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A spanning tree algorithm selects edges from a connected, undirected graph so that every vertex is included, the resulting subgraph stays connected, and no cycle is formed. “Spanning tree algorithm” can mean a traversal such as breadth-first search (BFS) or depth-first search (DFS) that builds any spanning tree; it does not automatically mean an algorithm for finding the cheapest one.
What is a spanning tree?
For a connected, undirected graph G = (V, E), a spanning tree is a subgraph T = (V, ET) that includes the graph’s full vertex set, uses only edges from the original graph, and is itself a tree. In plain language, it reaches every vertex without creating a cycle. e-PG Pathshala’s graph theory material describes a spanning tree as connected and acyclic.
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A tree with n vertices has exactly n − 1 edges. That is the minimum number of edges needed to keep those vertices connected: removing a tree edge disconnects it, while adding an extra edge between already-connected vertices would create a cycle.
How does a spanning tree algorithm work?
A basic method is to search the graph from a starting vertex and keep the edge that first discovers each new vertex. When the search has reached every vertex, the recorded discovery edges form a spanning tree. OpenStax illustrates this approach with BFS and DFS. Its graph-algorithm chapter also discusses minimum spanning tree methods.
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Breadth-first search
BFS explores outward in levels: it visits the starting vertex’s neighbors, then their as-yet-unvisited neighbors, and continues. A queue is commonly used to track vertices waiting to be explored. Each first-discovery edge is added to the tree.
Depth-first search
DFS follows one path as far as it can before backtracking to explore another. A stack, either explicit or supplied by recursion, can track the search. As with BFS, the edge that first reaches each new vertex becomes a tree edge.
The two searches can produce different spanning trees because their exploration order differs. Starting vertex and the order in which neighbors are considered can also change the result. A graph may have many valid spanning trees; there is not necessarily one unique answer.
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No. A spanning tree only needs to connect all vertices without cycles. If edges have weights representing costs, a minimum spanning tree (MST) is the spanning tree whose selected edge weights have the smallest possible total. BFS and DFS can construct a spanning tree without optimizing that total; Kruskal’s and Prim’s algorithms solve the weighted optimization problem.
| Algorithm type | Goal and method | Do edge weights matter? |
|---|---|---|
| BFS or DFS | Construct a spanning tree by recording first-discovery edges. BFS expands by levels; DFS follows paths and backtracks. | No. The traversal does not minimize total edge weight. |
| Kruskal | Build an MST by considering edges in nondecreasing weight order and accepting an edge only when it joins separate components. | Yes. Weights determine the order used to seek minimum total cost. |
| Prim | Build an MST by growing one tree, each time adding a least-weight edge that connects it to a vertex outside it. | Yes. It chooses the lightest edge crossing from the current tree to the rest. |
Kruskal must reject an edge whose endpoints are already in the same component, because adding it would make a cycle. Prim avoids that problem by choosing only an edge crossing from its current tree to a vertex outside it. Both are greedy MST algorithms, but their growth patterns differ. The University of Texas at Austin’s treatment of the MST problem discusses their implementation and running-time bounds.
What are the algorithms’ time complexities?
These are theoretical bounds, not measured performance. Exact running time depends on the graph representation and data structures used; the figures below reflect the cited teaching materials, whose pages do not state publication years.
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| Algorithm | Reported theoretical bound | Source and conditions |
|---|---|---|
| Kruskal | O(|E| log |E|) | OpenStax treatment, using disjoint sets to track components. |
| Kruskal | O(m log n), dominated by sorting; union-find operations add amortized O(m · α(n)) | University of Texas at Austin treatment; n denotes vertices, m edges, and α is the inverse Ackermann function. |
| Prim | O(|E| log |V| + |V| log |V|) | OpenStax treatment. |
| Prim | O((n + m) log n) with a binary heap; O(m + n log n) with a Fibonacci heap | University of Texas at Austin treatment; n denotes vertices and m edges. |
The notation differs between sources, but both analyses show why an MST runtime should be read alongside its implementation assumptions rather than treated as one universal figure.
What if the graph is disconnected?
A single spanning tree covering all vertices exists only when the graph is connected. If it has multiple connected components, searching each component produces a spanning forest: one spanning tree per component. For a weighted disconnected graph, finding a minimum spanning tree for each component gives a minimum spanning forest.
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