oj.uz

Submission #212023

# Submission time Handle Problem Language Result Execution time Memory
212023 2020-03-22T04:29:14 Z rama_pang Making Friends on Joitter is Fun (JOI20_joitter2) C++14
100 / 100
 913 ms 69752 KB 
// Solution:
// Let a component denote a set of vertices such that each vertex has an outgoing edge to every
// other vertex in the component
//
// There are 2 cases when adding an edge:
//
// If we add an edge from a vertex v to a component c, then there will form an edge from v to
// every node in component c.
//
// If we add an edge from a component a to a component b, and there exists an edge from b to a,
// then a and b will be merged into one component (a complete directed simple graph).
//
// We can maintain these edges and components using a set, merging them with the weighted union
// heuristic to achieve O(n log n) merges. Each merge takes O(log n).
// amortized complexity.
//
// Time: O(n log^2 n)
// Memory: O(n)

#include "bits/stdc++.h"
using namespace std;

const int MAXN = 100005;

long long Answer = 0;

namespace disjoint_set { // maintain disjoint set of components
  int p[MAXN];
  int component_size[MAXN];

  void Init() {
    iota(p, p + MAXN, 0);
    fill(component_size, component_size + MAXN, 1);
  }

  int Find(int x) {
    return p[x] == x ? x : p[x] = Find(p[x]);
  }
}

namespace ingoing_edges {
  map<int, set<int>> ingoing_edges_from_component[MAXN]; // set of ingoing edges from other components to component[] (a list oh them), thus number of edges = size() * component[].size()
  int total_size_ingoing_edges_from_component[MAXN]; // number of all element of elements of ingoing_edges_from_component[]

  void Insert(int base, int ingoing_component, int ingoing_id) {
    assert(ingoing_component == disjoint_set::Find(ingoing_id));
    if (ingoing_edges_from_component[base][ingoing_component].count(ingoing_id) == 0) {
      total_size_ingoing_edges_from_component[base]++;
      ingoing_edges_from_component[base][ingoing_component].emplace(ingoing_id);
    }
  }

  void Delete(int base, int ingoing_component, int ingoing_id) {
    assert(ingoing_component == disjoint_set::Find(ingoing_id));
    if (ingoing_edges_from_component[base][ingoing_component].count(ingoing_id) == 1) {
      total_size_ingoing_edges_from_component[base]--;
      ingoing_edges_from_component[base][ingoing_component].erase(ingoing_id);
    }
  }

  void Delete(int base, int ingoing_component) {
    total_size_ingoing_edges_from_component[base] -= ingoing_edges_from_component[base][ingoing_component].size();
    ingoing_edges_from_component[base].erase(ingoing_component);
  }
}

namespace outgoing_edges {
  map<int, int> edges_between_components[MAXN]; // count how many edges are there from component[] to another component divided by another component.size()
  int total_size_outgoing_edges_to_component[MAXN]; // sum of all edges_between_component[]'s value

  void Insert(int from, int to, int x) {
    total_size_outgoing_edges_to_component[from] += x;
    edges_between_components[from][to] += x;
  }

  void Delete(int from, int to, int x) {
    total_size_outgoing_edges_to_component[from] -= x;
    edges_between_components[from][to] -= x;
    if (edges_between_components[from][to] == 0) {
      edges_between_components[from].erase(to);
    }
  }

  void Delete(int from, int to) {
    total_size_outgoing_edges_to_component[from] -= edges_between_components[from][to];
    edges_between_components[from].erase(to);
  }
}

using ingoing_edges::ingoing_edges_from_component;
using ingoing_edges::total_size_ingoing_edges_from_component;

using outgoing_edges::edges_between_components;
using outgoing_edges::total_size_outgoing_edges_to_component;

bool IsThereAnEdgeBetweenComponent(int x, int y) {
  return (edges_between_components[x].count(y) == 1);
}

long long ComponentAnswer(int sz) { // count number of edges in one component (a component is a complete directed graph)
  return 1ll * sz * (sz - 1);
}

set<pair<int, int>> todo; // pending to use ConnectComponent(x, y)

void ConnectComponent(int x, int y) {
  x = disjoint_set::Find(x);
  y = disjoint_set::Find(y);
  if (x == y) return;

  int components_x_size = disjoint_set::component_size[x];
  int components_y_size = disjoint_set::component_size[y];

