DFS algorithm

8/4/2019 DFS algorithm

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Depth-First Search

● Depth-first search is another strategy for

exploring a graph i.e. is a systematic way to

find all the vertices reachable from a source

vertex.

■ 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



Depth-First Search

● Vertices initially colored white

● Then colored gray when discovered

●Then black when finished



DFS Example

sourcevertex



DFS Example

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DFS Example

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DFS Example

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DFS Example

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DFS Example

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DFS Example

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DFS Example

1 | 8 | |

|5 | 63 | 4

2 | 7 |

sourcevertex

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DFS Example

1 | 8 | |

|5 | 63 | 4

2 | 7 9 |

sourcevertex

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DFS Example

1 | 8 | |

|5 | 63 | 4

2 | 7 9 |10

sourcevertex

d f



DFS Example

1 | 8 |11 |

|5 | 63 | 4

2 | 7 9 |10

sourcevertex

d f



DFS Example

1 |12 8 |11 |

|5 | 63 | 4

2 | 7 9 |10

sourcevertex

d f



DFS Example

1 |12 8 |11 13|

|5 | 63 | 4

2 | 7 9 |10

sourcevertex

d f



DFS Example

1 |12 8 |11 13|

14|5 | 63 | 4

2 | 7 9 |10

sourcevertex

d f



DFS Example

1 |12 8 |11 13|

14|155 | 63 | 4

2 | 7 9 |10

sourcevertex

d f



DFS Example

1 |12 8 |11 13|16

14|155 | 63 | 4

2 | 7 9 |10

sourcevertex

d f



DFS Example

1 |12 8 |11 13|16

14|155 | 63 | 4

2 | 7 9 |10

sourcevertex

d f

Tree edges



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 grey vertex (grey to grey)



DFS Example

1 |12 8 |11 13|16

14|155 | 63 | 4

2 | 7 9 |10

sourcevertex

d f

Tree edges Back edges



DFS: Kinds of edges





■ Forward edge: from ancestor to descendent

○ Not a tree edge, though

○ From grey node to black node



DFS Example

1 |12 8 |11 13|16

14|155 | 63 | 4

2 | 7 9 |10

sourcevertex

d f

Tree edges Back edges Forward edges



DFS: Kinds of edges





■ Forward edge: from ancestor to descendent

■ Cross edge: between a tree or subtrees

○ From a grey node to a black node



DFS Example

1 |12 8 |11 13|16

14|155 | 63 | 4

2 | 7 9 |10

sourcevertex

d f

Tree edges Back edges Forward edges Cross edges



DFS Applications

David Luebke 26 4/13/2012



● Topological Sorting

● Strong connectedness




Topological Sort

● There are many problems involving a set of tasks in which some of the tasks must be donebefore others.

● For example, consider the problem of taking acourse only after taking its prerequisites.

● Is there any systematic way of linearly

arranging the courses in the order that theyshould be taken?

The Answer is Topological Sort




● Topological sort is a method of arranging the

vertices in a directed acyclic graph (DAG), as

a sequence, such that no vertex appear in the

sequence before its predecessor.




30

Topological Sort: DFS

F

C

G A B

D E

H



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F

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G A B

D E

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dfs(A)



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dfs(A)

dfs(D)

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D E

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dfs(A)

dfs(D)

dfs(E)

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D E

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dfs(A)

dfs(D)

dfs(E)

dfs(F)

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D E

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dfs(A)

dfs(D)

dfs(E)

dfs(F)

dfs(H)

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G A B

D E

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dfs(A)

dfs(D)

dfs(E)

dfs(F)

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G A B

D E



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dfs(A)

dfs(D)

dfs(E)

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A B

D E

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G



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dfs(A)

dfs(D)

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D E

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dfs(A)

dfs(D)

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D E

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dfs(A)

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dfs(A)

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4 5

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dfs(B)



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dfs(B)



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F

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D E

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3 2



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D E

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dfs(C)

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dfs(C)

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dfs(C)

dfs(G)

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D E

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dfs(C)

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F

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D E

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3 2 1

0

Topological order: C G B A D E F H



Strongly Connected Components

● Definition A strongly connected component of

a directed graph G is a maximal set of vertices

C V such that for every pair of vertices u and

v, there is a directed path from u to v and adirected path from v to u.

● Strongly-Connected-Components(G)




Disadvantages of DFS

● The disadvantage of Depth-First Search is that

there is a possibility that it may go down the

left-most path forever. Even a finite graph can

generate an infinite tree. One solution to thisproblem is to impose a cutoff depth on the

search.




● Depth-First Search is not guaranteed to find

the solution.

● And there is no guarantee to find a minimal

solution, if more than one solution exists.


BFS in Directed graph



BFS in Directed graph

● Pick a source vertex S to start.

● Find (or discover) the vertices that are

adjacent to S.

● Pick each child of S in turn and discover their

vertices adjacent to that child.

● Done when all children have been discovered

and examined.

● This results in a tree that is rooted at the source

vertex S.





Example



Initially, d[a] is set to 0 and the rest to ∞.Q [a].Remove head: Q [] children of a are c,b d[c]= ∞, d[b]= ∞ so d[c] d[a]+1=1, d[b] d[a]+1=1Q [c b]

Remove head: Q

[b] children of c are e,f d[e]= ∞, d[f]= ∞ so d[e] d[c]+1=2, d[f] d[c]+1=2

Q [b e f]



Remove head: Q [e f] children of b is f d[f] <> ∞, nothing done with it Remove head: Q [f]

children of e is d, i, h d[d]= ∞, d[i]= ∞, d[h]= ∞ so d[d] = d[i] = d[h] d[e]+1=3 Q [f d i h]

Remove head: Q [d i h] children of f is g,h d[g]= ∞, so d[g] d[f]+1 = 3 Q [d i h g]



Each of these has children that are

already has a value less than ∞, so these will not set any further values and we are done

with the BFS.



We can create tree out of order we visit the nodes



BFS always computes the

shortest path distance in d[I] between S and vertex I.



Pseudocode: Uses FIFO Queue Q

BFS(s) ; s is our source vertex for each u V - {s} ; Initialize unvisited vertices to

do d[u] ∞

d[s] 0 ; distance to source vertex is 0 Q {s} ; Queue of vertices to visit while Q<>0 do

remove u from Q for each v Adj[u] do ; Get adjacent vertices if d[v]= ∞ then d[v] d[u]+1 ; Increment depth

put v onto Q ; Add to nodes to explore

Breadth – first search



Breadth first search

Application

s



Testing bipartiteness

Finding all nodes within one connected



Finding all nodes within one connectedcomponent

The set of nodes reached by a BFS(breadth-first search) form the connected

component containing the starting node.



•Finding the shortest path between two nodes u and v

•Applications to image processing problems

Documents

DFS algorithm