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CS 332: Algorithms. Graph Algorithms . 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

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cs 332 algorithms

CS 332: Algorithms

Graph Algorithms

David Luebke 16/6/2014

review depth first search
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 26/6/2014

review dfs code
DFS(G)

{

for each vertex u  G->V

{

u->color = WHITE;

}

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;

}

Review: DFS Code

David Luebke 36/6/2014

dfs example
DFS Example

sourcevertex

David Luebke 46/6/2014

dfs example1
DFS Example

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David Luebke 56/6/2014

dfs example2
DFS Example

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David Luebke 66/6/2014

dfs example3
DFS Example

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David Luebke 76/6/2014

dfs example4
DFS Example

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David Luebke 86/6/2014

dfs example5
DFS Example

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David Luebke 96/6/2014

dfs example6
DFS Example

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David Luebke 106/6/2014

dfs example7
DFS Example

sourcevertex

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2 | 7

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David Luebke 116/6/2014

dfs example8
DFS Example

sourcevertex

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2 | 7

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David Luebke 126/6/2014

dfs example9
DFS Example

sourcevertex

d f

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2 | 7

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

David Luebke 136/6/2014

dfs example10
DFS Example

sourcevertex

d f

1 |

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2 | 7

9 |10

3 | 4

5 | 6

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David Luebke 146/6/2014

dfs example11
DFS Example

sourcevertex

d f

1 |

8 |11

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2 | 7

9 |10

3 | 4

5 | 6

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David Luebke 156/6/2014

dfs example12
DFS Example

sourcevertex

d f

1 |12

8 |11

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2 | 7

9 |10

3 | 4

5 | 6

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David Luebke 166/6/2014

dfs example13
DFS Example

sourcevertex

d f

1 |12

8 |11

13|

2 | 7

9 |10

3 | 4

5 | 6

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David Luebke 176/6/2014

dfs example14
DFS Example

sourcevertex

d f

1 |12

8 |11

13|

2 | 7

9 |10

3 | 4

5 | 6

14|

David Luebke 186/6/2014

dfs example15
DFS Example

sourcevertex

d f

1 |12

8 |11

13|

2 | 7

9 |10

3 | 4

5 | 6

14|15

David Luebke 196/6/2014

dfs example16
DFS Example

sourcevertex

d f

1 |12

8 |11

13|16

2 | 7

9 |10

3 | 4

5 | 6

14|15

David Luebke 206/6/2014

dfs kinds of edges
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 216/6/2014

dfs example17
DFS Example

sourcevertex

d f

1 |12

8 |11

13|16

2 | 7

9 |10

3 | 4

5 | 6

14|15

Tree edges

David Luebke 226/6/2014

dfs kinds of edges1
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)

David Luebke 236/6/2014

dfs example18
DFS Example

sourcevertex

d f

1 |12

8 |11

13|16

2 | 7

9 |10

3 | 4

5 | 6

14|15

Tree edges

Back edges

David Luebke 246/6/2014

dfs kinds of edges2
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 256/6/2014

dfs example19
DFS Example

sourcevertex

d f

1 |12

8 |11

13|16

2 | 7

9 |10

3 | 4

5 | 6

14|15

Tree edges

Back edges

Forward edges

David Luebke 266/6/2014

dfs kinds of edges3
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 276/6/2014

dfs example20
DFS Example

sourcevertex

d f

1 |12

8 |11

13|16

2 | 7

9 |10

3 | 4

5 | 6

14|15

Tree edges

Back edges

Forward edges

Cross edges

David Luebke 286/6/2014

dfs kinds of edges4
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 296/6/2014

dfs kinds of edges5
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?)

source

F?

David Luebke 306/6/2014

dfs kinds of edges6
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 treevertex, before other vertex is explored
      • So in fact the picture is wrong…bothlower tree edges cannot in fact betree edges

source

C?

David Luebke 316/6/2014

dfs and graph cycles
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 326/6/2014

dfs and cycles
DFS And Cycles
  • How would you modify the code to detect cycles?

DFS(G)

{

for each vertex u  G->V

{

u->color = WHITE;

}

time = 0;

for each vertex u  G->V

{

if (u->color == WHITE)

DFS_Visit(u);

}

}

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;

}

David Luebke 336/6/2014

dfs and cycles1
DFS And Cycles
  • What will be the running time?

DFS(G)

{

for each vertex u  G->V

{

u->color = WHITE;

}

time = 0;

for each vertex u  G->V

{

if (u->color == WHITE)

DFS_Visit(u);

}

}

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;

}

David Luebke 346/6/2014

dfs and cycles2
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

David Luebke 356/6/2014