Lecture 14 Graphs II: DFS 6.006 Fall 2011 Lecture 14: Graphs II: Depth-First Search Lecture Overview • Depth-First Search • Edge Classification • Cycle Testing • Topological Sort Recall: • graph search: explore a graph e.g., find a path from start vertex s to a desired vertex • adjacency lists: array Adj of jV j linked lists { for each vertex u 2 V , Adj[u] stores u's neighbors, i.e., fv 2 V j (u; v) 2 Eg (just outgoing edges if directed) For example: a a c b c a c b c b Adj Figure 1: Adjacency Lists Breadth-first Search (BFS): Explore level-by-level from s | find shortest paths 1 Lecture 14 Graphs II: DFS 6.006 Fall 2011 Depth-First Search (DFS) This is like exploring a maze. s Figure 2: Depth-First Search Frontier Depth First Search Algorithm • follow path until you get stuck • backtrack along breadcrumbs until reach unexplored neighbor • recursively explore • careful not to repeat a vertex parent = {s: None} search from DFS-visit (V, Adj, s): start vertex s start for v in Adj [s]: (only see v if v not in parent: stuff reachable parent [v] = s from s) nish DFS-visit (V, Adj, v) v } DFS (V, Adj) parent = { } explore for s in V: entire graph if s not in parent: parent [s] = None (could do same DFS-visit (V, Adj, s) } to extend BFS) Figure 3: Depth-First Search Algorithm 2 Lecture 14 Graphs II: DFS 6.006 Fall 2011 Example cross edge S 1 1 S2 a b c forward 8 edge 5 2 6 4 back d e f 7 edge 3 back edge Figure 4: Depth-First Traversal Edge Classification tree edges (formed by parent) nontree edges back edge: to ancestor forward edge: to descendant cross edge (to another subtree) Figure 5: Edge Classification • to compute this classification (back or not), mark nodes for duration they are \on the stack" • only tree and back edges in undirected graph Analysis • DFS-visit gets called with a vertex s only once (because then parent[s] set) X =) time in DFS-visit = jAdj[s]j = O(E) s2V • DFS outer loop adds just O(V ) =) O(V + E) time (linear time) 3 Lecture 14 Graphs II: DFS 6.006 Fall 2011 Cycle Detection Graph G has a cycle , DFS has a back edge Proof (<=) tree edges is a cycle back edge: to tree ancestor (=>) consider rst visit to cycle: v3 v2 vk v 1 v0 FIRST! • before visit to vi finishes, will visit vi+1 (& finish): will consider edge (vi; vi+1) =) visit vi+1 now or already did • =) before visit to v0 finishes, will visit vk (& didn't before) • =) before visit to vk (or v0) finishes, will see (vk; v0) as back edge Job scheduling Given Directed Acylic Graph (DAG), where vertices represent tasks & edges represent dependencies, order tasks without violating dependencies 4 Lecture 14 Graphs II: DFS 6.006 Fall 2011 8 7 9 G H I 4 3 2 1 A B C F D E 6 5 Figure 6: Dependence Graph: DFS Finishing Times Source: Source = vertex with no incoming edges = schedulable at beginning (A,G,I) Attempt: BFS from each source: • from A finds A, BH, C, F • from D finds D, BE, CF slow . and wrong! • from G finds G, H • from I finds I Topological Sort 8 > DFS-Visit(v) > > ... Reverse of DFS finishing times (time at which DFS-Visit(v) finishes) <> order.append(v) > > > order.reverse() :> 5 Lecture 14 Graphs II: DFS 6.006 Fall 2011 Correctness For any edge (u; v) | u ordered before v, i.e., v finished before u u v • if u visited before v: { before visit to u finishes, will visit v (via (u; v) or otherwise) { =) v finishes before u • if v visited before u: { graph is acyclic { =) u cannot be reached from v { =) visit to v finishes before visiting u 6 MIT OpenCourseWare http://ocw.mit.edu 6.006 Introduction to Algorithms Fall 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms..