Syntax error Difference Between Prim’s and Kruskal’s Algorithm

Difference Between Prim’s and Kruskal’s Algorithm

In this post, we will understand the differences between Prim's and Kruskal's algorithms.

Kruskal's algorithm for Mininum Spanning Tree (MST)

  • When a connected and undirected graph is given, a spanning tree of such a graph is the subgraph which is a tree that connects all of the vertices.
  • A single graph can have multiple spanning trees.
  • A minimum spanning tree (MST) (also known as minimum weight spanning tree) for a weighted, connected and undirected graph is a spanning tree that weighs less than or equal to the weight of every other spanning tree.
  • The weight of a spanning tree is determined by adding the weights associated with every edge of the spanning tree.
  • The Minimum Spanning Tree can be built from the vertex that has the minimum weight in the graph.
  • One node is traversed only once.
  • It runs quickly in sparse graphs.
  • The time complexity is O(E log V), where V is the number of vertices.
  • It can work with disconnected components too.

Steps to find MST using the Kruskal's algorithm:

  • Sort the edges in ascending order of their associated weight.
  • Select the smallest edge.
  • Check to see if it forms a cycle with the spanning-tree that has been formed until that point in time.
  • If the cycle hasn't been formed, this edge has to be included.
  • Otherwise, it can be discarded.
  • Steps 2,3,4 are repeated until the spanning tree contains V-1 edges.

Prim's algorithm Mininum Spanning Tree (MST)

  • This is similar to Kruskal's algorithm, i.e it is a greedy algorithm.
  • It begins with an empty spanning tree. Two sets of vertices are maintained.
  • The first set would contain the vertices which are already included in the MST, whereas the other set contains the vertices that haven't been included yet.
  • At every step, the algorithm considers all the edges that would connect the two sets. It then picks the minimum weight edge among these edges.
  • After this step, the algorithm moves to the other endpoint of the edge of the set that contains MST.
  • The Minimum Spanning Tree can be built from any vertex in the graph.
  • One node is travelled multiple times to get the minimum distance value.
  • It has a time complexity of O(V2), where V is the number of vertices. This time complexity can be enhanced to O(E + log V) using Fibonacci heaps.
  • It runs quickly in dense graphs.
  • It gives the connected components, and works with connected graph only.

Steps to find MST using the Prim's algorithm:

  • An mstSet is created, that keeps track of vertices that are already included in MST.
  • A key value is assigned to all vertices of the input graph.
  • The key values are initially assigned as 'INFINITE'.
  • The key value 0 is assigned to the first vertex so that it can be picked first.
  • When mstSet doesn't include all of the vertices, the below steps are followed.
    • A vertex 'u' is chosen that is not present in the mstSet and also has minimum key value.
    • This 'u' is now included in mstSet.
    • The key value of all the adjacent vertices of 'u' are updated.
    • This can be done by iterating through all the adjacent vertices.
    • For every adjacent vertex 'v', if the weight of edge 'u'-'v' is less than that of the previous key value of 'v', the key value is updated as the weight of 'u-v'.
Updated on: 2021-03-02T05:02:09+05:30

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