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답안 #212021

# 제출 시각 아이디 문제 언어 결과 실행 시간 메모리
212021 2020-03-22T03:59:48 Z rama_pang 조이터에서 친구를 만드는건 재밌어 (JOI20_joitter2) C++14
100 / 100
 943 ms 72184 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. If we use a map of sets, each merge takes O(log^2).
// This can be sped up using gp_hash_table of gp_hash_tables, speeding up each merge to O(1)
// amortized complexity.
//
// Time: O(n log n)
// Memory: O(n)

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

const int MAXN = 100005;
const int MAXM = 300005;

int N, M;
int A[MAXM], B[MAXM];

long long Answer = 0;

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

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

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;
    Answer += 1ll * total_size_ingoing_edges_from_component[y] * components_x_size;

    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()) {
      int f, s;
      tie(f, s) = *begin(todo);
      todo.erase(make_pair(f, s));
      ConnectComponent(f, s);
    }
  } else {
    if (ingoing_edges_from_component[y][x].count(real_x) == 0) { // if edge doesn't exist already
      Answer += disjoint_set::component_size[y];

      outgoing_edges::Insert(x, y, 1);
      ingoing_edges::Insert(y, x, real_x);
    }
  }
}

void init() {
  disjoint_set::Init();
}

void solve() {
  for (int i = 0; i < M; i++) {
    AddEdge(A[i], B[i]);
    cout << Answer << "\n";
  } 
}

int main() {
  read();
  init();
  solve();
  return 0;
}

Compilation message

joitter2.cpp: In function 'void AddEdge(int, int)':
joitter2.cpp:221:7: warning: unused variable 'real_y' [-Wunused-variable]
   int real_y = y;
       ^~~~~~

Subtask #1
1.0 / 1.0

# 결과 실행 시간 메모리 Grader output
1 Correct 10 ms 10496 KB Output is correct
2 Correct 10 ms 10528 KB Output is correct
3 Correct 10 ms 10496 KB Output is correct
4 Correct 10 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 12 ms 10752 KB Output is correct
8 Correct 11 ms 10624 KB Output is correct
9 Correct 11 ms 10624 KB Output is correct
10 Correct 10 ms 10496 KB Output is correct
11 Correct 10 ms 10496 KB Output is correct
12 Correct 11 ms 10624 KB Output is correct
13 Correct 11 ms 10496 KB Output is correct
14 Correct 11 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 10 ms 10496 KB Output is correct
18 Correct 10 ms 10624 KB Output is correct
19 Correct 11 ms 10496 KB Output is correct
20 Correct 11 ms 10496 KB Output is correct
21 Correct 11 ms 10624 KB Output is correct
22 Correct 11 ms 10624 KB Output is correct
23 Correct 11 ms 10624 KB Output is correct
24 Correct 10 ms 10496 KB Output is correct
25 Correct 12 ms 10624 KB Output is correct
26 Correct 10 ms 10496 KB Output is correct
27 Correct 10 ms 10624 KB Output is correct
28 Correct 10 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 11 ms 10496 KB Output is correct
32 Correct 12 ms 10496 KB Output is correct
33 Correct 11 ms 10624 KB Output is correct

Subtask #2
16.0 / 16.0

# 결과 실행 시간 메모리 Grader output
1 Correct 10 ms 10496 KB Output is correct
2 Correct 10 ms 10528 KB Output is correct
3 Correct 10 ms 10496 KB Output is correct
4 Correct 10 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 12 ms 10752 KB Output is correct
8 Correct 11 ms 10624 KB Output is correct
9 Correct 11 ms 10624 KB Output is correct
10 Correct 10 ms 10496 KB Output is correct
11 Correct 10 ms 10496 KB Output is correct
12 Correct 11 ms 10624 KB Output is correct
13 Correct 11 ms 10496 KB Output is correct
14 Correct 11 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 10 ms 10496 KB Output is correct
18 Correct 10 ms 10624 KB Output is correct
19 Correct 11 ms 10496 KB Output is correct
20 Correct 11 ms 10496 KB Output is correct
21 Correct 11 ms 10624 KB Output is correct
22 Correct 11 ms 10624 KB Output is correct
23 Correct 11 ms 10624 KB Output is correct
24 Correct 10 ms 10496 KB Output is correct
25 Correct 12 ms 10624 KB Output is correct
26 Correct 10 ms 10496 KB Output is correct
27 Correct 10 ms 10624 KB Output is correct
28 Correct 10 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 11 ms 10496 KB Output is correct
32 Correct 12 ms 10496 KB Output is correct
33 Correct 11 ms 10624 KB Output is correct
34 Correct 12 ms 10752 KB Output is correct
35 Correct 106 ms 16760 KB Output is correct
36 Correct 128 ms 19576 KB Output is correct
37 Correct 123 ms 19704 KB Output is correct
38 Correct 127 ms 19320 KB Output is correct
39 Correct 12 ms 10624 KB Output is correct
40 Correct 13 ms 10752 KB Output is correct
41 Correct 14 ms 10752 KB Output is correct
42 Correct 13 ms 10624 KB Output is correct
43 Correct 16 ms 10880 KB Output is correct
44 Correct 13 ms 10752 KB Output is correct
45 Correct 12 ms 10624 KB Output is correct
46 Correct 13 ms 10656 KB Output is correct
47 Correct 17 ms 10752 KB Output is correct
48 Correct 14 ms 10752 KB Output is correct
49 Correct 27 ms 11768 KB Output is correct
50 Correct 128 ms 19832 KB Output is correct
51 Correct 19 ms 11136 KB Output is correct
52 Correct 108 ms 17784 KB Output is correct
53 Correct 22 ms 11520 KB Output is correct
54 Correct 118 ms 18808 KB Output is correct
55 Correct 16 ms 11392 KB Output is correct
56 Correct 17 ms 11392 KB Output is correct
57 Correct 16 ms 11392 KB Output is correct
58 Correct 16 ms 11392 KB Output is correct
59 Correct 12 ms 10624 KB Output is correct
60 Correct 107 ms 15356 KB Output is correct
61 Correct 15 ms 10880 KB Output is correct
62 Correct 120 ms 19064 KB Output is correct

