Minimum Spanning Tree

1
Minimum Spanning Tree

• Input weighted graph G(V,E)
• Output A subset of E of minimum weight which
  forms a tree on V.
• Applications Network design problems
• The MST is the cheapest subset of edges that
  keeps the graph connected.

2
Minimum Spanning Tree

• Main idea behind the MST algorithms we will
  discuss
• At each step, add an edge to the intermediate
  spanning tree (in other words mark it as
  belonging to the spanning tree) so that
• it does not form a cycle, and
• it will result in a minimum spanning tree

3
MST Kruskal's algorithm

• Find the lightest edge in the graph and mark it.
• Find the lightest unmarked edge.
• if it does not close a cycle of marked edges,
  mark it
• Repeat step 2 until every vertex has been
  covered. The marked edges form the minimum
  spanning tree

4
MST Kruskal's algorithm

• Implementation details
• At any time during the algorithm, we have an
  intermediate forest. Whenever we mark an edge we
  are essentially combining two trees in one. In
  the end, we will have a single spanning tree.
• How do we check whether a candidate edge (u,v)
  closes a cycle?
• use a disjoint set data structure to maintain the
  vertices.
• if Find(u)Find(v), then u and v belong to the
  same tree.

5
MST Kruskal's algorithm

• tree ?
• for each vertex v
• MakeSet(v)
• sort the edges by non-decreasing weight
• for each edge (u,v) in order of non-decreasing
  weight
• if Find(u) ! Find(v)
• tree tree ?(u,v)
• Union(u,v)
• return tree

Running time O(E lgE)