How to Win Coding Competitions: Secrets of Champions

Week 4: Algorithms on GraphsLecture 6: Eulerian paths and Eulerian tours

Maxim BuzdalovSaint Petersburg 2016

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

Definitions

An Eulerian path is a path in a graphthat contains each edge of the graphexactly once

FBCEDFGCADBAE

A

B

C

D

E

F

G

An Eulerian tour is a Eulerian pathwhich starts and ends on the same vertex

FEABGCEDBCADF

A

B

C

D

E

F G

2 / 6

When do they exist?

When does Eulerian path exist?

I Undirected graph:I The graph is connectedI There are at most two vertices with odd degree

I Directed graph:I The graph is connected (when directions are removed)I At most one vertex u has deg+(u)− deg−(u) = +1I At most one vertex v has deg+(v)− deg−(v) = −1I All other vertices have deg+(x) = deg−(x)

When does Eulerian tour exist?I Undirected graph:

I The graph is connectedI All vertex degrees are even

I Directed graph:I The graph is strongly connectedI All vertices have deg+(x) = deg−(x)

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When do they exist?

When does Eulerian path exist?I Undirected graph:

I The graph is connectedI There are at most two vertices with odd degree

I Directed graph:I The graph is connected (when directions are removed)I At most one vertex u has deg+(u)− deg−(u) = +1I At most one vertex v has deg+(v)− deg−(v) = −1I All other vertices have deg+(x) = deg−(x)

When does Eulerian tour exist?I Undirected graph:

I The graph is connectedI All vertex degrees are even

I Directed graph:I The graph is strongly connectedI All vertices have deg+(x) = deg−(x)

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When do they exist?

When does Eulerian path exist?I Undirected graph:

I The graph is connectedI There are at most two vertices with odd degree

I Directed graph:I The graph is connected (when directions are removed)I At most one vertex u has deg+(u)− deg−(u) = +1I At most one vertex v has deg+(v)− deg−(v) = −1I All other vertices have deg+(x) = deg−(x)

When does Eulerian tour exist?I Undirected graph:

I The graph is connectedI All vertex degrees are even

I Directed graph:I The graph is strongly connectedI All vertices have deg+(x) = deg−(x)

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When do they exist?

When does Eulerian path exist?I Undirected graph:

I The graph is connectedI There are at most two vertices with odd degree

I Directed graph:I The graph is connected (when directions are removed)I At most one vertex u has deg+(u)− deg−(u) = +1I At most one vertex v has deg+(v)− deg−(v) = −1I All other vertices have deg+(x) = deg−(x)

When does Eulerian tour exist?

I Undirected graph:I The graph is connectedI All vertex degrees are even

I Directed graph:I The graph is strongly connectedI All vertices have deg+(x) = deg−(x)

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When do they exist?

When does Eulerian path exist?I Undirected graph:

I The graph is connectedI There are at most two vertices with odd degree

I Directed graph:I The graph is connected (when directions are removed)I At most one vertex u has deg+(u)− deg−(u) = +1I At most one vertex v has deg+(v)− deg−(v) = −1I All other vertices have deg+(x) = deg−(x)

When does Eulerian tour exist?I Undirected graph:

I The graph is connectedI All vertex degrees are even

I Directed graph:I The graph is strongly connectedI All vertices have deg+(x) = deg−(x)

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When do they exist?

When does Eulerian path exist?I Undirected graph:

I The graph is connectedI There are at most two vertices with odd degree

I Directed graph:I The graph is connected (when directions are removed)I At most one vertex u has deg+(u)− deg−(u) = +1I At most one vertex v has deg+(v)− deg−(v) = −1I All other vertices have deg+(x) = deg−(x)

When does Eulerian tour exist?I Undirected graph:

I The graph is connectedI All vertex degrees are even

I Directed graph:I The graph is strongly connectedI All vertices have deg+(x) = deg−(x)

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How to prove these existence theorems?

I Induction: assume that your graph has |E | edges,and the theorem was proven for all e < |E | numbers of edges

I Pick a vertexI Random when you don’t have “special” verticesI A “special” vertex if you have one

I The one with deg+(v) > deg−(v) if the graph is directed

I Traverse the graph and remove the traversed edgesI If you cannot do this anymore, what happens?

I Either there are no more edges → path/tour is foundI Some edges remain

I There are (maybe several) connected subgraphsI Find the paths/tours in them (can do this by induction)I Connect them with the path/tour constructed from removed edges

I That’s what Depth First Search can do!

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How to prove these existence theorems?

I Induction: assume that your graph has |E | edges,and the theorem was proven for all e < |E | numbers of edges

I Pick a vertexI Random when you don’t have “special” verticesI A “special” vertex if you have one

I The one with deg+(v) > deg−(v) if the graph is directed

I Traverse the graph and remove the traversed edgesI If you cannot do this anymore, what happens?

I Either there are no more edges → path/tour is foundI Some edges remain

I There are (maybe several) connected subgraphsI Find the paths/tours in them (can do this by induction)I Connect them with the path/tour constructed from removed edges

I That’s what Depth First Search can do!

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How to prove these existence theorems?

I Induction: assume that your graph has |E | edges,and the theorem was proven for all e < |E | numbers of edges

I Pick a vertexI Random when you don’t have “special” verticesI A “special” vertex if you have one

I The one with deg+(v) > deg−(v) if the graph is directed

I Traverse the graph and remove the traversed edgesI If you cannot do this anymore, what happens?