  { // delete edges between components to be connected
    Answer -= 1ll * edges_between_components[y][x] * components_x_size;
    Answer -= 1ll * edges_between_components[x][y] * components_y_size;

    outgoing_edges::Delete(x, y);
    outgoing_edges::Delete(y, x);

    ingoing_edges::Delete(x, y);
    ingoing_edges::Delete(y, x);
  }

  { // maintain weighted union heuristic and update disjoint set
    int total_size_x = total_size_ingoing_edges_from_component[x] + total_size_outgoing_edges_to_component[x];
    int total_size_y = total_size_ingoing_edges_from_component[y] + total_size_outgoing_edges_to_component[y];

    if (total_size_x < total_size_y) {
      swap(x, y); // this still is affected by the weighted union heuristic, thus the will take O(n log n) merges total
      swap(components_x_size, components_y_size);
    }

    // Union in disjoint set
    disjoint_set::p[y] = x;
    disjoint_set::component_size[x] += disjoint_set::component_size[y];
    disjoint_set::component_size[y] = 0;
  }

  { // update answer for full component
    Answer -= ComponentAnswer(components_x_size);
    Answer -= ComponentAnswer(components_y_size);
    Answer += ComponentAnswer(components_x_size + components_y_size);
  }

  { // handle all merges of ingoing edge of x and y
    Answer += 1ll * total_size_ingoing_edges_from_component[x] * components_y_size; // all vertices that can reach x can now reach y as well
    Answer += 1ll * total_size_ingoing_edges_from_component[y] * components_x_size; // all vertices that can reach y can now reach x as well

    while (!ingoing_edges_from_component[y].empty()) {
      auto ingoing_edges_from_component_y_key_value_pair = begin(ingoing_edges_from_component[y]);
      int current_ingoing_component = ingoing_edges_from_component_y_key_value_pair->first;

      for (auto &current_ingoer : ingoing_edges_from_component_y_key_value_pair->second) {
        if (ingoing_edges_from_component[x][current_ingoing_component].count(current_ingoer) == 1) {
          outgoing_edges::Delete(current_ingoing_component, y, 1); // when merging edge_between_components[][x] and [][y], this will be double counted, so we subtract it
          Answer -= components_x_size + components_y_size; // Answer is also double counted
        } else {
          ingoing_edges::Insert(x, current_ingoing_component, current_ingoer);
        }
      }

      if (IsThereAnEdgeBetweenComponent(x, current_ingoing_component)) {
        todo.emplace(x, current_ingoing_component);
      }

      outgoing_edges::Insert(current_ingoing_component, x, edges_between_components[current_ingoing_component][y]);
      outgoing_edges::Delete(current_ingoing_component, y);
      ingoing_edges::Delete(y, current_ingoing_component);
    }
  }

  { // handle all merges of outgoing edge of x and y
    while (!edges_between_components[y].empty()) {
      auto outgoing_edges_y_key_value_pair = begin(edges_between_components[y]);
      int nxt_component = outgoing_edges_y_key_value_pair->first;

      for (auto &current_ingoer : ingoing_edges_from_component[nxt_component][y]) {
        ingoing_edges::Insert(nxt_component, x, current_ingoer);
      }

      if (IsThereAnEdgeBetweenComponent(nxt_component, x)) {
        todo.emplace(x, nxt_component); // there is an edge from nxt_component to x and vice versa, so we need to merge them into one component
      }

      outgoing_edges::Insert(x, nxt_component, edges_between_components[y][nxt_component]);
      outgoing_edges::Delete(y, nxt_component);
      ingoing_edges::Delete(nxt_component, y);
    }
  }
}

