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![Depth-First Search
● Input:
■ G = (V, E) (No source vertex given!)
● Goal:
■ Explore the edges of G to “discover” every vertex in V starting at the most
current visited node
■ Search may be repeated from multiple sources
● Output:
■ 2 timestamps on each vertex:
○ d[v] = discovery time
○ f[v] = finishing time (done with examining v’s adjacency list)
■ Depth-first forest
1 2
5 4
3](https://image.slidesharecdn.com/dfs-190707173403/75/Depth-First-Search-DFS-2-2048.jpg)
![DFS Additional Data Structures
● Global variable: time-stamp
■ Incremented when nodes are discovered or finished
● color[u] – similar to BFS
■ White before discovery, gray while processing and black
when finished processing
● prev[u] – predecessor of u
● d[u], f[u] – discovery and finish times](https://image.slidesharecdn.com/dfs-190707173403/75/Depth-First-Search-DFS-4-2048.jpg)
![5
Depth-First Search: The Code
Data: color[V], time,
prev[V],d[V], f[V]
DFS(G) // where prog starts
{
for each vertex u ∈ V
{
color[u] = WHITE;
prev[u]=NIL;
f[u]=inf; d[u]=inf;
}
time = 0;
for each vertex u ∈ V
if (color[u] == WHITE)
DFS_Visit(u);
}
DFS_Visit(u)
{
color[u] = GREY;
time = time+1;
d[u] = time;
for each v ∈ Adj[u]
{
if(color[v] == WHITE){
prev[v]=u;
DFS_Visit(v);}
}
color[u] = BLACK;
time = time+1;
f[u] = time;
}
Initialize](https://image.slidesharecdn.com/dfs-190707173403/75/Depth-First-Search-DFS-5-2048.jpg)
![6
Depth-First Search: The Code
Data: color[V], time,
prev[V],d[V], f[V]
DFS(G) // where prog starts
{
for each vertex u ∈ V
{
color[u] = WHITE;
prev[u]=NIL;
f[u]=inf; d[u]=inf;
}
time = 0;
for each vertex u ∈ V
if (color[u] == WHITE)
DFS_Visit(u);
}
DFS_Visit(u)
{
color[u] = GREY;
time = time+1;
d[u] = time;
for each v ∈ Adj[u]
{
if(color[v] == WHITE){
prev[v]=u;
DFS_Visit(v);}
}
color[u] = BLACK;
time = time+1;
f[u] = time;
}
What does u[d] represent?](https://image.slidesharecdn.com/dfs-190707173403/75/Depth-First-Search-DFS-6-2048.jpg)
![7
Depth-First Search: The Code
Data: color[V], time,
prev[V],d[V], f[V]
DFS(G) // where prog starts
{
for each vertex u ∈ V
{
color[u] = WHITE;
prev[u]=NIL;
f[u]=inf; d[u]=inf;
}
time = 0;
for each vertex u ∈ V
if (color[u] == WHITE)
DFS_Visit(u);
}
DFS_Visit(u)
{
color[u] = GREY;
time = time+1;
d[u] = time;
for each v ∈ Adj[u]
{
if(color[v] == WHITE){
prev[v]=u;
DFS_Visit(v);}
}
color[u] = BLACK;
time = time+1;
f[u] = time;
}
What does f[d] represent?](https://image.slidesharecdn.com/dfs-190707173403/75/Depth-First-Search-DFS-7-2048.jpg)
![8
Depth-First Search: The Code
Data: color[V], time,
prev[V],d[V], f[V]
DFS(G) // where prog starts
{
for each vertex u ∈ V
{
color[u] = WHITE;
prev[u]=NIL;
f[u]=inf; d[u]=inf;
}
time = 0;
for each vertex u ∈ V
if (color[u] == WHITE)
DFS_Visit(u);
}
DFS_Visit(u)
{
color[u] = GREY;
time = time+1;
d[u] = time;
for each v ∈ Adj[u]
{
if(color[v] == WHITE){
prev[v]=u;
DFS_Visit(v);
}}
color[u] = BLACK;
time = time+1;
f[u] = time;
