWithHeight();

    // DP optimized with Segment Tree

    SegmentTree seg(N + 1);
    DisjointSet left_bound(N + 1), right_bound(N + 1); // left and right bound of components

    vector<bool> active(N + 1, false); // determine if column is already active (height > A[column])

    for (int height = 1; height <= N + 1; height++) {
      // Merge Components

      vector<pair<int, int>> LR_bounds_old; // to-be-merged components of height - 1
      vector<pair<int, int>> LR_bounds_new; // result of merged components of height
      
      for (auto &col : BuildingsWithHeight[height]) {
        if (col - 1 >= 1 && active[col - 1]) {
          int l = left_bound.Find(col - 1);
          int r = right_bound.Find(col - 1);
          assert(r == col - 1);
          LR_bounds_old.emplace_back(l, r);
        }
        if (col + 1 <= N && active[col + 1]) {
          int l = left_bound.Find(col + 1);
          int r = right_bound.Find(col + 1);
          assert(l == col + 1);
          LR_bounds_old.emplace_back(l, r);
        }
      }

      for (auto &col : BuildingsWithHeight[height]) {
        active[col] = true;
        if (col - 1 >= 1 && active[col - 1]) {
          left_bound.p[col] = col - 1;
          right_bound.p[col - 1] = col;
        }
        if (col + 1 <= N && active[col + 1]) {
          right_bound.p[col] = col + 1;
          left_bound.p[col + 1] = col;
        }
      }

      for (auto &col : BuildingsWithHeight[height]) {
        int l = col, r = col;
        if (col - 1 >= 1 && active[col - 1]) l = left_bound.Find(col - 1);
        if (col + 1 <= N && active[col + 1]) r = right_bound.Find(col + 1);  
        LR_bounds_new.emplace_back(l, r);
      }
      
      sort(begin(LR_bounds_old), end(LR_bounds_old));
      LR_bounds_old.resize(unique(begin(LR_bounds_old), end(LR_bounds_old)) - begin(LR_bounds_old));

      sort(begin(LR_bounds_new), end(LR_bounds_new));
      LR_bounds_new.resize(unique(begin(LR_bounds_new), end(LR_bounds_new)) - begin(LR_bounds_new));
      
      // Update the DP on Segment Tree

      // Add sum_of_maximums_of_previous_components - maximum_this_component
      int ptr = 0;
      for (auto &new_component : LR_bounds_new) {
        long long sum_of_maximums = 0;
        for (; ptr < LR_bounds_old.size(); ptr++) {
          if (new_component.first <= LR_bounds_old[ptr].first && LR_bounds_old[ptr].second <= new_component.second) {
            long long current_maximum = seg.RangeMaximumQuery(LR_bounds_old[ptr].first, LR_bounds_old[ptr].second);
            seg.RangeSumUpdate(LR_bounds_old[ptr].first, LR_bounds_old[ptr].second, -current_maximum);
            sum_of_maximums += current_maximum;
          } else if (new_component.second < LR_bounds_old[ptr].first) {
            break;
          }
        }
        seg.RangeSumUpdate(new_component.first, new_component.second, sum_of_maximums);
      }

      // Add StarDifference (Star[height][column] - Star[height - 1][column])
      for (auto &stars : StarDifference[height]) {
        int column = stars.first;
        int value = stars.second;
        seg.RangeSumUpdate(column, column, value);
      }
    }

    return SumAll() - seg.RangeMaximumQuery(1, N);
  }  

};

int main() {
  Full_Solution Solver;
  Solver.read();
  cout << Solver.solve() << "\n";
  return 0;
}

Compilation message

constellation3.cpp: In member function 'std::vector<std::vector<std::pair<int, int> > > Full_Solution::CalculateStarDifference()':
constellation3.cpp:212:25: warning: comparison between signed and unsigned integer expressions [-Wsign-compare]
       for (int j = 1; j < stars_per_column[i].size(); j++) {
                       ~~^~~~~~~~~~~~~~~~~~~~~~~~~~~~
constellation3.cpp:218:25: warning: comparison between signed and unsigned integer expressions [-Wsign-compare]
       for (int j = 0; j < stars_per_column[i].size(); j++) {
                       ~~^~~~~~~~~~~~~~~~~~~~~~~~~~~~
constellation3.cpp: In member function 'long long int Full_Solution::solve()':
constellation3.cpp:338:20: warning: comparison between signed and unsigned integer expressions [-Wsign-compare]
         for (; ptr < LR_bounds_old.size(); ptr++) {
                ~~~~^~~~~~~~~~~~~~~~~~~~~~