void AddEdge(int x, int y) {
  int real_x = x;
  int real_y = y;

  x = disjoint_set::Find(x);
  y = disjoint_set::Find(y);
  if (x == y) return;

  if (IsThereAnEdgeBetweenComponent(y, x)) {
    todo.emplace(x, y);
    while (!todo.empty()) {
      pair<int, int> cur = *begin(todo);
      todo.erase(cur);
      ConnectComponent(cur.first, cur.second);
    }
  } else if (ingoing_edges_from_component[y][x].count(real_x) == 0) { // if edge (real_x, real_y) doesn't already exist
    outgoing_edges::Insert(x, y, 1);
    ingoing_edges::Insert(y, x, real_x);
    Answer += disjoint_set::component_size[y]; // there forms an edge from real_x to all nodes in component y
  }
}

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

  disjoint_set::Init();

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

  for (int i = 0; i < M; i++) {
    int A, B; 
    cin >> A >> B;
    AddEdge(--A, --B);

    cout << Answer << '\n';
  }

  return 0;
}

Compilation message

joitter2.cpp: In function 'void AddEdge(int, int)':
joitter2.cpp:195:7: warning: unused variable 'real_y' [-Wunused-variable]
   int real_y = y;
       ^~~~~~

Subtask #1
1.0 / 1.0

# Verdict Execution time Memory Grader output
1 Correct 12 ms 10496 KB Output is correct
2 Correct 10 ms 10496 KB Output is correct
3 Correct 10 ms 10496 KB Output is correct
4 Correct 11 ms 10496 KB Output is correct
5 Correct 10 ms 10496 KB Output is correct
6 Correct 10 ms 10496 KB Output is correct
7 Correct 10 ms 10496 KB Output is correct
8 Correct 11 ms 10496 KB Output is correct
9 Correct 11 ms 10496 KB Output is correct
10 Correct 10 ms 10496 KB Output is correct
11 Correct 10 ms 10496 KB Output is correct
12 Correct 10 ms 10496 KB Output is correct
13 Correct 10 ms 10496 KB Output is correct
14 Correct 10 ms 10496 KB Output is correct
15 Correct 10 ms 10496 KB Output is correct
16 Correct 10 ms 10496 KB Output is correct
17 Correct 12 ms 10496 KB Output is correct
18 Correct 10 ms 10496 KB Output is correct
19 Correct 13 ms 10496 KB Output is correct
20 Correct 10 ms 10496 KB Output is correct
21 Correct 11 ms 10624 KB Output is correct
22 Correct 10 ms 10496 KB Output is correct
23 Correct 10 ms 10496 KB Output is correct
24 Correct 10 ms 10496 KB Output is correct
25 Correct 11 ms 10496 KB Output is correct
26 Correct 10 ms 10496 KB Output is correct
27 Correct 11 ms 10496 KB Output is correct
28 Correct 13 ms 10496 KB Output is correct
29 Correct 10 ms 10496 KB Output is correct
30 Correct 10 ms 10496 KB Output is correct
31 Correct 10 ms 10496 KB Output is correct
32 Correct 10 ms 10496 KB Output is correct
33 Correct 11 ms 10496 KB Output is correct

