bfs and dfs

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Group Name : Quantative

Depth First Search (DFS) Breadth First Search (BFS)

Topics

What is a graph?

1.Directed/Undirected

2.Weighted/Unweighted3.Cyclic/Acyclic

A set of vertices and edges

Breadth First Search (BFS)Start several paths at a time, and advance in each one step at a time

The breadth-first search uses a FIFO queue.

Depth First Search (DFS)Once a possible path is found, continue the search until the end of the path

The Depth-first search uses a LIFO Stack.

Graph Traversal

How It Works? 1.Pick a source vertex S to start. 2.Discover the vertices that are adjacent to S.

Depth-first: visit all neighbors of a neighbor before visiting your other neighbors

Breadth First: Pick each child of S

in turn and discover their vertices adjacent to that child.

Done when all children have been discovered and examined.

For a Graph G=(V, E) and n = |V| and m=|E|

When Adjacency List is used Complexity is O(m + n)

When Adjacency Matrix is used Scanning each row for checking the connectivity of a Vertex is in order O(n).

So, Complexity is O(n2 )

DFS uses space O(|V|) in the worst case to store the stack of vertices on the current search path as well as the set of already-visited vertices.

Analysis of DFS

Example:

Depth-first search: Directed graphs

The algorithm is essentially the same as for undirected graphs, the difference residing in the interpretation of the word "adjacent".

In a directed graph, node w is adjacent to node v if the directed edge (v, w) exists.

If (v, w) exists but (w, v) does not, then w is adjacent to v but v is not adjacent to w.

With this change of interpretation the procedures dfs and search apply equally well in the case of a directed graph.

DFS Example

Data Structure and Algorithm

sourcevertex

DFS Example


1 | | |

| | 3 |

2 | |

sourcevertex d f

DFS Example


1 | | |

| | 3 | 4

2 | |

sourcevertex d f

DFS Example


1 | | |

| 5 | 3 | 4

2 | |

sourcevertex d f

DFS Example


1 | | |

| 5 | 63 | 4

2 | |

sourcevertex d f

DFS Example


1 | | |

| 5 | 63 | 4

2 | 7 |

sourcevertex d f

DFS Example


1 | 8 | |

| 5 | 63 | 4

2 | 7 |

sourcevertex d f

DFS Example


1 | 8 | |

| 5 | 63 | 4

2 | 7 9 |

sourcevertex d f

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

Algorithm stepsStep:1 Push the root node in stack.Step:2 Loop until stack is empty.Step:3 Peek the node of the stack.Step:4 If the node has unvisited child nodes get the unvisited child node mark it has travers and push it on stack.


DFS: AlgorithmDFS(G) for each vertex u in V, color[u]=white; p[u]=NIL time=0; for each vertex u in V

if (color[u]=white) DFS-VISIT(u)


DFS-VISIT(u) color[u]=gray; time = time + 1; d[u] = time; for each v in Adj(u) do if (color[v] = white) p[v] = u; DFS-VISIT(v); color[u] = black; time = time + 1; f[u]= time;

DFS: Algorithm (Cont.)

sourcevertex

BFS:*Testing a graph for bipartitions*To find the shortest path from a vertex s to a vertex v in an un weighted graph*To find the length of such a path*To find out if a graph contains cycles *To construct a BSF tree/forest from a graph *Copying garbage collection

Applications

ApplicationsDFS:* Finding connected components.*Topological sorting. *Finding the bridges of a graph. *cycle Detecting *Finding strongly connected components. *Finding biconnectivity in graphs.

bfs and dfs

Engineering