Subtask #3
83.0 / 83.0

# 결과 실행 시간 메모리 Grader output
1 Correct 10 ms 10496 KB Output is correct
2 Correct 10 ms 10528 KB Output is correct
3 Correct 10 ms 10496 KB Output is correct
4 Correct 10 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 12 ms 10752 KB Output is correct
8 Correct 11 ms 10624 KB Output is correct
9 Correct 11 ms 10624 KB Output is correct
10 Correct 10 ms 10496 KB Output is correct
11 Correct 10 ms 10496 KB Output is correct
12 Correct 11 ms 10624 KB Output is correct
13 Correct 11 ms 10496 KB Output is correct
14 Correct 11 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 10 ms 10496 KB Output is correct
18 Correct 10 ms 10624 KB Output is correct
19 Correct 11 ms 10496 KB Output is correct
20 Correct 11 ms 10496 KB Output is correct
21 Correct 11 ms 10624 KB Output is correct
22 Correct 11 ms 10624 KB Output is correct
23 Correct 11 ms 10624 KB Output is correct
24 Correct 10 ms 10496 KB Output is correct
25 Correct 12 ms 10624 KB Output is correct
26 Correct 10 ms 10496 KB Output is correct
27 Correct 10 ms 10624 KB Output is correct
28 Correct 10 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 11 ms 10496 KB Output is correct
32 Correct 12 ms 10496 KB Output is correct
33 Correct 11 ms 10624 KB Output is correct
34 Correct 12 ms 10752 KB Output is correct
35 Correct 106 ms 16760 KB Output is correct
36 Correct 128 ms 19576 KB Output is correct
37 Correct 123 ms 19704 KB Output is correct
38 Correct 127 ms 19320 KB Output is correct
39 Correct 12 ms 10624 KB Output is correct
40 Correct 13 ms 10752 KB Output is correct
41 Correct 14 ms 10752 KB Output is correct
42 Correct 13 ms 10624 KB Output is correct
43 Correct 16 ms 10880 KB Output is correct
44 Correct 13 ms 10752 KB Output is correct
45 Correct 12 ms 10624 KB Output is correct
46 Correct 13 ms 10656 KB Output is correct
47 Correct 17 ms 10752 KB Output is correct
48 Correct 14 ms 10752 KB Output is correct
49 Correct 27 ms 11768 KB Output is correct
50 Correct 128 ms 19832 KB Output is correct
51 Correct 19 ms 11136 KB Output is correct
52 Correct 108 ms 17784 KB Output is correct
53 Correct 22 ms 11520 KB Output is correct
54 Correct 118 ms 18808 KB Output is correct
55 Correct 16 ms 11392 KB Output is correct
56 Correct 17 ms 11392 KB Output is correct
57 Correct 16 ms 11392 KB Output is correct
58 Correct 16 ms 11392 KB Output is correct
59 Correct 12 ms 10624 KB Output is correct
60 Correct 107 ms 15356 KB Output is correct
61 Correct 15 ms 10880 KB Output is correct
62 Correct 120 ms 19064 KB Output is correct
63 Correct 480 ms 72184 KB Output is correct
64 Correct 459 ms 72056 KB Output is correct
65 Correct 468 ms 71976 KB Output is correct
66 Correct 148 ms 14968 KB Output is correct
67 Correct 266 ms 23672 KB Output is correct
68 Correct 135 ms 14968 KB Output is correct
69 Correct 403 ms 24396 KB Output is correct
70 Correct 156 ms 14972 KB Output is correct
71 Correct 161 ms 15060 KB Output is correct
72 Correct 305 ms 23800 KB Output is correct
73 Correct 346 ms 23836 KB Output is correct
74 Correct 943 ms 37112 KB Output is correct
75 Correct 624 ms 31172 KB Output is correct
76 Correct 659 ms 37240 KB Output is correct
77 Correct 663 ms 37368 KB Output is correct
78 Correct 219 ms 23416 KB Output is correct
79 Correct 361 ms 26104 KB Output is correct
80 Correct 211 ms 23416 KB Output is correct
81 Correct 342 ms 26104 KB Output is correct
82 Correct 690 ms 51820 KB Output is correct
83 Correct 702 ms 51704 KB Output is correct
84 Correct 619 ms 51832 KB Output is correct
85 Correct 610 ms 51708 KB Output is correct
86 Correct 151 ms 14200 KB Output is correct
87 Correct 194 ms 16248 KB Output is correct
88 Correct 314 ms 23800 KB Output is correct
89 Correct 637 ms 36856 KB Output is correct