David Luebke

CS 332: Algorithms

Graph Algorithms

1 4/23/2020

Review: Depth-First Search

●

Depth-first search

is another strategy for exploring a graph ■ Explore “deeper” in the graph whenever possible ■ Edges are explored out of the most recently discovered vertex

v

that still has unexplored edges ■ When all of

v

’s edges have been explored, backtrack to the vertex from which

v

was discovered

David Luebke 2 4/23/2020

Review: DFS Code

DFS(G) { for each vertex u



{ u->color = WHITE; G->V } time = 0; for each vertex u



{ G->V if (u->color == WHITE) DFS_Visit(u); } } DFS_Visit(u) { u->color = YELLOW; time = time+1; u->d = time; for each v



{ u->Adj[] if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } David Luebke 3 4/23/2020

source vertex

DFS Example

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source vertex d f

1 | | |

DFS Example

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source vertex d f

1 | 2 | |

DFS Example

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source vertex d f

1 | 2 | 3 |

DFS Example

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source vertex d f

1 | 2 | 3 | 4

DFS Example

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source vertex d f

1 | 2 | 3 | 4

DFS Example

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source vertex d f

1 | 2 | 3 | 4

DFS Example

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source vertex d f

1 | 2 | 7 3 | 4

DFS Example

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source vertex d f

1 | 2 | 7 3 | 4

DFS Example

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DFS Example

source vertex d f

1 | 2 | 7 8 | | 9 | David Luebke 3 | 4 5 | 6 |

What is the structure of the yellow vertices? What do they represent?

13 4/23/2020

source vertex d f

1 | 2 | 7 3 | 4

DFS Example

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source vertex d f

1 | 2 | 7 3 | 4

DFS Example

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source vertex d f

1 |12 2 | 7 3 | 4

DFS Example

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source vertex d f

1 |12 2 | 7 3 | 4

DFS Example

8 |11 5 | 6 9 |10 13| | David Luebke 17 4/23/2020

source vertex d f

1 |12 2 | 7 3 | 4

DFS Example

8 |11 5 | 6 9 |10 13| 14| David Luebke 18 4/23/2020

source vertex d f

1 |12 2 | 7 3 | 4

DFS Example

8 |11 5 | 6 9 |10 13| 14|15 David Luebke 19 4/23/2020

source vertex d f

1 |12 2 | 7 3 | 4

DFS Example

8 |11 5 | 6 9 |10 13|16 14|15 David Luebke 20 4/23/2020

DFS: Kinds of edges

● DFS introduces an important distinction among edges in the original graph: ■

Tree edge

: encounter new (white) vertex ○ The tree edges form a spanning forest ○

Can tree edges form cycles? Why or why not?

David Luebke 21 4/23/2020

source vertex d f

1 |12 2 | 7

Tree edges

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DFS Example

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DFS: Kinds of edges

● DFS introduces an important distinction among edges in the original graph: ■

Tree edge

: encounter new (white) vertex ■

Back edge

: from descendent to ancestor ○ Encounter a yellow vertex (yellow to yellow)

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DFS Example

source vertex d f

1 |12 2 | 7 3 | 4

Tree edges Back edges

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DFS: Kinds of edges

● DFS introduces an important distinction among edges in the original graph: ■

Tree edge

: encounter new (white) vertex ■

Back edge

: from descendent to ancestor ■

Forward edge

: from ancestor to descendent ○ Not a tree edge, though ○ From yellow node to black node

David Luebke 25 4/23/2020

DFS Example

source vertex d f

1 |12 2 | 7 8 |11 3 | 4 5 | 6

Tree edges Back edges Forward edges

David Luebke 26 9 |10 13|16 14|15 4/23/2020

DFS: Kinds of edges

● DFS introduces an important distinction among edges in the original graph: ■

Tree edge

: encounter new (white) vertex ■

Back edge

: from descendent to ancestor ■

Forward edge

: from ancestor to descendent ■

Cross edge

: between a tree or subtrees ○ From a yellow node to a black node

David Luebke 27 4/23/2020

DFS Example

source vertex d f

1 |12 2 | 7 8 |11 13|16 9 |10 3 | 4 5 | 6 14|15

Tree edges Back edges Forward edges Cross edges

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DFS: Kinds of edges

● DFS introduces an important distinction among edges in the original graph: ■

Tree edge

: encounter new (white) vertex ■

Back edge

: from descendent to ancestor ■

Forward edge

: from ancestor to descendent ■

Cross edge

: between a tree or subtrees ● Note: tree & back edges are important; most algorithms don’t distinguish forward & cross

David Luebke 29 4/23/2020

DFS: Kinds Of Edges

● Thm 23.9: If G is undirected, a DFS produces only tree and back edges ● Proof by contradiction: ■ Assume there’s a forward edge ○ But F? edge must actually be a back edge (

why?

)

F?

source

David Luebke 30 4/23/2020

DFS: Kinds Of Edges

● Thm 23.9: If G is undirected, a DFS produces only tree and back edges ● Proof by contradiction: ■ Assume there’s a cross edge ○ But C? edge cannot be cross: ○ must be explored from one of the vertices it connects, becoming a tree vertex, before other vertex is explored ○ So in fact the picture is wrong…both lower tree edges cannot in fact be tree edges

source C?

David Luebke 31 4/23/2020

DFS And Graph Cycles

● Thm: An undirected graph is

acyclic

iff a DFS yields no back edges ■ If acyclic, no back edges (because a back edge implies a cycle ■ If no back edges, acyclic ○ No back edges implies only tree edges (

Why?

) ○ Only tree edges implies we have a tree or a forest ○ Which by definition is acyclic ● Thus, can run DFS to find whether a graph has a cycle

David Luebke 32 4/23/2020

DFS And Cycles

●

How would you modify the code to detect cycles?

DFS(G) { for each vertex u



{ u->color = WHITE; G->V } time = 0; for each vertex u



{ G->V if (u->color == WHITE) DFS_Visit(u); } } David Luebke 33 DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; for each v



{ u->Adj[] if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } 4/23/2020

DFS And Cycles

●

What will be the running time?

DFS(G) { for each vertex u



{ u->color = WHITE; G->V } time = 0; for each vertex u



{ G->V if (u->color == WHITE) DFS_Visit(u); } } David Luebke 34 DFS_Visit(u) { u->color = GREY; time = time+1; u->d = time; for each v



{ u->Adj[] if (v->color == WHITE) DFS_Visit(v); } u->color = BLACK; time = time+1; u->f = time; } 4/23/2020

DFS And Cycles

●

What will be the running time?

● A: O(V+E) ● We can actually determine if cycles exist in O(V) time: ■ In an undirected acyclic forest, |E|  |V| - 1 ■ So count the edges: if ever see |V| distinct edges, must have seen a back edge along the way

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