1. Spanning Trees Longin Jan Latecki Temple University based on slides by
David Matuszek, UPenn,
Rose Hoberman, CMU,
Bing Liu, U. of Illinois,
Boting Yang, U. of Regina

2. Spanning trees Suppose you have a connected undirected graph
Connected: every node is reachable from every other node
Undirected: edges do not have an associated direction
...then a spanning tree of the graph is a connected subgraph with the same nodes in which there are no cycles
A connected, undirected graph
Four of the spanning trees of the graph

3. Theorem A simple graph is connected iff it has a spanning tree. Proof.
We assume that a graph G is connected.
If G has a simple circuit, we remove one edge from it.
We repeat this step, until there is no more circuits in G.
Let T be the obtained subgraph.
T has the same nodes as G, since we did not remove any node.
T has no circuits, since we removed all circuits.
T is connected, since in every circuit there was an alternative path.
The proof in the opposite direction is very simple.
Since the presented algorithm to construct a spanning tree is very inefficient, we need a better one.

4. Finding a spanning tree
To find a spanning tree of a graph,
pick an initial node and call it part of the spanning tree
do a search from the initial node:
each time you find a node that is not in the spanning tree, add to the spanning tree both the new node and the edge you followed to get to it
An undirected graph
One possible result of a BFS starting from top
One possible result of a DFS starting from top

5. Graph Traversal Algorithm
To traverse a tree, we use tree traversal algorithms like pre-order, in-order, and post-order to visit all the nodes in a tree
Similarly, graph traversal algorithm tries to visit all the nodes it can reach.
If a graph is disconnected, a graph traversal that begins at a node v will visit only a subset of nodes, that is, the connected component containing v.

6. Two basic traversal algorithms
Two basic graph traversal algorithms:
Depth-first-search (DFS), also called backtracking
After visiting node v, DFS proceeds along a path from v as deeply into the graph as possible before backing up
Based on stack
Breadth-first-search (BFS)
After visit node v, BFS visits every node adjacent to v before visiting any other nodes
Based on queue

7. Depth-first search (DFS)
DFS strategy looks similar to pre-order. From a given node v, it first visits itself. Then, recursively visit its unvisited neighbors one by one.
DFS can be defined recursively as follows.
procedure DFS(G: connected graph with vertices v1, v2, …, vn)
T:=tree consisting only of the vertex v1
visit(v1)
mark v1 as visited;
procedure visit(v: vertex of G)
for each node w adjacent to v not yet in T
begin
mark w as visited;
add vertex w and edge {v, w} to T
visit(w);
end

8. x x x x x DFS example v3 v2 v2 v3 v1 v1 v4 v5 v4 G v5 Start from v3 1

9. Backtracking (animation)
dead end ?
dead end
dead end ?
start ? ?
dead end
dead end ?
success!

10. Coloring a map You wish to color a map with not more than four colors
red, yellow, green, blue
Adjacent countries must be in different colors
You don’t have enough information to choose colors
Each choice leads to another set of choices
One or more sequences of choices may (or may not) lead to a solution
Many coloring problems can be solved with backtracking

11. Full example: Map coloring
The Four Color Theorem states that any map on a plane can be colored with no more than four colors, so that no two countries with a common border are the same color
For most maps, finding a legal coloring is easy
For some maps, it can be fairly difficult to find a legal coloring

12. Backtacking
We go through all the countries recursively, starting with country zero
At each country we decide a color
It must be different from all adjacent countries
If we cannot find a legal color, we report failure
If we find a color, we use it and recurred with the next country
If we ran out of countries (colored them all), we reported success
When we returned from the topmost call, we were done

13. Solution for this map
Solution found = true map[0] is red map[1] is yellow map[2] is green map[3] is blue map[4] is yellow map[5] is green map[6] is blue map[7] is red

14. Breadth-first search (BFS)
BFS strategy looks similar to level-order. From a given node v, it first visits itself. Then, it visits every node adjacent to v before visiting any other nodes.
1. Visit v
2. Visit all v’s neighbors
3. Visit all v’s neighbors’ neighbors …
Similar to level-order, BFS is based on a queue.

15. Algorithm for BFS
procedure BFS(G: connected graph with vertices v1, v2, …, vn)
T:= tree consisting only of vertex v1
L:= empty list
put v1 in the list L of unprocessed vertices;
while L is not empty
begin
remove the first vertex v from L
for each neighbor w of v
if w is not in L and not in T then
add w to the end of the list L
add w and edge {v, w} to T
end

16. x x x x x BFS example v5 v2 v3 v1 v3 v4 v4 v2 v5 G v1 Start from v5 1
Visit
Queue (front to back) v5
empty v3 v4
v3, v4 v2
v4, v2 v1 1 v5 v1 v4 v3 v5 v2 G v3 2 v4 3 x x x v2 4 x x v1 5

17. Backtracking Applications
Use backtracking to find a subset of {31, 27, 15, 11, 7, 5} with sum equal to 39.