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Reference: introduction to algorithm

Kruscal Algorithm O(ElgV) by Binary Heaps

Prim's algorithm O(E + VlgV) by Fibonacci heaps

A problem os a spanning tree

for connected, undirected graph G=(V,E) with a weight function w: E->R

Kruskal'a algorithm

Implementation by C++

#include 
   
    #include 
    
     #include 
     
      #include 
      
       #include 
       
        using namespace std;class Solution{    // implementation by Weighted Quick Union.    struct DisjSets    {        vector
        
          nodes; vector
         
           sizes; int count; // the default five member for DisjSets. DisjSets(int Count) : nodes(Count, -1), sizes(Count, 1), count(Count) {} DisjSets(const DisjSets &rhs) = default; DisjSets(DisjSets &&rhs) = default; DisjSets &operator=(const DisjSets &rhs) = default; DisjSets &operator=(DisjSets &&rhs) = default; // weighted quick union void unionSet(int p, int q) { int i = find(p); int j = find(q); if (i == j) return; if (sizes[i] < sizes[j]) { nodes[i] = j; sizes[j] += sizes[i]; } else { nodes[j] = i; sizes[i] += sizes[j]; } --count; } int find(int p) { while (nodes[p] != -1) p = nodes[p]; return p; } }; struct cmp { bool operator()(const tuple
          
            & l, const tuple
           
             & r) { return get<2>(l) > get<2>(r); } };public:/* Output: return a vector of vector. for the sub vector[0] for start point, vector[1] for the end point for the input sub vector Input: verticle the number of verticles. edges the input edges. start: edges[n][0] end: edges[n][1] weight: edges[n][3] */ vector
            
             
              > mstKruskal(int verticle, vector
              
               
                > &edges) { DisjSets Sets(verticle); vector
                
                 
                  > minSpanTree; // the tuple store the start, end ,and weight for the edges. priority_queue
                  
                   >, cmp> minHeap; for(auto & edge : edges) { minHeap.push(make_tuple(edge[0], edge[1], edge[2])); } while(!minHeap.empty()) { auto edge = minHeap.top(); int u = get<0>(edge), v = get<1>(edge); if(Sets.find(u) != Sets.find(v)) { minSpanTree.push_back(vector
                   
                    {u, v}); Sets.unionSet(u, v); } minHeap.pop(); } return minSpanTree; }};int main(){ vector
                    
                     
                      > edges = { vector
                      
                       {0, 1, 5}, vector
                       
                        {1, 2, 4}, vector
                        
                         {2, 3, 5}, vector
                         
                          {3, 4, 3}, vector
                          
                           {4, 5, 6}, vector
                           
                            {5, 0, 8}, vector
                            
                             {0, 2, 1}, vector
                             
                              {0, 3, 5}, vector
                              
                               {0, 4, 3} }; Solution sol; auto minSpanTree = sol.mstKruskal(6, edges); return 0;}
                              
                             
                            
                           
                          
                         
                        
                       
                      
                     
                    
                   
                  
                 
                
               
              
             
            
           
          
         
        
       
      
     
    
   

Prim's Algorithm

Prim implementation by C++ priority_queue

#include 
   
    #include 
    
     #include 
     
      #include 
      
       #include 
       
        #include 
        
         using namespace std;class Solution {private: struct cmp{ bool operator ()(const pair
         
           & l , const pair
          
            &r) { return l.second > r.second; } };public:/* 1) Initialize keys of all vertices as infinite and parent of every vertex as -1.2) Create an empty priority_queue pq. Every item of pq is a pair (weight, vertex). Weight (or key) is used used as first item of pair as first item is by default used to compare two pairs.3) Initialize all vertices as not part of MST yet. We use boolean array inMST[] for this purpose. This array is required to make sure that an already considered vertex is not included in pq again. This is where Ptim's implementation differs from Dijkstra. In Dijkstr's algorithm, we didn't need this array as distances always increase. We require this array here because key value of a processed vertex may decrease if not checked.4) Insert source vertex into pq and make its key as 0.5) While either pq doesn't become empty a) Extract minimum key vertex from pq. Let the extracted vertex be u. b) Include u in MST using inMST[u] = true. c) Loop through all adjacent of u and do following for every vertex v. // If weight of edge (u,v) is smaller than // key of v and v is not already in MST If inMST[v] = false && key[v] > weight(u, v) (i) Update key of v, i.e., do key[v] = weight(u, v) (ii) Insert v into the pq (iv) parent[v] = u 6) Print MST edges using parent array. */ vector
           
            
             > mstPrim(int verticle, vector
             
              
               > &edges) { constexpr int MAX_INT = std::numeric_limits
               
                ::max(); vector
                
                 
                  > results; vector
                  
                    Keys(verticle, MAX_INT); // store the weight of each node vector
                   
                     Parents(verticle, -1); // -1 means that the node doesn't have parent. vector
                    
                      inMST(verticle, false); // at initialization we add all node to the Priority. // the pair.first for adjacent node, pair.second for the weight unordered_map
                     
                      >> adjacents; // build the adjacents for(auto & edge : edges) { int u = edge[0], v = edge[1], weight = edge[2]; adjacents[u].push_back(make_pair(v, weight)); adjacents[v].push_back(make_pair(u, weight)); } Keys[0] = 0; //Assume node 0 is the root. // pair.first for node id, second for the weight. priority_queue
                      
                       
                        , vector
                        
                         
                          >, cmp> minHeap; minHeap.push(make_pair(0, 0)); while(!minHeap.empty()) { auto u = minHeap.top().first; minHeap.pop(); inMST[u] = true; for(auto & adj : adjacents[u]) { int v = adj.first; int weight = adj.second; if(inMST[v] == false && Keys[v] > weight) { // update key of v Keys[v] = weight; minHeap.push(make_pair(v, Keys[v])); Parents[v] = u; } } } for(int i = 0; i < Parents.size(); ++i) { if(Parents[i] != -1) { results.push_back(vector
                          
                           {i, Parents[i]}); } } return results; }};int main(){ vector
                           
                            
                             > edges = { vector
                             
                              {0, 1, 5}, vector
                              
                               {1, 2, 4}, vector
                               
                                {2, 3, 5}, vector
                                
                                 {3, 4, 3}, vector
                                 
                                  {4, 5, 6}, vector
                                  
                                   {5, 0, 8}, vector
                                   
                                    {0, 2, 1}, vector
                                    
                                     {0, 3, 5}, vector
                                     
                                      {0, 4, 3} }; Solution sol; auto minSpanTree = sol.mstPrim(6, edges); return 0;}
                                     
                                    
                                   
                                  
                                 
                                
                               
                              
                             
                            
                           
                          
                         
                        
                       
                      
                     
                    
                   
                  
                 
                
               
              
             
            
           
          
         
        
       
      
     
    
   

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