union_find_example.C

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/*
                          Aleph_w
 
  Data structures & Algorithms
  version 2.0.0b
  https://github.com/lrleon/Aleph-w
 
  This file is part of Aleph-w library
 
  Copyright (c) 2002-2026 Leandro Rabindranath Leon
 
  Permission is hereby granted, free of charge, to any person obtaining a copy
  of this software and associated documentation files (the "Software"), to deal
  in the Software without restriction, including without limitation the rights
  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
  copies of the Software, and to permit persons to whom the Software is
  furnished to do so, subject to the following conditions:
 
  The above copyright notice and this permission notice shall be included in all
  copies or substantial portions of the Software.
 
  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
  SOFTWARE.
*/
 
 
#include <iostream>
#include <string>
#include <vector>
#include <tpl_union.H>
 
using namespace std;
using namespace Aleph;
 
// =============================================================================
// Example 1: Basic Union-Find Operations
// =============================================================================
 
void demo_basic_operations()
{
  cout << "\n";
  cout << "╔══════════════════════════════════════════════════════════════════╗\n";
  cout << "║          EXAMPLE 1: Basic Union-Find Operations                  ║\n";
  cout << "╚══════════════════════════════════════════════════════════════════╝\n\n";
 
  // Fixed_Relation: For elements identified by integers 0..n-1
  const size_t n = 10;
  Fixed_Relation uf(n);
 
  cout << "Created Union-Find with " << n << " elements (0 to 9)\n";
  cout << "Initially, each element is in its own singleton set.\n\n";
 
  // Show initial state - demonstrate which elements are connected
  cout << "Initial state: Each element is isolated in its own set.\n";
  cout << "Number of disjoint sets: " << uf.get_num_blocks() << "\n";
 
  cout << "\n--- Performing unions ---\n\n";
 
  // Union some sets
  cout << "join(0, 1): Merge sets containing 0 and 1\n";
  uf.join(0, 1);
 
  cout << "join(2, 3): Merge sets containing 2 and 3\n";
  uf.join(2, 3);
 
  cout << "join(4, 5): Merge sets containing 4 and 5\n";
  uf.join(4, 5);
 
  cout << "join(0, 2): Merge sets {0,1} and {2,3} into {0,1,2,3}\n";
  uf.join(0, 2);
 
  cout << "\n--- Checking connectivity ---\n\n";
 
  // Check if elements are in the same set
  auto check = [&uf](size_t a, size_t b) {
    bool same = uf.are_connected(a, b);
    cout << "  " << a << " and " << b << " are " 
         << (same ? "in the SAME set" : "in DIFFERENT sets") << "\n";
  };
 
  cout << "Connectivity queries:\n";
  check(0, 3);  // Should be same (both in {0,1,2,3})
  check(4, 5);  // Should be same (both in {4,5})
  check(1, 4);  // Should be different
  check(6, 7);  // Should be different (both singletons)
 
  cout << "\n--- Final state ---\n\n";
  
  // Show connectivity relationships
  cout << "Elements in same set as 0: ";
  for (size_t i = 0; i < n; ++i)
    if (uf.are_connected(0, i))
      cout << i << " ";
  cout << "\n";
 
  cout << "Elements in same set as 4: ";
  for (size_t i = 0; i < n; ++i)
    if (uf.are_connected(4, i))
      cout << i << " ";
  cout << "\n";
 
  cout << "\nNumber of disjoint sets: " << uf.get_num_blocks() << "\n";
}
 
// =============================================================================
// Example 2: Network Connectivity Problem
// =============================================================================
 
void demo_network_connectivity()
{
  cout << "\n";
  cout << "╔══════════════════════════════════════════════════════════════════╗\n";
  cout << "║          EXAMPLE 2: Network Connectivity Problem                 ║\n";
  cout << "╚══════════════════════════════════════════════════════════════════╝\n\n";
 
  cout << "Problem: A network has 8 computers. Given a list of direct\n";
  cout << "connections, determine which computers can communicate.\n\n";
 
  const size_t num_computers = 8;
  Fixed_Relation network(num_computers);
 
