#### Boruvkas Minimum Spanning Tree

J
```﻿/**
* @author [Jason Nardoni](https://github.com/JNardoni)
* @file
*
* @brief
* [Borůvkas Algorithm](https://en.wikipedia.org/wiki/Borůvka's_algorithm) to
*find the Minimum Spanning Tree
*
*
* @details
* Boruvka's algorithm is a greepy algorithm to find the MST by starting with
*small trees, and combining them to build bigger ones.
*	1. Creates a group for every vertex.
*	2. looks through each edge of every vertex for the smallest weight. Keeps
*track of the smallest edge for each of the current groups.
*  3. Combine each group with the group it shares its smallest edge, adding the
*smallest edge to the MST.
*  4. Repeat step 2-3 until all vertices are combined into a single group.
*
* It assumes that the graph is connected. Non-connected edges can be
*represented using 0 or INT_MAX
*
*/

#include <cassert>   /// for assert
#include <climits>   /// for INT_MAX
#include <iostream>  /// for IO operations
#include <vector>    /// for std::vector

/**
* @namespace greedy_algorithms
* @brief Greedy Algorithms
*/
namespace greedy_algorithms {
/**
* @namespace boruvkas_minimum_spanning_tree
* @brief Functions for the [Borůvkas
* Algorithm](https://en.wikipedia.org/wiki/Borůvka's_algorithm) implementation
*/
namespace boruvkas_minimum_spanning_tree {
/**
* @brief Recursively returns the vertex's parent at the root of the tree
* @param parent the array that will be checked
* @param v vertex to find parent of
* @returns the parent of the vertex
*/
int findParent(std::vector<std::pair<int, int>> parent, const int v) {
if (parent[v].first != v) {
parent[v].first = findParent(parent, parent[v].first);
}

return parent[v].first;
}

/**
* @brief the implementation of boruvka's algorithm
* @returns the MST as 2d vectors
*/
size_t total_groups = size;

if (size <= 1) {
}

// Stores the current Minimum Spanning Tree. As groups are combined, they
// are added to the MST
std::vector<std::vector<int>> MST(size, std::vector<int>(size, INT_MAX));
for (int i = 0; i < size; i++) {
MST[i][i] = 0;
}

// Step 1: Create a group for each vertex

// Stores the parent of the vertex and its current depth, both initialized
// to 0
std::vector<std::pair<int, int>> parent(size, std::make_pair(0, 0));

for (int i = 0; i < size; i++) {
parent[i].first =
i;  // Sets parent of each vertex to itself, depth remains 0
}

// Repeat until all are in a single group
while (total_groups > 1) {
std::vector<std::pair<int, int>> smallest_edge(
size, std::make_pair(-1, -1));  // Pairing: start node, end node

// Step 2: Look throught each vertex for its smallest edge, only using
// the right half of the adj matrix
for (int i = 0; i < size; i++) {
for (int j = i + 1; j < size; j++) {
if (adj[i][j] == INT_MAX || adj[i][j] == 0) {  // No connection
continue;
}

// Finds the parents of the start and end points to make sure
// they arent in the same group
int parentA = findParent(parent, i);
int parentB = findParent(parent, j);

if (parentA != parentB) {
// Grabs the start and end points for the first groups
// current smallest edge
int start = smallest_edge[parentA].first;
int end = smallest_edge[parentA].second;

// If there is no current smallest edge, or the new edge is
// smaller, records the new smallest
smallest_edge[parentA].first = i;
smallest_edge[parentA].second = j;
}

// Does the same for the second group
start = smallest_edge[parentB].first;
end = smallest_edge[parentB].second;

smallest_edge[parentB].first = j;
smallest_edge[parentB].second = i;
}
}
}
}

// Step 3: Combine the groups based off their smallest edge

for (int i = 0; i < size; i++) {
// Makes sure the smallest edge exists
if (smallest_edge[i].first != -1) {
// Start and end points for the groups smallest edge
int start = smallest_edge[i].first;
int end = smallest_edge[i].second;

// Parents of the two groups - A is always itself
int parentA = i;
int parentB = findParent(parent, end);

// Makes sure the two nodes dont share the same parent. Would
// happen if the two groups have been
// merged previously through a common shortest edge
if (parentA == parentB) {
continue;
}

// Tries to balance the trees as much as possible as they are
// merged. The parent of the shallower
// tree will be pointed to the parent of the deeper tree.
if (parent[parentA].second < parent[parentB].second) {
parent[parentB].first = parentA;  // New parent
parent[parentB].second++;         // Increase depth
} else {
parent[parentA].first = parentB;
parent[parentA].second++;
}
// Add the connection to the MST, using both halves of the adj
// matrix
total_groups--;  // one fewer group
}
}
}
return MST;
}

/**
* @brief counts the sum of edges in the given tree
* @returns the int size of the tree
*/
int sum = 0;

// Moves through one side of the adj matrix, counting the sums of each edge
for (int i = 0; i < size; i++) {
for (int j = i + 1; j < size; j++) {
}
}
}
return sum;
}
}  // namespace boruvkas_minimum_spanning_tree
}  // namespace greedy_algorithms

/**
* @brief Self-test implementations
* @returns void
*/
static void tests() {
std::cout << "Starting tests...\n\n";
std::vector<std::vector<int>> graph = {
{0, 5, INT_MAX, 3, INT_MAX}, {5, 0, 2, INT_MAX, 5},
{INT_MAX, 2, 0, INT_MAX, 3}, {3, INT_MAX, INT_MAX, 0, INT_MAX},
{INT_MAX, 5, 3, INT_MAX, 0},
};
std::vector<std::vector<int>> MST =
greedy_algorithms::boruvkas_minimum_spanning_tree::boruvkas(graph);
assert(greedy_algorithms::boruvkas_minimum_spanning_tree::test_findGraphSum(
MST) == 13);
std::cout << "1st test passed!" << std::endl;

graph = {{0, 2, 0, 6, 0},
{2, 0, 3, 8, 5},
{0, 3, 0, 0, 7},
{6, 8, 0, 0, 9},
{0, 5, 7, 9, 0}};
MST = greedy_algorithms::boruvkas_minimum_spanning_tree::boruvkas(graph);
assert(greedy_algorithms::boruvkas_minimum_spanning_tree::test_findGraphSum(
MST) == 16);
std::cout << "2nd test passed!" << std::endl;
}

/**
* @brief Main function
* @returns 0 on exit
*/
int main() {
tests();  // run self-test implementations
return 0;
}
```