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Depth First Search Algorithm in 🧠 DFS in Artificial Intelligence (AI)
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- 2. Outline and Reading Depth-First Search 2 4Definitions (§6.1) ■ Subgraph ■ Connectivity ■ Spanning trees and forests 4 Depth-first search (§6.3.1) ■ Algorithm ■ Example • Properties • Analysis 4 Applications of DFS (§6.5) ■ Path finding • Cycle finding
- 3. Subgraphs Depth-First Search 3 0 A subgraphS of a graph G is a graph such that ■The vertices of S are a subset of the vertices of G ■ The edges of S are a subset of the edges of G 4 A spanning subgraph of Gis a subgraph thatcontains all the vertices of G Spanning subgraph Subgraph
- 4. Connectivi@ Depth-First Search 4 ¥A graph is connectedif there is a path between every pair of vertices ¥A connected component of a graph G is a maximal connected subgraph of G Connected graph Non connected graph with two connected component
- 5. Trees and Forests Depth-First Search 5 4A (free) tree is an undirected graph T such that • T is connected • T has no cycles This definition of tree is different from the one of a rooted tree 4 A forest is an undirected graph without cycles 4 The connected components of a forest are trees Tree Forest
- 6. Spanning Trees andForests Depth-First Search 6 6 A spanning tree of aconnected graph is a spanning subgraph that is a tree 6 A spanning tree is not unique unless the graph is a tree 6 Spanning trees have applications to the designof communication neWorks 4 A spanning forest of agraph is a spanningsubgraph that is a forest Grap h Spanning tree
- 7. Depth-First Search Depth-First Search 7 4 Depth-firstsearch (DFS) is a general technique for traversing a graph 4 A DFS traversal of a graph G ■Visits all the vertices and edges of G • Determines whether G is connected • Computes the connected components of G ■Computes a spanning 0 DFS on a graph with n velices and m edges takes O(n +m ) time 4 DFS can be further extended to solve other graph problems • Find and report a path between two given vertices ■ Find a cycle in the graph 0 Depth-first search is to graphs what Euler tour is to binary trees
- 8. DFS Algorithm 6 Thealgorithm uses a mechanism for setting and getting “labels” of vertices and edges AIgorithmDFS(G) Input gGraph Output labeling of the edges of G .as discovery edges and back edges for all u e G.rerfices() setLabel(u, UNEXPLORED) for all e z G.edges() setLabel{e, UNEXPLORED) for all r e G.vertices() U getLabel v) —— AlgorithmfiPS(G, v) Input gGraph and a start vertexGr of Output labeling of the edges of G in the connected component of r as discovery edges and back edges setLabel{v, WSSUED) Nez all e e G.incidentEdges(v) if getLabel(e) —— UNEXPLORED • ‹- pPOsite{v,e) if getLabel(w) —— UNEXPLORED setLabel(e, DISCOVERY) DFS(G, w) else Depth-First Search
- 9. Example 9 Depth-First Search unexplored velex visited velex unexplorededge discovery edge • back edge
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- 11. DFS and NazeTraversal Depth-First Search 11 4 The DFS algorithm is similar to a classic strategy for exploring a maze ■ We mark each intersection, corner and dead end (velex) visited • We mark each corridor (edge ) traversed ■ We keep track of the path back to the entrance (start vertex) by means of a rope (recursion stack)
- 12. Prope%ies DFS Depth-First Search 12 Property 1 DFS(G, »)visits all the velices and edges in the connected component of v Propert:y 2 The discovery edges labeled by DFS(G, r) form a spanning tree of the connected component of v
- 13. Analysis DFS Depth-First Search 13 y n '-edge label takes O(1) time : ::, eex ,g:b*eeedx • once as UNEXPLORED ■ once as VISITED 0 Each edge is labeled twice • once as UNEXPLORED • once as DISCOVERY or BACK 4 Method incidentEdges is called once for each velex 0 DFS runs in O(zi +m) time provided the graph is represented by the adjacency list structure ■ Recall that Z, deg(v) = 2nz
- 14. Path Finding Depth-First Search 14 We canspecialize the DFS algorithm to find a path between two given velices u and z using the template method pattern We call DFS(C, u) with u as the start vertex We use a stack S to keep track of the path between the start vertex and the current vertex As soon as destination velex z is encountered, we return the path as the contents of the stack Algorithmpa@DNS(G, r, z) setLabel(v, VISITED) S.puch(v) if r return S.elements{) for all e z G.incidentEdges(v) iI getLabel(e) ——UNF XPLORED W *Rp' $ite(v,e) if getLabel(w) UNEXPLORED setLabel(e, DISCOVER ) S.push(e) pathDFS(G, w, z) S.R•p{e) else setLabel(e, BACK) S.p*R(•)
- 15. Cycle Finding Depth-First Search 15 We canspecialize the DFS algorithm to find a simple cycle using the template method pahern We use a stack S to keeptrack of the path between the stal vertex and the current velex + As soon as a back edge (v, w) is encountered, we return the cycle as the portion of the stack from the top to vertex ›r Algorithm cycleDFS{G, v, z) setLabel‹v, VISITED for all e e G.incidentEdges(v) if getLabelie) -- UNEXPLORED w ‹- opposite[v,e) S.push(e) if getLabel(w j -- UNEXPLORE.D setLabel(e, DISCOVERY) patliDFS(G, w, z) S.pop{e) else fi ‹- new empty stack repeat o +- S.pop ) -R ffsh{O) until o -- w return T.elements()