Graphs bfs dfs
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The document discusses Breadth First Search (BFS) and Depth First Search (DFS) algorithms for graphs. It defines key graph terminology and describes how to represent graphs using adjacency lists and matrices. It then explains how BFS and DFS work by exploring the graph in different ways, keeping track of vertex colors and paths/trees produced. Both algorithms run in O(V+E) time and have different applications like routing, sorting, and finding connected components.
![BFS - algorithm
BFS(G, s)
// G is the graph and s is the starting node
1 for each vertex u ∈ V [G] - {s}
2
do color[u] ← WHITE
// color of vertex u
3
d[u] ← ∞
// distance from source s to vertex u
4
π[u] ← NIL
// predecessor of u
5 color[s] ← GRAY
6 d[s] ← 0
7 π[s] ← NIL
8 Q←Ø
// Q is a FIFO - queue
9 ENQUEUE(Q, s)
10 while Q ≠ Ø
// iterates as long as there are gray vertices. Lines 10-18
11
do u ← DEQUEUE(Q)
12
for each v ∈ Adj [u]
13
do if color[v] = WHITE
// discover the undiscovered adjacent vertices
14
then color[v] ← GRAY
// enqueued whenever painted gray
15
d[v] ← d[u] + 1
16
π[v] ← u
17
ENQUEUE(Q, v)
18
color[u] ← BLACK
// painted black whenever dequeued
10/27/13
NITTTR CHD
10](https://image.slidesharecdn.com/graphsbfsdfs-131027013414-phpapp01/85/Graphs-bfs-dfs-10-320.jpg)
![Depth first search
It searches ‘deeper’ the graph when possible.
Starts at the selected node and explores as far as
possible along each branch before backtracking.
Vertices go through white, gray and black stages of
color.
White – initially
Gray – when discovered first
Black – when finished i.e. the adjacency list of the vertex is
completely examined.
Also records timestamps for each vertex
10/27/13
d[v]
f[v]
when the vertex is first discovered
when the vertex is finished
NITTTR CHD
13](https://image.slidesharecdn.com/graphsbfsdfs-131027013414-phpapp01/85/Graphs-bfs-dfs-13-320.jpg)
![Depth first search - algorithm
DFS(G)
1 for each vertex u ∈ V [G]
2
do color[u] ← WHITE
3
π[u] ← NIL
4 time ← 0
5 for each vertex u ∈ V [G]
6
do if color[u] = WHITE
7
then DFS-VISIT(u)
// color all vertices white, set their parents NIL
// zero out time
// call only for unexplored vertices
// this may result in multiple sources
DFS-VISIT(u)
1 color[u] ← GRAY ▹White vertex u has just been discovered.
2 time ← time +1
3 d[u] time
// record the discovery time
4 for each v ∈ Adj[u]
▹Explore edge(u, v).
5
do if color[v] = WHITE
6
then π[v] ← u
// set the parent value
7
DFS-VISIT(v)
// recursive call
8 color[u] BLACK
▹ Blacken u; it is finished.
9 f [u] ▹ time ← time +1
10/27/13
NITTTR CHD
14](https://image.slidesharecdn.com/graphsbfsdfs-131027013414-phpapp01/85/Graphs-bfs-dfs-14-320.jpg)
Graphs bfs dfs
- 1. Graphs Breadth First Search & Depth First Search Submitted By: Jailalita Gautam
- 2. Contents 10/27/13 Overview of Graph terminology. Graph representation. Breadth first search. Depth first search. Applications of BFS and DFS. References. NITTTR CHD 2
- 3. Graph terminology - overview A graph consists of set of vertices V = {v1, v2, ….. vn} set of edges that connect the vertices E ={e1, e2, …. em} Two vertices in a graph are adjacent if there is an edge connecting the vertices. Two vertices are on a path if there is a sequences of vertices beginning with the first one and ending with the second one Graphs with ordered edges are directed. For directed graphs, vertices have in and out degrees. Weighted Graphs have values associated with edges. 10/27/13 NITTTR CHD 3
- 4. Graph representation There are two standard ways to represent a graph G=(V,E) : as collection of adjacency list or as an adjacency matrix. Adjacency list preferred for sparse graphs- those for which |E| much less than |V|^2. Adjacency matrix preferred for dense graphs- |E| is close to |V|^2. 10/27/13 NITTTR CHD 4
- 5. Graph representation – undirected graph 10/27/13 Adjacency list NITTTR CHD Adjacency matrix 5
- 6. Graph representation – directed graph 10/27/13 Adjacency list NITTTR CHD Adjacency matrix 6
- 7. Breadth first search Given a graph G=(V,E) – set of vertices and edges a distinguished source vertex s Breadth first search systematically explores the edges of G to discover every vertex that is reachable from s. It also produces a ‘breadth first tree’ with root s that contains all the vertices reachable from s. For any vertex v reachable from s, the path in the breadth first tree corresponds to the shortest path in graph G from s to v. It works on both directed and undirected graphs. However, we will explore only directed graphs. 10/27/13 NITTTR CHD 7
