Books / Analysis of Algorithms / Chapter 20

Minimum Spanning Trees - Prim's Algorithm

An alternative algorithm for finding MST’s is Prim’s algorithm which runs asymptotically faster than Kruskal’s algorithm, but at the price of requiring a queue data structure.

Prim’s Algorithm

Prim’s algorithm finds MST’s by growing the MST by adding light edges between the current tree and other vertices. The vertices are stored in a minimum priority queue based on the smallest weigh edge connecting vertices currently not in the tree with a vertex in the tree.

Algorithm

MST-PRIM(G,w,r)
1.  for each u ∈ G.V
2.     u.key = ∞
3.     u.pi = NIL
4.  r.key = 0
5.  Q = G.V
6.  while Q ≠ ∅
7.     u = EXTRACT-MIN(Q)
8.     for each v ∈ G.Adj[u]
9.        if v ∈ Q and w(u,v) < v.key
10.          v.pi = u
11.          v.key = w(u,v)

Basically the algorithm works as follows:

  1. Initialize Q and set the source (root) key to 0
  2. While Q is not empty, dequeue the vertex with minimum weight edge and add it to the tree by adding edge (u.π,u) to T
  3. For each vertex v in Adj[u] that is still in Q, check if w(u,v) (the edge weights from u for all vertices not in T) are less than the current v.key (the current smallest edge weight) and if so update the predecessor and key fields

The run time of Prim’s algorithm depends on the implementation of the priority queue, but can be made to run in O(E + V lg V) using a Fibonacci heap for the priority queue.

Example

Using the same undirected graph from chapter 19

image

We will (arbitrarily) initialize Q with vertex 1 giving

image

Step 1: Dequeue vertex 1 and update Q (and reprioritizing) by changing u3.key = 2 (edge (u1,u3)), u2.key = 3 (edge (u1,u2)), u4.key = 6 (edge (u1,u4))

image

Step 2: Dequeue vertex 3 (adding edge (u1,u3) to T) and update Q (and reprioritizing) by changing u4.key = 4 (edge (u3,u4))

image

Step 3: Dequeue vertex 2 (adding edge (u1,u2) to T) and update Q (and reprioritizing) by changing u5.key = 2 (edge (u2,u5))

image

Step 4: Dequeue vertex 5 (adding edge (u2,u5) to T) with no updates to Q

image

Step 5: Dequeue vertex 4 (adding edge (u3,u4) to T) with no updates to Q

image

At this point Q = ∅ giving the final MST

image

with total weight 11.


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