/** * @file * @brief [Disjoint union](https://en.wikipedia.org/wiki/Disjoint_union) * * @details * The Disjoint union is the technique to find connected component in graph efficiently. * * ### Algorithm * In Graph, if you have to find out the number of connected components, there are 2 options * 1. Depth first search * 2. Disjoint union * 1st option is inefficient, Disjoint union is the most optimal way to find this. */ #include /// for io operations #include /// for std::set #include /// for std::vector /** * @namespace graph * @brief Graph Algorithms */ namespace graph { /** * @namespace disjoint_union * @brief Function for [Disjoint union] (https://en.wikipedia.org/wiki/Disjoint_union) implementation */ namespace disjoint_union { int64_t number_of_nodes; // denotes number of nodes std::vector parent; // parent of each node std::vector connected_set_size; // size of each set /** * @brief function the initialize every node as it's own parent * @returns void */ void make_set() { for (int64_t i = 1; i <= number_of_nodes; i++) { parent[i] = i; connected_set_size[i] = 1; } } /** * @brief To find the component where following node belongs to * @param val parent of val should be found * @return parent of val */ int64_t find_set(int64_t val) { while (parent[val] != val) { parent[val] = parent[parent[val]]; val = parent[val]; } return val; } /** * @brief To join 2 components to belong to one * @param node1 1st component * @param node2 2nd component * @returns void */ void union_sets(int64_t node1, int64_t node2) { node1 = find_set(node1); // find the parent of node1 node2 = find_set(node2); // find the parent of node2 // If parents of both nodes are not same, combine them if (node1 != node2) { if (connected_set_size[node1] < connected_set_size[node2]) { std::swap(node1, node2); // swap both components } parent[node2] = node1; // make node1 as parent of node2. connected_set_size[node1] += connected_set_size[node2]; // sum the size of both as they combined } } /** * @brief To find total no of connected components * @return Number of connected components */ int64_t no_of_connected_components() { std::set temp; // temp set to count number of connected components for (int64_t i = 1; i <= number_of_nodes; i++) temp.insert(find_set(i)); return temp.size(); // return the size of temp set } /** * @brief Test Implementations * @returns void */ static void test() { std::cin >> number_of_nodes; parent.resize(number_of_nodes + 1); connected_set_size.resize(number_of_nodes + 1); make_set(); int64_t edges = 0; std::cin >> edges; // no of edges in the graph while (edges--) { int64_t node_a = 0, node_b = 0; std::cin >> node_a >> node_b; union_sets(node_a, node_b); } std::cout << no_of_connected_components() << std::endl; } } // namespace disjoint_union } // namespace graph /** * @brief Main function * @returns 0 on exit */ int main() { graph::disjoint_union::test(); // Execute the tests return 0; }