- 8. Breadth first search - concepts To keep track of progress, it colors each vertex - white, gray or black. All vertices start white. A vertex discovered first time during the search becomes nonwhite. All vertices adjacent to black ones are discovered. Whereas, gray ones may have some white adjacent vertices. 10/27/13 NITTTR CHD 8
- 9. BFS – How it produces a Breadth first tree The tree initially contains only root. – s Whenever a vertex v is discovered in scanning adjacency list of vertex u 10/27/13 Vertex v and edge (u,v) are added to the tree. NITTTR CHD 9
- 10. BFS - algorithm BFS(G, s) // G is the graph and s is the starting node 1 for each vertex u ∈ V [G] - {s} 2 do color[u] ← WHITE // color of vertex u 3 d[u] ← ∞ // distance from source s to vertex u 4 π[u] ← NIL // predecessor of u 5 color[s] ← GRAY 6 d[s] ← 0 7 π[s] ← NIL 8 Q←Ø // Q is a FIFO - queue 9 ENQUEUE(Q, s) 10 while Q ≠ Ø // iterates as long as there are gray vertices. Lines 10-18 11 do u ← DEQUEUE(Q) 12 for each v ∈ Adj [u] 13 do if color[v] = WHITE // discover the undiscovered adjacent vertices 14 then color[v] ← GRAY // enqueued whenever painted gray 15 d[v] ← d[u] + 1 16 π[v] ← u 17 ENQUEUE(Q, v) 18 color[u] ← BLACK // painted black whenever dequeued 10/27/13 NITTTR CHD 10
- 11. Breadth First Search - example 10/27/13 NITTTR CHD 11
- 12. Breadth first search - analysis Enqueue and Dequeue happen only once for each node. - O(V). Total time spent in scanning adjacency lists is O(E) . Initialization overhead O(V) Total runtime O(V+E) 10/27/13 NITTTR CHD 12
- 13. Depth first search It searches ‘deeper’ the graph when possible. Starts at the selected node and explores as far as possible along each branch before backtracking. Vertices go through white, gray and black stages of color. White – initially Gray – when discovered first Black – when finished i.e. the adjacency list of the vertex is completely examined. Also records timestamps for each vertex 10/27/13 d[v] f[v] when the vertex is first discovered when the vertex is finished NITTTR CHD 13
- 14. Depth first search - algorithm DFS(G) 1 for each vertex u ∈ V [G] 2 do color[u] ← WHITE 3 π[u] ← NIL 4 time ← 0 5 for each vertex u ∈ V [G] 6 do if color[u] = WHITE 7 then DFS-VISIT(u) // color all vertices white, set their parents NIL // zero out time // call only for unexplored vertices // this may result in multiple sources DFS-VISIT(u) 1 color[u] ← GRAY ▹White vertex u has just been discovered. 2 time ← time +1 3 d[u] time // record the discovery time 4 for each v ∈ Adj[u] ▹Explore edge(u, v). 5 do if color[v] = WHITE 6 then π[v] ← u // set the parent value 7 DFS-VISIT(v) // recursive call 8 color[u] BLACK ▹ Blacken u; it is finished. 9 f [u] ▹ time ← time +1 10/27/13 NITTTR CHD 14
- 15. Depth first search – example 10/27/13 NITTTR CHD 15
- 16. Depth first search - analysis Lines 1-3, initialization take time Θ(V). Lines 5-7 take time Θ(V), excluding the time to call the DFS-VISIT. DFS-VISIT is called only once for each node (since it’s called only for white nodes and the first step in it is to paint the node gray). Loop on line 4-7 is executed |Adj(v)| times. Since ∑vєV |Adj(v)| = Ө (E), the total cost of executing lines 4-7 of DFS-VISIT is θ(E). The total cost of DFS is θ(V+E) 10/27/13 NITTTR CHD 16
- 17. BFS and DFS - comparison Space complexity of DFS is lower than that of BFS. Time complexity of both is same – O(|V|+|E|). The behavior differs for graphs where not all the vertices can be reached from the given vertex s. Predecessor subgraphs produced by DFS may be different than those produced by BFS. The BFS product is just one tree whereas the DFS product may be multiple trees. 10/27/13 NITTTR CHD 17
- 18. BFS and DFS – possible applications Possible to use in routing / exploration. E.g., I want to explore all the nearest pizza places and want to go to the nearest one with only two intersections. Find distance from my factory to every delivery center. Most of the mapping software (GOOGLE maps, YAHOO(?) maps) should be using these algorithms. Applications of DFS Topologically sorting a directed acyclic graph. Finding the strongly connected components of a directed graph. 10/27/13 List the graph elements in such an order that all the nodes are listed before nodes to which they have outgoing edges. List all the sub graphs of a strongly connected graph which themselves are strongly connected. NITTTR CHD 18
- 19. References Data structures with C++ using STL by Ford, William; Topp, William; Prentice Hall. Introduction to Algorithms by Cormen, Thomas et. al., The MIT press. http://en.wikipedia.org/wiki/Graph_theory http://en.wikipedia.org/wiki/Depth_first_search 10/27/13 NITTTR CHD 19