Minimal Spanning Trees


















Spanning Trees

What's a Spanning Tree for a connected undirected graph?

	A subgraph that connects all the nodes with no cycles.

	1. all the nodes from the graph

	2. a subset of the edges from the graph
		a. the edges must connect all the nodes
		b. the edges must form no cycles

Find a Spanning Tree for the undirected graph.
(graph15.pdf)

	V = { a, b, c, d, e, f }
	E = { {a,b}, {a,d}, {b,c}, {b,d}, {b,e},
	      {c,e}, {c,f}, {d,e}, {e,f} }

	a---b---c
	| / | / |
	d---e---f


















Minimal Spanning Trees

What's a Minimal Spanning Tree?

	A spanning tree that has a smallest sum of edge weights.

What are applications of Minimal Spanning Trees?
(graph16.pdf)

	Find the lowest cost to connect all sites in a communication network.

	$35  Atlanta  Chicago
	$70  Atlanta  Denver
	$40  Atlanta  New York
	$65  Chicago  Denver
	$50  Chicago  New York
	$60  Chicago  San Francisco
	$45  Denver   San Francisco

	S---C---N
	 \ / \ /
	  D---A


















Kruskal's Algorithm

How do you construct a minimal spanning tree?

	Start with all the nodes and none of the edges.
	Initialize sets of nodes that are connected.
	  (Make a separate set for each node.)

How do you add the edges?

	Add edges one at a time in order from smallest to largest.
	Only add an edge if it doesn't create a cycle in the graph.
	Also combine the sets of nodes connected by the edge.

How do you know if adding an edge will create a cycle?

	Connecting two nodes in the same set creates a cycle.


Kruskal's Algorithm Example

Use Kruskal's algorithm to find a minimal spanning tree.
For each iteration, show which edge is being considered,
indicate if the edge is added or not, and
give the current sets of connected nodes.
(graph16.pdf)

	$35  Atlanta  Chicago
	$70  Atlanta  Denver
	$40  Atlanta  New York
	$65  Chicago  Denver
	$50  Chicago  New York
	$60  Chicago  San Francisco
	$45  Denver   San Francisco

	S---C---N
	 \ / \ /
	  D---A


















Classwork

You may work with a partner.

Use Kruskal's algorithm to find a minimal spanning tree.
For each iteration, show which edge is being considered,
indicate if the edge is added or not, and
give the current sets of connected nodes.
(graph17.pdf)

	V = { a, b, c, d, e, f }
	E = { {a,b}, {a,d}, {b,c}, {b,d}, {b,f}, {c,f}, {d,e}, {e,f} }

	Use these edge weights.
	{a,b}	8	{a,d}	11
	{b,c}	7	{b,d}	6
	{b,f}	4	{c,f}	14
	{d,e}	1	{e,f}	2

	a---b---c
	| /   \ |
	d---e---f


















Kruskal's Algorithm Code

	Kruskal()

	   sort the edges in the graph by increasing weight

	   initialize T to be an empty set of edges
	   for each node n, make a set that contains n

	   for each edge {x,y} in the sorted list of edges

	      if x and y are in different sets (no cycle with {x,y})
	         add edge {x,y} to T
		 combine sets containing x and y into a single set
	      end if

	   end for

	   T contains the edges in the tree

	end Kruskal

What's the Big-Oh for the running time of Kruskal's Algorithm?

	1. Sort the edges by weight in O(e log e) time.
	2. Add the edges in O(e log n) time.
	3. Total is O(e log e + e log n) = O(e log e) = O(e log n).