  // Network topology (direct connections)
  vector<pair<size_t, size_t>> connections = {
    {0, 1}, {1, 2},           // Computers 0-1-2 connected
    {3, 4}, {4, 5}, {5, 3},   // Computers 3-4-5 connected (triangle)
    {6, 7}                     // Computers 6-7 connected
  };
 
  cout << "Network topology (direct connections):\n";
  cout << "  Cluster A: 0 — 1 — 2\n";
  cout << "  Cluster B: 3 — 4 — 5 (triangle)\n";
  cout << "  Cluster C: 6 — 7\n";
  cout << "  Isolated:  (none)\n\n";
 
  // Add connections
  cout << "Adding connections:\n";
  for (auto& [a, b] : connections) {
    cout << "  Connect " << a << " ↔ " << b << "\n";
    network.join(a, b);
  }
 
  cout << "\n--- Communication queries ---\n\n";
 
  auto can_communicate = [&](size_t a, size_t b) {
    bool can = network.are_connected(a, b);
    cout << "  Computer " << a << " and " << b << ": "
         << (can ? "✓ CAN communicate" : "✗ CANNOT communicate") << "\n";
  };
 
  cout << "Testing if computers can communicate:\n";
  can_communicate(0, 2);   // Yes (same cluster)
  can_communicate(3, 5);   // Yes (same cluster)
  can_communicate(0, 3);   // No (different clusters)
  can_communicate(6, 7);   // Yes (same cluster)
  can_communicate(2, 6);   // No (different clusters)
 
  cout << "\n--- Adding a bridge connection ---\n\n";
 
  cout << "Adding connection 2 ↔ 3 (bridges Cluster A and B)\n";
  network.join(2, 3);
 
  cout << "\nUpdated connectivity:\n";
  can_communicate(0, 5);   // Now yes!
  can_communicate(1, 4);   // Now yes!
 
  cout << "\nTotal isolated networks: " << network.size() << "\n";
}
 
// =============================================================================
// Example 3: Kruskal's MST (Simplified)
// =============================================================================
 
struct Edge {
  size_t u, v;
  double weight;
  
  bool operator<(const Edge& other) const {
    return weight < other.weight;
  }
};
 
void demo_kruskal_simulation()
{
  cout << "\n";
  cout << "╔══════════════════════════════════════════════════════════════════╗\n";
  cout << "║          EXAMPLE 3: Kruskal's MST Algorithm Simulation           ║\n";
  cout << "╚══════════════════════════════════════════════════════════════════╝\n\n";
 
  cout << "Kruskal's algorithm finds the Minimum Spanning Tree by:\n";
  cout << "  1. Sort edges by weight\n";
  cout << "  2. For each edge (u,v): add to MST if u,v in different sets\n";
  cout << "  3. Union-Find tracks which vertices are connected\n\n";
 
  // Graph with 6 vertices
  const size_t V = 6;
  
  // Edges: (u, v, weight)
  vector<Edge> edges = {
    {0, 1, 4.0}, {0, 2, 2.0}, {1, 2, 1.0}, {1, 3, 5.0},
    {2, 3, 8.0}, {2, 4, 10.0}, {3, 4, 2.0}, {3, 5, 6.0}, {4, 5, 3.0}
  };
 
  cout << "Graph edges:\n";
  for (const auto& e : edges)
    cout << "  " << e.u << " —(" << e.weight << ")— " << e.v << "\n";
 
  // Sort edges by weight
  sort(edges.begin(), edges.end());
 
  cout << "\nSorted edges by weight:\n";
  for (const auto& e : edges)
    cout << "  " << e.u << " —(" << e.weight << ")— " << e.v << "\n";
 