Subtask #2
16.0 / 16.0

# Verdict Execution time Memory Grader output
1 Correct 12 ms 10496 KB Output is correct
2 Correct 10 ms 10496 KB Output is correct
3 Correct 10 ms 10496 KB Output is correct
4 Correct 11 ms 10496 KB Output is correct
5 Correct 10 ms 10496 KB Output is correct
6 Correct 10 ms 10496 KB Output is correct
7 Correct 10 ms 10496 KB Output is correct
8 Correct 11 ms 10496 KB Output is correct
9 Correct 11 ms 10496 KB Output is correct
10 Correct 10 ms 10496 KB Output is correct
11 Correct 10 ms 10496 KB Output is correct
12 Correct 10 ms 10496 KB Output is correct
13 Correct 10 ms 10496 KB Output is correct
14 Correct 10 ms 10496 KB Output is correct
15 Correct 10 ms 10496 KB Output is correct
16 Correct 10 ms 10496 KB Output is correct
17 Correct 12 ms 10496 KB Output is correct
18 Correct 10 ms 10496 KB Output is correct
19 Correct 13 ms 10496 KB Output is correct
20 Correct 10 ms 10496 KB Output is correct
21 Correct 11 ms 10624 KB Output is correct
22 Correct 10 ms 10496 KB Output is correct
23 Correct 10 ms 10496 KB Output is correct
24 Correct 10 ms 10496 KB Output is correct
25 Correct 11 ms 10496 KB Output is correct
26 Correct 10 ms 10496 KB Output is correct
27 Correct 11 ms 10496 KB Output is correct
28 Correct 13 ms 10496 KB Output is correct
29 Correct 10 ms 10496 KB Output is correct
30 Correct 10 ms 10496 KB Output is correct
31 Correct 10 ms 10496 KB Output is correct
32 Correct 10 ms 10496 KB Output is correct
33 Correct 11 ms 10496 KB Output is correct
34 Correct 13 ms 10624 KB Output is correct
35 Correct 97 ms 14456 KB Output is correct
36 Correct 146 ms 17272 KB Output is correct
37 Correct 127 ms 17528 KB Output is correct
38 Correct 124 ms 17016 KB Output is correct
39 Correct 13 ms 10496 KB Output is correct
40 Correct 14 ms 10752 KB Output is correct
41 Correct 16 ms 10752 KB Output is correct
42 Correct 13 ms 10624 KB Output is correct
43 Correct 14 ms 10752 KB Output is correct
44 Correct 14 ms 10752 KB Output is correct
45 Correct 13 ms 10624 KB Output is correct
46 Correct 12 ms 10624 KB Output is correct
47 Correct 14 ms 10752 KB Output is correct
48 Correct 14 ms 10752 KB Output is correct
49 Correct 26 ms 11520 KB Output is correct
50 Correct 131 ms 17580 KB Output is correct
51 Correct 18 ms 11008 KB Output is correct
52 Correct 115 ms 15456 KB Output is correct
53 Correct 23 ms 11520 KB Output is correct
54 Correct 120 ms 16504 KB Output is correct
55 Correct 19 ms 11392 KB Output is correct
56 Correct 17 ms 11264 KB Output is correct
57 Correct 17 ms 11264 KB Output is correct
58 Correct 16 ms 11392 KB Output is correct
59 Correct 12 ms 10496 KB Output is correct
60 Correct 107 ms 12948 KB Output is correct
61 Correct 16 ms 10880 KB Output is correct
62 Correct 124 ms 16760 KB Output is correct