Subtask #1
14.0 / 14.0

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

Subtask #2
21.0 / 21.0

# Verdict Execution time Memory Grader output
1 Correct 5 ms 384 KB Output is correct
2 Correct 5 ms 384 KB Output is correct
3 Correct 5 ms 384 KB Output is correct
4 Correct 5 ms 384 KB Output is correct
5 Correct 5 ms 384 KB Output is correct
6 Correct 5 ms 384 KB Output is correct
7 Correct 5 ms 384 KB Output is correct
8 Correct 5 ms 384 KB Output is correct
9 Correct 5 ms 384 KB Output is correct
10 Correct 5 ms 384 KB Output is correct
11 Correct 5 ms 384 KB Output is correct
12 Correct 5 ms 384 KB Output is correct
13 Correct 6 ms 384 KB Output is correct
14 Correct 5 ms 384 KB Output is correct
15 Correct 5 ms 384 KB Output is correct
16 Correct 5 ms 384 KB Output is correct
17 Correct 5 ms 384 KB Output is correct
18 Correct 5 ms 384 KB Output is correct
19 Correct 5 ms 384 KB Output is correct
20 Correct 5 ms 384 KB Output is correct
21 Correct 5 ms 384 KB Output is correct
22 Correct 5 ms 384 KB Output is correct
23 Correct 8 ms 768 KB Output is correct
24 Correct 8 ms 768 KB Output is correct
25 Correct 8 ms 768 KB Output is correct
26 Correct 8 ms 768 KB Output is correct
27 Correct 8 ms 768 KB Output is correct
28 Correct 8 ms 768 KB Output is correct
29 Correct 8 ms 768 KB Output is correct
30 Correct 8 ms 768 KB Output is correct
31 Correct 8 ms 768 KB Output is correct
32 Correct 8 ms 768 KB Output is correct
33 Correct 8 ms 768 KB Output is correct
34 Correct 8 ms 768 KB Output is correct
35 Correct 9 ms 768 KB Output is correct
36 Correct 7 ms 768 KB Output is correct
37 Correct 6 ms 768 KB Output is correct
38 Correct 7 ms 768 KB Output is correct
39 Correct 7 ms 640 KB Output is correct
40 Correct 8 ms 768 KB Output is correct
41 Correct 7 ms 640 KB Output is correct
42 Correct 7 ms 768 KB Output is correct
43 Correct 8 ms 768 KB Output is correct
44 Correct 7 ms 640 KB Output is correct

Subtask #3
65.0 / 65.0

# Verdict Execution time Memory Grader output
1 Correct 5 ms 384 KB Output is correct
2 Correct 5 ms 384 KB Output is correct
3 Correct 5 ms 384 KB Output is correct
4 Correct 5 ms 384 KB Output is correct
5 Correct 5 ms 384 KB Output is correct
6 Correct 5 ms 384 KB Output is correct
7 Correct 5 ms 384 KB Output is correct
8 Correct 5 ms 384 KB Output is correct
9 Correct 5 ms 384 KB Output is correct
10 Correct 5 ms 384 KB Output is correct
11 Correct 5 ms 384 KB Output is correct
12 Correct 5 ms 384 KB Output is correct
13 Correct 6 ms 384 KB Output is correct
14 Correct 5 ms 384 KB Output is correct
15 Correct 5 ms 384 KB Output is correct
16 Correct 5 ms 384 KB Output is correct
17 Correct 5 ms 384 KB Output is correct
18 Correct 5 ms 384 KB Output is correct
19 Correct 5 ms 384 KB Output is correct
20 Correct 5 ms 384 KB Output is correct
21 Correct 5 ms 384 KB Output is correct
22 Correct 5 ms 384 KB Output is correct
23 Correct 8 ms 768 KB Output is correct
24 Correct 8 ms 768 KB Output is correct
25 Correct 8 ms 768 KB Output is correct
26 Correct 8 ms 768 KB Output is correct
27 Correct 8 ms 768 KB Output is correct
28 Correct 8 ms 768 KB Output is correct
29 Correct 8 ms 768 KB Output is correct
30 Correct 8 ms 768 KB Output is correct
31 Correct 8 ms 768 KB Output is correct
32 Correct 8 ms 768 KB Output is correct
33 Correct 8 ms 768 KB Output is correct
34 Correct 8 ms 768 KB Output is correct
35 Correct 9 ms 768 KB Output is correct
36 Correct 7 ms 768 KB Output is correct
37 Correct 6 ms 768 KB Output is correct
38 Correct 7 ms 768 KB Output is correct
39 Correct 7 ms 640 KB Output is correct
40 Correct 8 ms 768 KB Output is correct
41 Correct 7 ms 640 KB Output is correct
42 Correct 7 ms 768 KB Output is correct
43 Correct 8 ms 768 KB Output is correct
44 Correct 7 ms 640 KB Output is correct
45 Correct 616 ms 43140 KB Output is correct
46 Correct 604 ms 42604 KB Output is correct
47 Correct 589 ms 42400 KB Output is correct
48 Correct 600 ms 42992 KB Output is correct
49 Correct 597 ms 41936 KB Output is correct
50 Correct 571 ms 41776 KB Output is correct
51 Correct 573 ms 41904 KB Output is correct
52 Correct 614 ms 42988 KB Output is correct
53 Correct 615 ms 42840 KB Output is correct
54 Correct 593 ms 45448 KB Output is correct
55 Correct 555 ms 44920 KB Output is correct
56 Correct 536 ms 44208 KB Output is correct
57 Correct 530 ms 43924 KB Output is correct
58 Correct 327 ms 41160 KB Output is correct
59 Correct 317 ms 41224 KB Output is correct
60 Correct 558 ms 44120 KB Output is correct
61 Correct 392 ms 40552 KB Output is correct
62 Correct 531 ms 41932 KB Output is correct
63 Correct 345 ms 40176 KB Output is correct
64 Correct 379 ms 39352 KB Output is correct
65 Correct 552 ms 42052 KB Output is correct
66 Correct 360 ms 39928 KB Output is correct