| From | To | Weight |
|---|---|---|
| A | C | 2 |
| B | C | 1 |
| D | E | 2 |
| B | D | 5 |
| A | B | 4 |
These edges form one valid minimum spanning tree for the built-in example graph.
A minimum spanning tree (MST) is defined on a weighted graph G = (V, E), where V is the set of nodes and E is the set of weighted links. The MST is a subset of edges T \u2286 E that connects all nodes without cycles and minimizes the total weight.
The objective is: Minimize \( W(T) = \sum_{(u,v)\in T} w(u,v) \) subject to: (1) all nodes are connected, (2) \(|T| = |V|-1\), and (3) no cycles exist.
- Choose Edge List or Adjacency Matrix input mode.
- Enter your weighted links using node labels and numeric weights.
- Select an algorithm: Kruskal or Prim.
- Keep undirected enabled for standard spanning trees.
- Click Calculate to see results above the form.
- Use Download CSV or Download PDF to export.
1) What a Minimum Spanning Tree Represents
A minimum spanning tree (MST) selects a set of links that connects every node while keeping the total cost as small as possible. If your graph has N nodes, the MST contains exactly N−1 edges and never forms a cycle. This calculator reports the chosen edges and the final weight sum, letting you compare algorithms with the same input.
2) Why MST Ideas Appear in Physics
Many physics problems can be expressed as networks: sensor nodes in an experiment, lattice sites in a discretized model, or stations in a measurement pipeline. When you assign each link a “cost” such as cable length, attenuation, latency, or construction effort, an MST provides a baseline wiring plan that preserves global connectivity with minimal total cost.
3) Choosing Meaningful Edge Weights
The MST solution is only as good as the weight definition. In practice, weights often combine multiple terms, for example w = a·L + b·A where L is distance and A is expected loss. Use consistent units and ensure larger weights truly mean “less desirable” links. If two parallel options exist, keep the smallest weight to reflect the best route.
4) Kruskal and Prim in One View
Kruskal sorts all edges by weight and adds them if they do not create a cycle. Prim grows a single tree from a start node, repeatedly taking the cheapest edge that expands the visited set. For connected, undirected graphs, both return the same minimum total weight, but their intermediate steps can differ and are useful for learning.
5) Connectivity, Disconnected Graphs, and Forests
An MST requires the graph to be connected. If your inputs contain isolated components, the output becomes a minimum spanning forest: each component gets its own tree, and the calculator warns that a single spanning tree cannot cover all nodes. Fix this by adding physical links or by restricting the analysis to a connected region.
6) Complexity and Practical Input Size
Kruskal is dominated by sorting edges, typically O(E log E). Prim can be implemented in different ways; this calculator uses a clear method suitable for small and medium datasets. For dense graphs, an adjacency matrix can be convenient, while an edge list is compact for sparse networks.
7) Recommended Workflow for Reliable Results
Start with a cleaned edge list, confirm node labels, then run both algorithms as a consistency check. Enable the step log if you want to see why certain edges are skipped (cycle prevention) or chosen (lowest frontier cost). Adjust rounding only for display; the underlying selection uses the original numeric weights.
8) Example Interpretation with Real Numbers
Suppose five nodes represent detector stations and weights represent cable meters: the MST might choose four edges totaling 11.0 m instead of a fully connected design that exceeds 20 m. This does not claim optimal signal quality; it provides the minimal-cost backbone. You can refine weights to include loss or shielding to better match experimental constraints.
1) What is the difference between an MST and the shortest path?
An MST connects all nodes with minimum total weight. A shortest path minimizes the distance between two specific nodes. You can have a short path that does not minimize the overall network cost.
2) Why does the MST always have N−1 edges?
Any cycle is unnecessary for connectivity. Removing one edge from a cycle keeps nodes connected and reduces or preserves total weight. A cycle-free connected graph with N nodes must have exactly N−1 edges.
3) Can this calculator handle directed graphs?
Classical MST is defined for undirected graphs. If you uncheck “undirected,” the interpretation becomes nonstandard. For directed networks, you usually need an arborescence method rather than MST.
4) What if multiple MSTs exist with the same total weight?
This happens when several edges share equal weights. The calculator will output one valid MST based on sorting and tie behavior. The total weight remains the same across all equivalent MST solutions.
5) How should I choose a start node for Prim?
For connected undirected graphs, the final MST weight is independent of the start node. Pick any node that is convenient. Steps may look different, but the minimum total cost is unchanged.
6) Why do I get a “disconnected graph” warning?
Your inputs contain at least one node that cannot be reached from others using the provided edges. Add missing links, correct labels, or analyze each connected component separately to obtain a full spanning tree.
7) Are negative weights allowed?
Yes. Kruskal and Prim still work with negative weights, and the MST may prefer those edges. Ensure negative values truly represent lower cost in your model to avoid unintended selections.