  // Kruskal's algorithm using Union-Find
  Fixed_Relation uf(V);
  vector<Edge> mst;
  double total_weight = 0;
 
  cout << "\n--- Running Kruskal's algorithm ---\n\n";
 
  for (const auto& e : edges) {
    if (!uf.are_connected(e.u, e.v)) {
      cout << "  ADD edge " << e.u << " —(" << e.weight << ")— " << e.v
           << " (connects different components)\n";
      uf.join(e.u, e.v);
      mst.push_back(e);
      total_weight += e.weight;
    } else {
      cout << "  SKIP edge " << e.u << " —(" << e.weight << ")— " << e.v
           << " (would create cycle)\n";
    }
 
    // MST has V-1 edges
    if (mst.size() == V - 1)
      break;
  }
 
  cout << "\n--- Minimum Spanning Tree ---\n\n";
  cout << "MST edges:\n";
  for (const auto& e : mst)
    cout << "  " << e.u << " —(" << e.weight << ")— " << e.v << "\n";
  cout << "\nTotal MST weight: " << total_weight << "\n";
}
 
// =============================================================================
// Example 4: Path Compression Visualization
// =============================================================================
 
void demo_path_compression()
{
  cout << "\n";
  cout << "╔══════════════════════════════════════════════════════════════════╗\n";
  cout << "║          EXAMPLE 4: Path Compression Effect                      ║\n";
  cout << "╚══════════════════════════════════════════════════════════════════╝\n\n";
 
  cout << "Path compression flattens the tree during find operations,\n";
  cout << "making subsequent finds much faster.\n\n";
 
  const size_t n = 8;
  Fixed_Relation uf(n);
 
  // Create a chain structure by sequential unions
  cout << "Creating connected components by sequential unions:\n";
  for (size_t i = 0; i < n - 1; ++i) {
    cout << "  join(" << i << ", " << (i + 1) << ")\n";
    uf.join(i, i + 1);
  }
 
  cout << "\nAfter all joins:\n";
  cout << "  All elements 0-" << (n-1) << " are now in the same set.\n";
  cout << "  Number of sets: " << uf.get_num_blocks() << "\n";
 
  cout << "\nVerifying all elements are connected:\n";
  for (size_t i = 1; i < n; ++i) {
    bool conn = uf.are_connected(0, i);
    cout << "  0 connected to " << i << ": " << (conn ? "YES" : "NO") << "\n";
  }
 
  cout << "\nPath compression happens automatically during are_connected().\n";
  cout << "The internal tree structure is flattened, making future\n";
  cout << "operations nearly O(1) - this is the key to Union-Find's speed!\n";
}
 
// =============================================================================
// Main
// =============================================================================
 
int main()
{
  cout << "\n";
  cout << "╔══════════════════════════════════════════════════════════════════╗\n";
  cout << "║     UNION-FIND (Disjoint Set Union) Data Structure Demo          ║\n";
  cout << "║                                                                  ║\n";
  cout << "║     Aleph-w Library - https://github.com/lrleon/Aleph-w          ║\n";
  cout << "╚══════════════════════════════════════════════════════════════════╝\n";
 
  demo_basic_operations();
  demo_network_connectivity();
  demo_kruskal_simulation();
  demo_path_compression();
 
  cout << "\n";
  cout << "╔══════════════════════════════════════════════════════════════════╗\n";
  cout << "║                         Summary                                  ║\n";
  cout << "╠══════════════════════════════════════════════════════════════════╣\n";
  cout << "║  Union-Find is one of the most elegant data structures:          ║\n";
  cout << "║                                                                  ║\n";
  cout << "║  • Nearly O(1) operations via union-by-rank & path compression  ║\n";
  cout << "║  • Essential for Kruskal's MST algorithm                        ║\n";
  cout << "║  • Used in network connectivity, image processing, and more     ║\n";
  cout << "║                                                                  ║\n";
  cout << "║  Aleph-w provides Fixed_Relation for integer elements and       ║\n";
  cout << "║  Relation for arbitrary types with hash-based element lookup.   ║\n";
  cout << "╚══════════════════════════════════════════════════════════════════╝\n\n";
 
  return 0;
}