}Will all vertices eventually be colored black?](https://image.slidesharecdn.com/dfs-190707173403/75/Depth-First-Search-DFS-8-2048.jpg)
![26
Depth-First Search: The Code
Data: color[V], time,
prev[V],d[V], f[V]
DFS(G) // where prog starts
{
for each vertex u ∈ V
{
color[u] = WHITE;
prev[u]=NIL;
f[u]=inf; d[u]=inf;
}
time = 0;
for each vertex u ∈ V
if (color[u] == WHITE)
DFS_Visit(u);
}
DFS_Visit(u)
{
color[u] = GREY;
time = time+1;
d[u] = time;
for each v ∈ Adj[u]
{
if (color[v] == WHITE)
prev[v]=u;
DFS_Visit(v);
}
color[u] = BLACK;
time = time+1;
f[u] = time;
}What will be the running time?](https://image.slidesharecdn.com/dfs-190707173403/75/Depth-First-Search-DFS-26-2048.jpg)
![27
Depth-First Search: The Code
Data: color[V], time,
prev[V],d[V], f[V]
DFS(G) // where prog starts
{
for each vertex u ∈ V
{
color[u] = WHITE;
prev[u]=NIL;
f[u]=inf; d[u]=inf;
}
time = 0;
for each vertex u ∈ V
if (color[u] == WHITE)
DFS_Visit(u);
}
DFS_Visit(u)
{
color[u] = GREY;
time = time+1;
d[u] = time;
for each v ∈ Adj[u]
{
if (color[v] == WHITE)
prev[v]=u;
DFS_Visit(v);
}
color[u] = BLACK;
time = time+1;
f[u] = time;
}Running time: O(V2
) because call DFS_Visit on each vertex,
and the loop over Adj[] can run as many as |V| times
O(V)
O(V)
O(V)](https://image.slidesharecdn.com/dfs-190707173403/75/Depth-First-Search-DFS-27-2048.jpg)
![28
Depth-First Search: The Code
Data: color[V], time,
prev[V],d[V], f[V]
DFS(G) // where prog starts
{
for each vertex u ∈ V
{
color[u] = WHITE;
prev[u]=NIL;
f[u]=inf; d[u]=inf;
}
time = 0;
for each vertex u ∈ V
if (color[u] == WHITE)
DFS_Visit(u);
}
DFS_Visit(u)
{
color[u] = GREY;
time = time+1;
d[u] = time;
for each v ∈ Adj[u]
{
if (color[v] == WHITE)
prev[v]=u;
DFS_Visit(v);
}
color[u] = BLACK;
time = time+1;
f[u] = time;
}
BUT, there is actually a tighter bound.
How many times will DFS_Visit() actually be called?](https://image.slidesharecdn.com/dfs-190707173403/75/Depth-First-Search-DFS-28-2048.jpg)
![29
Depth-First Search: The Code
Data: color[V], time,
prev[V],d[V], f[V]
DFS(G) // where prog starts
{
for each vertex u ∈ V
{
color[u] = WHITE;
prev[u]=NIL;
f[u]=inf; d[u]=inf;
}
time = 0;
for each vertex u ∈ V
if (color[u] == WHITE)
DFS_Visit(u);
}
DFS_Visit(u)
{
color[u] = GREY;
time = time+1;
d[u] = time;
for each v ∈ Adj[u]
{
if (color[v] == WHITE)
prev[v]=u;
DFS_Visit(v);
}
color[u] = BLACK;
time = time+1;
f[u] = time;
}
So, running time of DFS = O(V+E)](https://image.slidesharecdn.com/dfs-190707173403/75/Depth-First-Search-DFS-29-2048.jpg)
![Reference
● Cormen –
■ Chapter 22 (Elementary Graph Algorithms)
● Exercise –
■ 22.3-4 –Detect edge using d[u], d[v], f[u], f[v]
■ 22.3-11 – Connected Component
■ 22.3-12 – Singly connected
39](https://image.slidesharecdn.com/dfs-190707173403/75/Depth-First-Search-DFS-39-2048.jpg)
More Related Content
![Depth first search [dfs]](https://cdn.slidesharecdn.com/ss_thumbnails/depthfirstsearchdfs-190926145304-thumbnail.jpg?width=640&height=640&fit=bounds)
Similar to Depth First Search ( DFS )
Depth First Search ( DFS )
- 1.