Subtask #3
83.0 / 83.0

# Verdict Execution time Memory Grader output
1 Correct 12 ms 10496 KB Output is correct
2 Correct 10 ms 10496 KB Output is correct
3 Correct 10 ms 10496 KB Output is correct
4 Correct 11 ms 10496 KB Output is correct
5 Correct 10 ms 10496 KB Output is correct
6 Correct 10 ms 10496 KB Output is correct
7 Correct 10 ms 10496 KB Output is correct
8 Correct 11 ms 10496 KB Output is correct
9 Correct 11 ms 10496 KB Output is correct
10 Correct 10 ms 10496 KB Output is correct
11 Correct 10 ms 10496 KB Output is correct
12 Correct 10 ms 10496 KB Output is correct
13 Correct 10 ms 10496 KB Output is correct
14 Correct 10 ms 10496 KB Output is correct
15 Correct 10 ms 10496 KB Output is correct
16 Correct 10 ms 10496 KB Output is correct
17 Correct 12 ms 10496 KB Output is correct
18 Correct 10 ms 10496 KB Output is correct
19 Correct 13 ms 10496 KB Output is correct
20 Correct 10 ms 10496 KB Output is correct
21 Correct 11 ms 10624 KB Output is correct
22 Correct 10 ms 10496 KB Output is correct
23 Correct 10 ms 10496 KB Output is correct
24 Correct 10 ms 10496 KB Output is correct
25 Correct 11 ms 10496 KB Output is correct
26 Correct 10 ms 10496 KB Output is correct
27 Correct 11 ms 10496 KB Output is correct
28 Correct 13 ms 10496 KB Output is correct
29 Correct 10 ms 10496 KB Output is correct
30 Correct 10 ms 10496 KB Output is correct
31 Correct 10 ms 10496 KB Output is correct
32 Correct 10 ms 10496 KB Output is correct
33 Correct 11 ms 10496 KB Output is correct
34 Correct 13 ms 10624 KB Output is correct
35 Correct 97 ms 14456 KB Output is correct
36 Correct 146 ms 17272 KB Output is correct
37 Correct 127 ms 17528 KB Output is correct
38 Correct 124 ms 17016 KB Output is correct
39 Correct 13 ms 10496 KB Output is correct
40 Correct 14 ms 10752 KB Output is correct
41 Correct 16 ms 10752 KB Output is correct
42 Correct 13 ms 10624 KB Output is correct
43 Correct 14 ms 10752 KB Output is correct
44 Correct 14 ms 10752 KB Output is correct
45 Correct 13 ms 10624 KB Output is correct
46 Correct 12 ms 10624 KB Output is correct
47 Correct 14 ms 10752 KB Output is correct
48 Correct 14 ms 10752 KB Output is correct
49 Correct 26 ms 11520 KB Output is correct
50 Correct 131 ms 17580 KB Output is correct
51 Correct 18 ms 11008 KB Output is correct
52 Correct 115 ms 15456 KB Output is correct
53 Correct 23 ms 11520 KB Output is correct
54 Correct 120 ms 16504 KB Output is correct
55 Correct 19 ms 11392 KB Output is correct
56 Correct 17 ms 11264 KB Output is correct
57 Correct 17 ms 11264 KB Output is correct
58 Correct 16 ms 11392 KB Output is correct
59 Correct 12 ms 10496 KB Output is correct
60 Correct 107 ms 12948 KB Output is correct
61 Correct 16 ms 10880 KB Output is correct
62 Correct 124 ms 16760 KB Output is correct
63 Correct 484 ms 69752 KB Output is correct
64 Correct 473 ms 69600 KB Output is correct
65 Correct 476 ms 69752 KB Output is correct
66 Correct 158 ms 13432 KB Output is correct
67 Correct 271 ms 22136 KB Output is correct
68 Correct 145 ms 13432 KB Output is correct
69 Correct 409 ms 22904 KB Output is correct
70 Correct 156 ms 13432 KB Output is correct
71 Correct 169 ms 13688 KB Output is correct
72 Correct 315 ms 22264 KB Output is correct
73 Correct 331 ms 22392 KB Output is correct
74 Correct 913 ms 34808 KB Output is correct
75 Correct 576 ms 28772 KB Output is correct
76 Correct 662 ms 34808 KB Output is correct
77 Correct 666 ms 34936 KB Output is correct
78 Correct 236 ms 22264 KB Output is correct
79 Correct 388 ms 23840 KB Output is correct
80 Correct 232 ms 22216 KB Output is correct
81 Correct 359 ms 23672 KB Output is correct
82 Correct 704 ms 50184 KB Output is correct
83 Correct 704 ms 50552 KB Output is correct
84 Correct 615 ms 50168 KB Output is correct
85 Correct 611 ms 50168 KB Output is correct
86 Correct 178 ms 12664 KB Output is correct
87 Correct 187 ms 13688 KB Output is correct
88 Correct 331 ms 22392 KB Output is correct
89 Correct 658 ms 34424 KB Output is correct