Definitions

# Prim's algorithm

Prim's algorithm is an algorithm in graph theory that finds a minimum spanning tree for a connected weighted 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 discovered in 1930 by mathematician Vojtěch Jarník and later independently by computer scientist Robert C. Prim in 1957 and rediscovered by Edsger Dijkstra in 1959. Therefore it is sometimes called the DJP algorithm, the Jarník algorithm, or the Prim-Jarník algorithm.

## Description

The algorithm continuously increases the size of a tree starting with a single vertex until it spans all the vertices.

• Input: A connected weighted graph with vertices V and edges E.
• Initialize: Vnew = {x}, where x is an arbitrary node (starting point) from V, Enew= {}
• repeat until Vnew=V:
• Choose edge (u,v) from E with minimal weight such that u is in Vnew and v is not (if there are multiple edges with the same weight, choose arbitrarily)
• Add v to Vnew, add (u,v) to Enew
• Output: Vnew and Enew describe the minimal spanning tree

## Time complexity

Minimum edge weight data structure Time complexity (total)
adjacency matrix, searching O(V2)
binary heap (as in pseudocode below) 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 using an adjacency matrix graph representation and searching an array of weights to find the minimum weight edge to add requires O(V2) running time. Using a simple binary heap data structure and an adjacency list representation, Prim's algorithm can be shown to run in time which is 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 significantly faster when the graph is dense enough that E is $Omega$(V log V).

## Example

Image Description
This is our original weighted graph. The numbers near the arcs indicate their weight.
Vertex D has been arbitrarily chosen as a starting point. Vertices A, B, E and F are connected to D through a single edge. A is the vertex nearest to D and will be chosen as the second vertex along with the edge AD.
The next vertex chosen is the vertex nearest to either D or A. B is 9 away from D and 7 away from A, E is 15, and F is 6. F is the smallest distance away, so we highlight the vertex F and the arc DF.
The algorithm carries on as above. Vertex B, which is 7 away from A, is highlighted.
In this case, we can choose between C, E, and G. C is 8 away from B, E is 7 away from B, and G is 11 away from F. E is nearest, so we highlight the vertex E and the arc EB.
Here, the only vertices available are C and G. C is 5 away from E, and G is 9 away from E. C is chosen, so it is highlighted along with the arc EC.
Vertex G is the only remaining vertex. It is 11 away from F, and 9 away from E. E is nearer, so we highlight it and the arc EG.
Now all the vertices have been selected and the minimum spanning tree is shown in green. In this case, it has weight 39.

## Pseudocode

### Min-heap

Initialization

inputs: A graph, a function returning edge weights weight-function, and an initial vertex
initial placement of all vertices in the 'not yet seen' set, set initial vertex to be added to the tree, and place all vertices in a min-heap to allow for removal of the min distance from the minimum graph.
`for each vertex in graph`
`   set min_distance of vertex to ∞`
`   set parent of vertex to null`
`   set minimum_adjacency_list of vertex to empty list`
`   set is_in_Q of vertex to true`
`set distance of initial vertex to zero`
`add to minimum-heap Q all vertices in graph.`
Algorithm In the algorithm description above,
nearest vertex is Q[0], now latest addition
fringe is v in Q where distance of v < ∞ after nearest vertex is removed
not seen is v in Q where distance of v = ∞ after nearest vertex is removed
The while loop will fail when remove minimum returns null. The adjacency list is set to allow a directional graph to be returned.
time complexity: V for loop, log(V) for the remove function
`while latest_addition = remove minimum in Q`
`    set is_in_Q of latest_addition to false`
`    add latest_addition to (minimum_adjacency_list of (parent of latest_addition))`
`    add (parent of latest_addition) to (minimum_adjacency_list of latest_addition)`
time complexity: E/V, the average number of vertices
`    for each adjacent of latest_addition`
`    if (is_in_Q of adjacent) and (weight-function(latest_addition, adjacent) < min_distance of adjacent)`
`        set parent of adjacent to latest_addition`
`        set min_distance of adjacent to weight-function(latest_addition, adjacent)`
time complexity: log(V), the height of the heap
`        update adjacent in Q, order by min_distance`

## 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 Y are connected. Let Y1 be a minimum spanning tree of P. If Y1=Y then Y is a minimum spanning tree. Otherwise, let e be the first edge added during the construction of Y that is not in Y1, and V be the set of vertices connected by the edges added before e. Then one endpoint of e is in V and the other is not. Since Y1 is a spanning tree of P, there is a path in Y1 joining the two endpoints. As one travels along the path, one must encounter an edge f joining a vertex in V to one that is not in V. Now, at the iteration when e was added to Y, f could also have been added and it would be added instead of e if its weight was less than e. Since f was not added, we conclude that

w(f) ≥ w(e).

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

Other algorithms for this problem include Kruskal's algorithm and Borůvka's algorithm.

## 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 generalisations. In: Bell System Technical Journal, 36 (1957), pp. 1389–1401
• D. Cherition and R. E. Tarjan: Finding minimum spanning trees. In: SIAM Journal of Computing, 5 (Dec. 1976), pp. 724–741
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 23.2: The algorithms of Kruskal and Prim, pp.567–574.

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