- 2. Depth-First Search ● Input: ■G = (V, E) (No source vertex given!) ● Goal: ■ Explore the edges of G to “discover” every vertex in V starting at the most current visited node ■ Search may be repeated from multiple sources ● Output: ■ 2 timestamps on each vertex: ○ d[v] = discovery time ○ f[v] = finishing time (done with examining v’s adjacency list) ■ Depth-first forest 1 2 5 4 3
- 3. Depth-First Search ● Search“deeper” in the graph whenever possible ● Edges are explored out of the most recently discovered vertex v that still has unexplored edges • After all edges of v have been explored, the search “backtracks” from the parent of v • The process continues until all vertices reachable from the original source have been discovered • If undiscovered vertices remain, choose one of them as a new source and repeat the search from that vertex • DFS creates a “depth-first forest” 1 2 5 4 3
- 4. DFS Additional DataStructures ● Global variable: time-stamp ■ Incremented when nodes are discovered or finished ● color[u] – similar to BFS ■ White before discovery, gray while processing and black when finished processing ● prev[u] – predecessor of u ● d[u], f[u] – discovery and finish times
- 5. 5 Depth-First Search: TheCode Data: color[V], time, prev[V],d[V], f[V] DFS(G) // where prog starts { for each vertex u ∈ V { color[u] = WHITE; prev[u]=NIL; f[u]=inf; d[u]=inf; } time = 0; for each vertex u ∈ V if (color[u] == WHITE) DFS_Visit(u); } DFS_Visit(u) { color[u] = GREY; time = time+1; d[u] = time; for each v ∈ Adj[u] { if(color[v] == WHITE){ prev[v]=u; DFS_Visit(v);} } color[u] = BLACK; time = time+1; f[u] = time; } Initialize
- 6. 6 Depth-First Search: TheCode Data: color[V], time, prev[V],d[V], f[V] DFS(G) // where prog starts { for each vertex u ∈ V { color[u] = WHITE; prev[u]=NIL; f[u]=inf; d[u]=inf; } time = 0; for each vertex u ∈ V if (color[u] == WHITE) DFS_Visit(u); } DFS_Visit(u) { color[u] = GREY; time = time+1; d[u] = time; for each v ∈ Adj[u] { if(color[v] == WHITE){ prev[v]=u; DFS_Visit(v);} } color[u] = BLACK; time = time+1; f[u] = time; } What does u[d] represent?
- 7. 7 Depth-First Search: TheCode Data: color[V], time, prev[V],d[V], f[V] DFS(G) // where prog starts { for each vertex u ∈ V { color[u] = WHITE; prev[u]=NIL; f[u]=inf; d[u]=inf; } time = 0; for each vertex u ∈ V if (color[u] == WHITE) DFS_Visit(u); } DFS_Visit(u) { color[u] = GREY; time = time+1; d[u] = time; for each v ∈ Adj[u] { if(color[v] == WHITE){ prev[v]=u; DFS_Visit(v);} } color[u] = BLACK; time = time+1; f[u] = time; } What does f[d] represent?
- 8. 8 Depth-First Search: TheCode Data: color[V], time, prev[V],d[V], f[V] DFS(G) // where prog starts { for each vertex u ∈ V { color[u] = WHITE; prev[u]=NIL; f[u]=inf; d[u]=inf; } time = 0; for each vertex u ∈ V if (color[u] == WHITE) DFS_Visit(u); } DFS_Visit(u) { color[u] = GREY; time = time+1; d[u] = time; for each v ∈ Adj[u] { if(color[v] == WHITE){ prev[v]=u; DFS_Visit(v); }} color[u] = BLACK; time = time+1; f[u] = time; }Will all vertices eventually be colored black?
- 9.
- 10. 10 DFS Example 1 || | ||| | | source vertex d fS A B C D E F G
- 11. 11 DFS Example 1 || | ||| 2 | | source vertex d fS A B C D E F G
- 12. 12 DFS Example 1 || | ||3 | 2 | | source vertex d fS A B C D E F G
- 13. 13 DFS Example 1 || | ||3 | 4 2 | | source vertex d fS A B C D E F G
- 14. 14 DFS Example 1 || | |5 |3 | 4 2 | | source vertex d fS A B C D E F G
- 15. 15 DFS Example 1 || | |5 | 63 | 4 2 | | source vertex d fS A B C D E F G
- 16. 16 DFS Example 1 || | |5 | 63 | 4 2 | 7 | source vertex d fS A B C D E F G
- 17. 17 DFS Example 1 |8 | | |5 | 63 | 4 2 | 7 | source vertex d fS A B C D E F G
- 18. 18 DFS Example 1 |8 | | |5 | 63 | 4 2 | 7 9 | source vertex d f What is the structure of the grey vertices? What do they represent? S A B C D E F G
- 19. 19 DFS Example 1 |8 | | |5 | 63 | 4 2 | 7 9 |10 source vertex d fS A B C D E F G
- 20. 20 DFS Example 1 |8 |11 | |5 | 63 | 4 2 | 7 9 |10 source vertex d fS A B C D E F G
- 21. 21 DFS Example 1 |128 |11 | |5 | 63 | 4 2 | 7 9 |10 source vertex d fS A B C D E F G
- 22. 22 DFS Example 1 |128 |11 13| |5 | 63 | 4 2 | 7 9 |10 source vertex d fS A B C D E F G
- 23. 23 DFS Example 1 |128 |11 13| 14|5 | 63 | 4 2 | 7 9 |10 source vertex d fS A B C D E F G
- 24. 24 DFS Example 1 |128 |11 13| 14|155 | 63 | 4 2 | 7 9 |10 source vertex d fS A B C D E F G
- 25. 25 DFS Example 1 |128 |11 13|16 14|155 | 63 | 4 2 | 7 9 |10 source vertex d fS A B C D E F G
- 26. 26 Depth-First Search: TheCode Data: color[V], time, prev[V],d[V], f[V] DFS(G) // where prog starts { for each vertex u ∈ V { color[u] = WHITE; prev[u]=NIL; f[u]=inf; d[u]=inf; } time = 0; for each vertex u ∈ V if (color[u] == WHITE) DFS_Visit(u); } DFS_Visit(u) { color[u] = GREY; time = time+1; d[u] = time; for each v ∈ Adj[u] { if (color[v] == WHITE) prev[v]=u; DFS_Visit(v); } color[u] = BLACK; time = time+1; f[u] = time; }What will be the running time?
