Spanning Trees

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

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.
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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

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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.

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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
• Breadth-first-search (BFS)
• After visit node v, BFS visits every node
  adjacent to v before visiting any other nodes

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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 DSF(G connected graph with vertices
  v1, v2, , vn)
• Ttree 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

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DFS example

• Start from v3

1
v3
2
v2
v2
v3
v1
x
x
x
4
3
v1
v4
x
x
v5
v4
5
v5
G
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Backtracking (animation)
dead end
?
dead end
dead end
?
start
?
?
dead end
dead end
?
success!
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Coloring a map

• You wish to color a map withnot more than four
  colors
• red, yellow, green, blue
• Adjacent countries must be indifferent colors
• You dont 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

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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

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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

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Solution for this map

• Solution found truemap0 is redmap1 is
  yellowmap2 is greenmap3 is bluemap4 is
  yellowmap5 is greenmap6 is bluemap7 is
  red

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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 vs neighbors
• 3. Visit all vs neighbors neighbors
• Similar to level-order, BFS is based on a queue.

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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
• begin
• add w to the end of the list L
• add w and edge v, w to T
• end
• end

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BFS example

• Start from v5

Visit Queue (front to back)
v5 v5
empty
v3 v3
v4 v3, v4
v4
v2 v4, v2
v2
empty
v1 v1
empty
1
v5
x
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Backtracking Applications

• Use backtracking to find a subset of31, 27, 15,
  11, 7, 5with sum equal to 39.