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Depth-First Search Lecture 24 COMP171 Fall 2006. Graph / Slide 2 Depth-First Search (DFS) * DFS is another popular graph search strategy n Idea is similar.

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Presentation on theme: "Depth-First Search Lecture 24 COMP171 Fall 2006. Graph / Slide 2 Depth-First Search (DFS) * DFS is another popular graph search strategy n Idea is similar."— Presentation transcript:

1 Depth-First Search Lecture 24 COMP171 Fall 2006

2 Graph / Slide 2 Depth-First Search (DFS) * DFS is another popular graph search strategy n Idea is similar to pre-order traversal (visit node, then visit children recursively) * DFS can provide certain information about the graph that BFS cannot n It can tell whether we have encountered a cycle or not

3 Graph / Slide 3 DFS Algorithm * DFS will continue to visit neighbors in a recursive pattern n Whenever we visit v from u, we recursively visit all unvisited neighbors of v. Then we backtrack (return) to u. n Note: it is possible that w2 was unvisited when we recursively visit w1, but became visited by the time we return from the recursive call. u v w1 w2 w3

4 Graph / Slide 4 DFS Algorithm Flag all vertices as not visited Flag yourself as visited For unvisited neighbors, call RDFS(w) recursively We can also record the paths using pred[ ].

5 Graph / Slide 5 Example 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) F F F F F F F F F F Initialize visited table (all False) Initialize Pred to -1 - - - - - - - - - - Pred

6 Graph / Slide 6 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) F F T F F F F F F F Mark 2 as visited - - - - - - - - - - Pred RDFS( 2 ) Now visit RDFS(8)

7 Graph / Slide 7 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) F F T F F F F F T F Mark 8 as visited mark Pred[8] - - - - - - - - 2 - Pred RDFS( 2 ) RDFS(8) 2 is already visited, so visit RDFS(0) Recursive calls

8 Graph / Slide 8 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T F T F F F F F T F Mark 0 as visited Mark Pred[0] 8 - - - - - - - 2 - Pred RDFS( 2 ) RDFS(8) RDFS(0) -> no unvisited neighbors, return to call RDFS(8) Recursive calls

9 Graph / Slide 9 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T F T F F F F F T F 8 - - - - - - - 2 - Pred RDFS( 2 ) RDFS(8) Now visit 9 -> RDFS(9) Recursive calls Back to 8

10 Graph / Slide 10 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T F T F F F F F T T Mark 9 as visited Mark Pred[9] 8 - - - - - - - 2 8 Pred RDFS( 2 ) RDFS(8) RDFS(9) -> visit 1, RDFS(1) Recursive calls

11 Graph / Slide 11 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T F F F F F T T Mark 1 as visited Mark Pred[1] 8 9 - - - - - - 2 8 Pred RDFS( 2 ) RDFS(8) RDFS(9) RDFS(1) visit RDFS(3) Recursive calls

12 Graph / Slide 12 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T F F F F T T Mark 3 as visited Mark Pred[3] 8 9 - 1 - - - - 2 8 Pred RDFS( 2 ) RDFS(8) RDFS(9) RDFS(1) RDFS(3) visit RDFS(4) Recursive calls

13 Graph / Slide 13 RDFS( 2 ) RDFS(8) RDFS(9) RDFS(1) RDFS(3) RDFS(4)  STOP all of 4’s neighbors have been visited return back to call RDFS(3) 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T F F F T T Mark 4 as visited Mark Pred[4] 8 9 - 1 3 - - - 2 8 Pred Recursive calls

14 Graph / Slide 14 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T F F F T T 8 9 - 1 3 - - - 2 8 Pred RDFS( 2 ) RDFS(8) RDFS(9) RDFS(1) RDFS(3) visit 5 -> RDFS(5) Recursive calls Back to 3

15 Graph / Slide 15 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T T F F T T 8 9 - 1 3 3 - - 2 8 Pred RDFS( 2 ) RDFS(8) RDFS(9) RDFS(1) RDFS(3) RDFS(5) 3 is already visited, so visit 6 -> RDFS(6) Recursive calls Mark 5 as visited Mark Pred[5]