- 27. 27 Depth-First Search: TheCode Data: color[V], time, prev[V],d[V], f[V] DFS(G) // where prog starts { for each vertex u ∈ V { color[u] = WHITE; prev[u]=NIL; f[u]=inf; d[u]=inf; } time = 0; for each vertex u ∈ V if (color[u] == WHITE) DFS_Visit(u); } DFS_Visit(u) { color[u] = GREY; time = time+1; d[u] = time; for each v ∈ Adj[u] { if (color[v] == WHITE) prev[v]=u; DFS_Visit(v); } color[u] = BLACK; time = time+1; f[u] = time; }Running time: O(V2 ) because call DFS_Visit on each vertex, and the loop over Adj[] can run as many as |V| times O(V) O(V) O(V)
- 28. 28 Depth-First Search: TheCode Data: color[V], time, prev[V],d[V], f[V] DFS(G) // where prog starts { for each vertex u ∈ V { color[u] = WHITE; prev[u]=NIL; f[u]=inf; d[u]=inf; } time = 0; for each vertex u ∈ V if (color[u] == WHITE) DFS_Visit(u); } DFS_Visit(u) { color[u] = GREY; time = time+1; d[u] = time; for each v ∈ Adj[u] { if (color[v] == WHITE) prev[v]=u; DFS_Visit(v); } color[u] = BLACK; time = time+1; f[u] = time; } BUT, there is actually a tighter bound. How many times will DFS_Visit() actually be called?
- 29. 29 Depth-First Search: TheCode Data: color[V], time, prev[V],d[V], f[V] DFS(G) // where prog starts { for each vertex u ∈ V { color[u] = WHITE; prev[u]=NIL; f[u]=inf; d[u]=inf; } time = 0; for each vertex u ∈ V if (color[u] == WHITE) DFS_Visit(u); } DFS_Visit(u) { color[u] = GREY; time = time+1; d[u] = time; for each v ∈ Adj[u] { if (color[v] == WHITE) prev[v]=u; DFS_Visit(v); } color[u] = BLACK; time = time+1; f[u] = time; } So, running time of DFS = O(V+E)
- 30. 30 DFS: Kinds ofedges ● 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? No
- 31. 31 DFS Example 1 |128 |11 13|16 14|155 | 63 | 4 2 | 7 9 |10 source vertex d f Tree edges
- 32. 32 DFS: Kinds ofedges ● 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) ○ Self loops are considered as to be back edge.
- 33. 33 DFS Example 1 |128 |11 13|16 14|155 | 63 | 4 2 | 7 9 |10 source vertex d f Tree edges Back edges
- 34. 34 DFS: Kinds ofedges ● 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 grey node to black node
- 35. 35 DFS Example 1 |128 |11 13|16 14|155 | 63 | 4 2 | 7 9 |10 source vertex d f Tree edges Back edges Forward edges
- 36. 36 DFS: Kinds ofedges ● 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 grey node to a black node
- 37. 37 DFS Example 1 |128 |11 13|16 14|155 | 63 | 4 2 | 7 9 |10 source vertex d f Tree edges Back edges Forward edges Cross edges
- 38. 38 DFS: Kinds ofedges ● 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
- 39. Reference ● Cormen – ■Chapter 22 (Elementary Graph Algorithms) ● Exercise – ■ 22.3-4 –Detect edge using d[u], d[v], f[u], f[v] ■ 22.3-11 – Connected Component ■ 22.3-12 – Singly connected 39