Prim's Algorithm
Minimum Spanning Tree
A spanning tree of a graph is a subgraph that includes all the vertices of the original graph and is also a tree (connected and acyclic). In simpler terms, it’s a way to connect all the points in your graph without any loops, using the fewest possible edges. If your graph is weighted, a minimum spanning tree is a spanning tree with the smallest possible total edge weight. For better understanding of minimum spanning tree and Prim’s Algorithm watch this video.
Prim's Algorithm
Watching this video, you might you under the way Prim’s algorithm work. Also refer this algorithm.
Prim’s algorithm is a greedy algorithm used to find a minimum spanning tree for a weighted, undirected graph. Here’s how it works:
- Initialization:
- Start with an arbitrary vertex in the graph and mark it as visited.
- Create an empty set to store the edges of the minimum spanning tree.
- Iteration:
- Find the edge with the smallest weight that connects a visited vertex to an unvisited vertex.
- Add this edge to the minimum spanning tree set.
- Mark the newly connected vertex as visited.
- Termination:
- Repeat step 2 until all vertices in the graph are visited.
The result is a minimum spanning tree for the graph
Prim's Algorithm and Greedy Strategy
Prim’s algorithm uses a greedy strategy to find the minimum spanning tree. “Greedy” means that at each step, the algorithm makes the locally optimal choice without considering the overall, long-term consequences. In Prim’s algorithm, this translates to always picking the edge with the smallest weight that connects a vertex already in the spanning tree to a vertex not yet included.
Here’s how the greedy strategy plays out in Prim’s algorithm:
- Local Optimization: At each iteration, the algorithm only considers the edges connected to the vertices currently in the spanning tree. It selects the edge with the minimum weight among these, regardless of how this choice might affect future edge selections. This is a purely local decision.
- No Backtracking: Once an edge is added to the spanning tree, it’s never removed. The algorithm doesn’t reconsider previous choices, even if they later turn out to be suboptimal in the context of the entire tree. This lack of backtracking is a hallmark of greedy algorithms.
- Global Optimality: While each step is locally optimal, the greedy strategy, in the case of Prim’s algorithm, remarkably leads to a globally optimal solution, i.e., a minimum spanning tree. This isn’t always the case with greedy algorithms; for many other problems, greedy approaches only find approximate solutions.
It’s important to note that the greedy strategy doesn’t guarantee the absolute minimum spanning tree in cases where multiple minimum spanning trees exist with the same total weight. Prim’s algorithm will find one of these minimum spanning trees, but not necessarily all of them.
