Transcript Euler Circuit - Cleveland State University

Euler Circuit
Chapter 5
Fleury’s Algorithm
• Euler’s theorems are very useful to find if a graph
has an Euler circuit or an Euler path when the
graph is simple.
• However, for a complicated graph with hundreds of
vertices and edges, we need an algorithm.
• Algorithm: A set of procedural rules
• Examples
– The instruction of assembling a new bike,
– Recipe for baking an apple pie
Fleury’s Algorithm
• Finds an Euler circuit in a connected graph
with no odd vertices.
• Finds an Euler path in a connected graph
with two odd vertices.
• The idea behind the algorithm: Don’t burn
your bridges behind you.
A
B
Bridge
You would only want to cross that bridge if you know
that all edges in A have been traveled.
Fleury’s Algorithm
• Bridges are the last edges we want to cross to find an
Euler circuit.
• The graph whose bridges we are supposed to avoid is
not necessarily the original graph of the problem. Instead
it is that part of the original graph which has yet to be
traveled.
• Once we travel along an edge, we are done with it.
• Each time we traverse an edge, the untraveled part of
the graph changes and its bridges are changing.
• We start with 2 copies of the graph: copy 1 for making
decisions and copy 2 for record keeping.
Fleury’s Theorem
• Every time we traverse another edge, we erase
it from copy 1 but mark (red) and level it with the
appropriate number on copy 2.
• Copy 1 gets smaller and cop2 gets redder.
• Copy 1 helps us decide where to go next; copy 2
helps us reconstruct our trip.
Copy 2
Copy 1
A
B
C
E
D
F
B
C
E
Start at F (arbitrarily)
D
F
Fleury’s Theorem
Copy 1
Copy 2
A
A
B
B
C
C
D
D
E
E
F
Step 1: Travel from F to C
Could have also travel
from F to D
F
Fleury’s Theorem
Copy 1
Copy 2
A
A
B
B
2
C
C
D
D
1
E
E
F
Step 2: Travel from C to D
Could have also travel
to A or to E
F
Fleury’s Theorem
Copy 1
Copy 2
A
A
B
C
D
B
3
C
2
D
1
E
E
F
F
Step 3: Travel from D to A
Could have also travel
to B but not to F – DF is a bridge!
Fleury’s Theorem
Copy 1
Copy 2
A
A
B
3
4
2
C
C
B
D
D
1
E
E
F
F
Step 4: Travel from A to C
Could have also travel
to E but not to B – AB is a bridge!
Fleury’s Theorem
Copy 1
Copy 2
A
A
B
C
E
D
3
4
2
C
5
B
D
1
E
F
Step 5: Travel from C to E
There is no choice!
F
Fleury’s Theorem
Copy 1
Copy 2
A 7
3
4
6
B
2
C
5
E
1
8
D
9
F
Steps 6, 7, 8, and 9: Only one way to go at each step
Fleury’s Algorithm for finding Euler
circuit
• First make sure that the graph is connected and
all the vertices have even degree.
• Pick any vertex as the stating point
• When you have a choice, always choose to
travel along an edge that is not a bridge of the
yet-to-be-traveled part of the graph
• Label the edges in the order in which you travel
them
• When you cannot travel any more, stop. You are
done.
Fleury’s Algorithm for finding Euler
path
• First make sure that the graph is connected and have
exactly two vertices of odd degree.
• Pick any vertex of the two vertices of odd degree as the
stating point.
• When you have a choice, always choose to travel along
an edge that is not a bridge of the yet-to-be-traveled part
of the graph
• Label the edges in the order in which you travel them
• When you cannot travel any more, stop. You are done.
• When steps are followed properly, the trip will
guaranteed to end at the other vertex of odd degree.