Lecture 14 Minimum Spanning Tree (cont’d)

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).

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)

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’

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.

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.

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.

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.

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.

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)

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.