I Either there are no more edges → path/tour is foundI Some edges remain

I There are (maybe several) connected subgraphsI Find the paths/tours in them (can do this by induction)I Connect them with the path/tour constructed from removed edges

I That’s what Depth First Search can do!

4 / 6

How to prove these existence theorems?

I Induction: assume that your graph has |E | edges,and the theorem was proven for all e < |E | numbers of edges

I Pick a vertexI Random when you don’t have “special” verticesI A “special” vertex if you have one

I The one with deg+(v) > deg−(v) if the graph is directed

I Traverse the graph and remove the traversed edges

I If you cannot do this anymore, what happens?I Either there are no more edges → path/tour is foundI Some edges remain

I There are (maybe several) connected subgraphsI Find the paths/tours in them (can do this by induction)I Connect them with the path/tour constructed from removed edges

I That’s what Depth First Search can do!

4 / 6

How to prove these existence theorems?

I Induction: assume that your graph has |E | edges,and the theorem was proven for all e < |E | numbers of edges

I Pick a vertexI Random when you don’t have “special” verticesI A “special” vertex if you have one

I The one with deg+(v) > deg−(v) if the graph is directed

I Traverse the graph and remove the traversed edgesI If you cannot do this anymore, what happens?

I Either there are no more edges → path/tour is foundI Some edges remain

I There are (maybe several) connected subgraphsI Find the paths/tours in them (can do this by induction)I Connect them with the path/tour constructed from removed edges

I That’s what Depth First Search can do!

4 / 6

How to prove these existence theorems?

I Induction: assume that your graph has |E | edges,and the theorem was proven for all e < |E | numbers of edges

I Pick a vertexI Random when you don’t have “special” verticesI A “special” vertex if you have one

I The one with deg+(v) > deg−(v) if the graph is directed

I Traverse the graph and remove the traversed edgesI If you cannot do this anymore, what happens?

I Either there are no more edges → path/tour is foundI Some edges remain

I There are (maybe several) connected subgraphsI Find the paths/tours in them (can do this by induction)I Connect them with the path/tour constructed from removed edges

I That’s what Depth First Search can do!

4 / 6

DFS modification to solve Eulerian paths

G = 〈V ,E 〉A(v) = {u | (v , u) ∈ E}R ← [ ]procedure DFS(v)

for u ∈ A(v) doRemove u from A(v)if graph is undirected then

Remove v from A(u)end ifDFS(u)

end forR ← [v ] + R

end procedure

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DFS modification to solve Eulerian paths

G = 〈V ,E 〉A(v) = {u | (v , u) ∈ E} . No U set is used: can visit a vertex more than once!R ← [ ]procedure DFS(v)

for u ∈ A(v) doRemove u from A(v)if graph is undirected then

Remove v from A(u)end ifDFS(u)

end forR ← [v ] + R

end procedure

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DFS modification to solve Eulerian paths

G = 〈V ,E 〉A(v) = {u | (v , u) ∈ E} . No U set is used: can visit a vertex more than once!R ← [ ] . The result storageprocedure DFS(v)

for u ∈ A(v) doRemove u from A(v)if graph is undirected then

Remove v from A(u)end ifDFS(u)

end forR ← [v ] + R

end procedure

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DFS modification to solve Eulerian paths

G = 〈V ,E 〉A(v) = {u | (v , u) ∈ E} . No U set is used: can visit a vertex more than once!R ← [ ] . The result storageprocedure DFS(v) . Should be called on a non-regular vertex, if any

for u ∈ A(v) doRemove u from A(v)if graph is undirected then

Remove v from A(u)end ifDFS(u)

end forR ← [v ] + R

end procedure

5 / 6

DFS modification to solve Eulerian paths

G = 〈V ,E 〉A(v) = {u | (v , u) ∈ E} . No U set is used: can visit a vertex more than once!R ← [ ] . The result storageprocedure DFS(v) . Should be called on a non-regular vertex, if any

for u ∈ A(v) doRemove u from A(v) . Will never follow the (v , u) edge anymoreif graph is undirected then

Remove v from A(u)end ifDFS(u)

end forR ← [v ] + R

end procedure

5 / 6

DFS modification to solve Eulerian paths

G = 〈V ,E 〉A(v) = {u | (v , u) ∈ E} . No U set is used: can visit a vertex more than once!R ← [ ] . The result storageprocedure DFS(v) . Should be called on a non-regular vertex, if any

for u ∈ A(v) doRemove u from A(v) . Will never follow the (v , u) edge anymoreif graph is undirected then

Remove v from A(u) . If undirected, the anti-edge must be removedend ifDFS(u)

end forR ← [v ] + R

end procedure

5 / 6

DFS modification to solve Eulerian paths

G = 〈V ,E 〉A(v) = {u | (v , u) ∈ E} . No U set is used: can visit a vertex more than once!R ← [ ] . The result storageprocedure DFS(v) . Should be called on a non-regular vertex, if any

for u ∈ A(v) doRemove u from A(v) . Will never follow the (v , u) edge anymoreif graph is undirected then

Remove v from A(u) . If undirected, the anti-edge must be removedend ifDFS(u)

end forR ← [v ] + R . Prepend v to the answer

end procedure

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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Example: DFS finds Eulerian path

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