Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

procedure depth_first( G = (V,E): connected graph )
T := tree with only some single v,V
visit( v, T )
{ T is a spanning tree. }
procedure breadth_first( G = (V,E): connected graph )
T := tree with only some single v,V
L := v
while L - /
Remove first vertex v,L
for each neighbor w,V of v
if w6L and w6T then
Add w to the end of L
Add w and edge {v,w} to T
{ T is a spanning tree. }
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Theorem: Let G = (V,E) be a connected graph with |V| = n. Then either depth
first or breadth first takes O(e), or O(n 2 ), steps to construct a spanning tree.
Proof: For a simple graph, |E| ; n(n41)/2.
Bactracking applications:
• Graph coloring: can a graph be colored in n colors
• n-Queens problem: find places on a n$n board so n queens are toothless
n
! $
• Sums of subsets: Given " x
# i
% , where xi ,N, find a subset whose sum is M
&i=1
• Web crawlers: search all hyperlinks on a network efficiently
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