Prim's algorithm

Graph and tree search algorithms
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In computer science, Prim's algorithm is a greedy algorithm that finds a minimum spanning tree for a connected weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. The algorithm was developed in 1930 by Czech mathematician Vojtěch Jarník and later independently by computer scientist Robert C. Prim in 1957 and rediscovered by Edsger Dijkstra in 1959. Therefore it is also sometimes called the DJP algorithm, the Jarník algorithm, or the Prim–Jarník algorithm.

Other algorithms for this problem include Kruskal's algorithm and Borůvka's algorithm. These algorithms find the minimum spanning forest in a possibly disconnected graph. By running Prim's algorithm for each connected component of the graph, it can also be used to find the minimum spanning forest.

Prim's algorithm starting at vertex A. In the second step, BD is chosen to add to the tree instead of AB arbitrarily, as both have weight 2. Afterwards, AB is excluded because it is between two nodes that are already in the tree.

Description

Informal

Initialize a tree with a single vertex, chosen arbitrarily from the graph.
Grow the tree by one edge: of the edges that connect the tree to vertices not yet in the tree, find the minimum-weight edge, and transfer it to the tree.
Repeat step 2 (until all vertices are in the tree).

Technical

If a graph is empty then the algorithm terminates immediately. Thus, we assume otherwise.

The algorithm starts with a tree consisting of a single vertex, and continuously increases its size one edge at a time, until it spans all vertices.

Input: A non-empty connected weighted graph with vertices V and edges E (the weights can be negative).
Initialize: V_new = {x}, where x is an arbitrary node (starting point) from V, E_new = {}
Repeat until V_new = V:
- Choose an edge {u, v} with minimal weight such that u is in V_new and v is not (if there are multiple edges with the same weight, any of them may be picked)
- Add v to V_new, and {u, v} to E_new
Output: V_new and E_new describe a minimal spanning tree

Time complexity

Prim's algorithm has many applications, such as in the generation of this maze, which applies Prim's algorithm to a randomly weighted grid graph.

Minimum edge weight data structure	Time complexity (total)
adjacency matrix, searching	O(\|V\|²)
binary heap and adjacency list	O((\|V\| + \|E\|) log \|V\|) = O(\|E\| log \|V\|)
Fibonacci heap and adjacency list	O(\|E\| + \|V\| log \|V\|)

A simple implementation of Prim's, using an adjacency matrix graph representation and linearly searching an array of weights to find the minimum weight edge, to add requires O(|V|²) running time. Switching to an adjacency list representation brings this down to O(|V||E|), which is strictly better for sparse graphs. However, this running time can be greatly improved further by using heaps to implement finding minimum weight edges in the algorithm's inner loop.

A first improved version uses a heap to store all edges of the input graph, ordered by their weight. This leads to an O(|E| log |E|) worst-case running time. But storing vertices instead of edges can improve it still further. The heap should order the vertices by the smallest edge-weight that connects them to any vertex in the partially constructed minimum spanning tree (MST) (or infinity if no such edge exists). Every time a vertex v is chosen and added to the MST, a decrease-key operation is performed on all vertices w outside the partial MST such that v is connected to w, setting the key to the minimum of its previous value and the edge cost of (v,w).

Using a simple binary heap data structure, Prim's algorithm can now be shown to run in time O(|E| log |V|) where |E| is the number of edges and |V| is the number of vertices. Using a more sophisticated Fibonacci heap, this can be brought down to O(|E| + |V| log |V|), which is asymptotically faster when the graph is dense enough that |E| is ω(|V|).

Demonstration of proof. In this case, the graph Y₁ = Y − f + e is already equal to Y. In general, the process may need to be repeated.

Proof of correctness

Let P be a connected, weighted graph. At every iteration of Prim's algorithm, an edge must be found that connects a vertex in a subgraph to a vertex outside the subgraph. Since P is connected, there will always be a path to every vertex. The output Y of Prim's algorithm is a tree, because the edge and vertex added to tree Y are connected. Let Y₁ be a minimum spanning tree of graph P. If Y₁=Y then Y is a minimum spanning tree. Otherwise, let e be the first edge added during the construction of tree Y that is not in tree Y₁, and V be the set of vertices connected by the edges added before edge e. Then one endpoint of edge e is in set V and the other is not. Since tree Y₁ is a spanning tree of graph P, there is a path in tree Y₁ joining the two endpoints. As one travels along the path, one must encounter an edge f joining a vertex in set V to one that is not in set V. Now, at the iteration when edge e was added to tree Y, edge f could also have been added and it would be added instead of edge e if its weight was less than e, and since edge f was not added, we conclude that

w(f) \ge w(e).

Let tree Y₂ be the graph obtained by removing edge f from and adding edge e to tree Y₁. It is easy to show that tree Y₂ is connected, has the same number of edges as tree Y₁, and the total weights of its edges is not larger than that of tree Y₁, therefore it is also a minimum spanning tree of graph P and it contains edge e and all the edges added before it during the construction of set V. Repeat the steps above and we will eventually obtain a minimum spanning tree of graph P that is identical to tree Y. This shows Y is a minimum spanning tree.

Explanation for Time Complexity

In the method that uses binary heaps, we can observe that the traversal is executed O(V+E) times (similar to BFS). Each traversal has operation which takes O(LogV) time. So overall time complexity is O(E+V)*O(LogV) which is O((E+V)*LogV) = O(E*LogV) (For a connected graph, V = O(E)).

References

V. Jarník: O jistém problému minimálním [About a certain minimal problem], Práce Moravské Přírodovědecké Společnosti, 6, 1930, pp. 57–63. (in Czech)
R. C. Prim: Shortest connection networks and some generalizations. In: Bell System Technical Journal, 36 (1957), pp. 1389–1401
D. Cheriton and R. E. Tarjan: Finding minimum spanning trees. In: SIAM Journal on Computing, 5 (Dec. 1976), pp. 724–741
Dijkstra, E. W. (1959). "A note on two problems in connexion with graphs" (PDF). Numerische Mathematik 1: 269–271. doi:10.1007/BF01386390.

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