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# DFS ppt.pdf

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Depth first search [dfs]
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# DFS ppt.pdf

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### DFS ppt.pdf

1. 1. DEPTH FIRST SEARCH [DFS] PRESENTED BY : ANKIT KUMAR BRANCH : ECE UNI. ROLL : 12500319018
2. 2. INTRODUCTION  A depth-first search (DFS) explores a path all the way to a leaf before backtracking and exploring another path.  For example, after searching A, then B, then D, the search backtracks and tries another path from B.  Node are explored in the order A B D E H L M N I O P C F G J K Q. L M N O P N will be found before J.
3. 3. INTRO….  To keep track of progress DFS colors each vertex white, gray or black. Initially all the vertices are colored white. Then they are colored gray when discovered. Finally colored black when finished.  Besides creating depth first forest DFS also timestamps each vertex. Each vertex goes through two time stamps:  Discover time d[u]: when u is first discovered Finish time f[u]: when backtrack from u or finished u f[u] > d[u]
4. 4. DFS: ALGORITHM DFS: Algorithm DFS(G) 1. for each vertex u in G 2. color[u]=white 3. ᴨ[u]=NIL 4. time=0 5. for each vertex u in G 6. if (color[u]==white) 7. DFS-VISIT(G,u)
5. 5. ALGO(CONT…) DFS-VISIT(u) 1. time = time + 1 2. d[u] = time 3. color[u]=gray 4. for each v € Adj(u) in G do 5.if (color[v] = =white) 6. ᴨ[v] = u; 7. DFS-VISIT(G,v); 8. color[u] = black 9. time = time + 1; 10. f[u]= time;
6. 6. DFS: COMPLEXITY ANALYSIS  Initialization complexity is O(V)  DFS_VISIT is called exactly once for each vertex  And DFS_VISIT scans all the edges which causes cost of O(E)  Thus overall complexity is O(V + E)
7. 7. DFS: APPLICATION  Topological Sort  Strongly Connected Component
8. 8. CLASSIFICATION OF EDGES  Tree edge: Edge (u,v) is a tree edge if v was first discovered by exploring edge (u,v). White color indicates tree edge.  Back edge: Edge (u,v) is a back edge if it connects a vertex u to a ancestor v in a depth first tree. Gray color indicates back edge.  Forward edge: Edge (u,v) is a forward edge if it is non-tree edge and it connects a vertex u to a descendant v in a depth first tree. Black color indicates forward edge.  Cross edge: rest all other edges are called the cross edge. Black color indicates forward edge.
9. 9. EXAMPLE OF EDGES
10. 10. THEOREM DERIVED FROM DFS  Theorem 1: In a depth first search of an undirected graph G, every edge of G is either a tree edge or back edge.  Theorem 2: A directed graph G is acyclic if and only if a depth-first search of G yields no back edges.