16 Graph / Slide 16 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T T T F T T 8 9 - 1 3 3 5 - 2 8 Pred RDFS( 2 ) RDFS(8) RDFS(9) RDFS(1) RDFS(3) RDFS(5) RDFS(6) visit 7 -> RDFS(7) Recursive calls Mark 6 as visited Mark Pred[6]

17 Graph / Slide 17 2 4 3 5 1 7 6 9 8 0 Adjacency List source 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T T T T T T 8 9 - 1 3 3 5 6 2 8 Pred RDFS( 2 ) RDFS(8) RDFS(9) RDFS(1) RDFS(3) RDFS(5) RDFS(6) RDFS(7) -> Stop no more unvisited neighbors Recursive calls Mark 7 as visited Mark Pred[7]

18 Graph / Slide 18 Adjacency List 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T T T T T T 8 9 - 1 3 3 5 6 2 8 Pred RDFS( 2 ) RDFS(8) RDFS(9) RDFS(1) RDFS(3) RDFS(5) RDFS(6) -> Stop Recursive calls 2 4 3 5 1 7 6 9 8 0 source

19 Graph / Slide 19 Adjacency List 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T T T T T T 8 9 - 1 3 3 5 6 2 8 Pred RDFS( 2 ) RDFS(8) RDFS(9) RDFS(1) RDFS(3) RDFS(5) -> Stop Recursive calls 2 4 3 5 1 7 6 9 8 0 source

20 Graph / Slide 20 Adjacency List 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T T T T T T 8 9 - 1 3 3 5 6 2 8 Pred RDFS( 2 ) RDFS(8) RDFS(9) RDFS(1) RDFS(3) -> Stop Recursive calls 2 4 3 5 1 7 6 9 8 0 source

21 Graph / Slide 21 Adjacency List 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T T T T T T 8 9 - 1 3 3 5 6 2 8 Pred RDFS( 2 ) RDFS(8) RDFS(9) RDFS(1) -> Stop Recursive calls 2 4 3 5 1 7 6 9 8 0 source

22 Graph / Slide 22 Adjacency List 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T T T T T T 8 9 - 1 3 3 5 6 2 8 Pred RDFS( 2 ) RDFS(8) RDFS(9) -> Stop Recursive calls 2 4 3 5 1 7 6 9 8 0 source

23 Graph / Slide 23 Adjacency List 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T T T T T T 8 9 - 1 3 3 5 6 2 8 Pred RDFS( 2 ) RDFS(8) -> Stop Recursive calls 2 4 3 5 1 7 6 9 8 0 source

24 Graph / Slide 24 Example Finished Adjacency List 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T T T T T T 8 9 - 1 3 3 5 6 2 8 Pred RDFS( 2 ) -> Stop Recursive calls finished 2 4 3 5 1 7 6 9 8 0 source

25 Graph / Slide 25 DFS Path Tracking Adjacency List 0 1 2 3 4 5 6 7 8 9 Visited Table (T/F) T T T T T T T T T T 8 9 - 1 3 3 5 6 2 8 Pred Try some examples. Path(0) -> Path(6) -> Path(7) -> DFS find out path too 2 4 3 5 1 7 6 9 8 0 source

26 Graph / Slide 26 DFS Tree Resulting DFS-tree. Notice it is much “deeper” than the BFS tree. Captures the structure of the recursive calls - when we visit a neighbor w of v, we add w as child of v - whenever DFS returns from a vertex v, we climb up in the tree from v to its parent

27 Graph / Slide 27 Time Complexity of DFS (Using adjacency list) * We never visited a vertex more than once * We had to examine all edges of the vertices n We know Σ vertex v degree(v) = 2m where m is the number of edges * So, the running time of DFS is proportional to the number of edges and number of vertices (same as BFS) n O(n + m) * You will also see this written as: n O(|v|+|e|)|v| = number of vertices (n) |e| = number of edges (m)

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