1. Lecture 14 Minimum Spanning Tree (cont’d)

2. Trees and cycles
Adding any non-tree edge to a tree will introduce a cycle
For a tree with a cycle, removing any edge in the cycle results in a tree.
Basic operation: add an edge and remove another edge in the cycle created (swap).

3. General Algorithm for MST
Initialize the tree to be empty. (initially a subset of some MST)
Repeat
Find a cut that does not have any edge in the current tree
Add the min cost edge of the cut to the tree (by Key Lemma: still a subset of some MST)
Until the tree has n-1 edges (Now we already have a tree, so it must be a MST)

4. Key Property
Key Lemma: Suppose F is a set of edges that is inside some MST T, if 𝑆, 𝑆 is a cut that does not contain any edge in F and e is the minimum cost edge in the cut, then it is safe to add e to F.
𝐹∪ 𝑒 ⊂𝑇′ for some MST T’

5. Designing a MST algorithm
Goal: do the two operations efficiently
1. How to find a cut that does not go through any edges we have chosen
2. How to find a minimum cost edge in the cut.

6. Prim’s algorithm Starting from an arbitrary vertex s.
Make sure that the edges we select form a connected component including s.
Choosing the cut: The set of vertices connected to s.
Finding the minimum cost edge: Use a data-structure similar to Dijkstra.

7. Prim’s algorithm Prim(s)
initialize dis[u] to be all infinity, prev[u] to be NULL
For neighbors of s, initialize dis[u] = w[s,u], prev[u] = s
Mark s as visited
FOR i = 2 to n
Among all vertices that are not visited, find the one with smallest distance, call it u.
Mark u as visited
FOR all edges (u,v)
IF w[u,v] < dis[v] THEN
dis[v] = w[u,v]
prev[v] = u.

8. Dijkstra’s algorithm Dijkstra(s)
initialize dis[u] to be all infinity, prev[u] to be NULL
For neighbors of s, initialize dis[u] = w[s,u], prev[u] = s
Mark s as visited
FOR i = 2 to n
Among all vertices that are not visited, find the one with smallest distance, call it u.
Mark u as visited
FOR all edges (u,v)
IF dis[u]+w[u,v] < dis[v] THEN
dis[v] = dis[u]+w[u,v]
prev[v] = u.

9. Kruskal’s algorithm
Idea: don’t try to find a cut, directly find the shortest edge.
Algorithm: Sort the edges in ascending order of weight.
For each edge, if adding it does not create a cycle, then add the edge to the tree.
Checking the cycle needs a special data structure that we will talk about later.

10. Running time of Kruskal
Sorting takes O(mlog m) time.
For each edge, we need to check whether adding the edge creates a cycle.
This can be done very efficiently (o(log m)).
Total running time O(m log m)

11. Prim vs. Kruskal
Running Time: Prim seems to be always faster O(m+nlog n) vs. O(m log m)
However, the O(m+nlog n) version of Prim is not very easy to implement, and has a large hidden constant
If you use a regular binary heap, the running time are the same, and Kruskal is usually faster in practice/easier to implement.
If the graph is dense, O(n2) version of Prim is easy to implement and faster than Kruskal.