6.006 13. Searching II: breadth-first search and depth-first search

MIT OpenCourseWare http://ocw.mit.edu6.006Introduction to AlgorithmsSpring 2008For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture 13 Searching II 6.006 Spring 2008Lecture 13: Searching II: Breadth-First Search and Depth-First Search Lecture Overview: Search 2 of 3 Breadth-First Search • Shortest Paths • • Depth-First Search • Edge Classification Readings CLRS 22.2-22.3 Recall: graph search: explore a graphe.g., find a path from start vertices to a desired vertexadjacency lists: array Adj of | V | linked lists • for each vertex u�V, Adj[u] stores u’s neighbors, i.e. {v�V | (u, v)�E} v -just outgoing edges if directed abcabcccbaAdjFigure 1: Adjacency Lists 1Lecture 13 Searching II 6.006 Spring 2008. . .level Øslevel 1level 2last levelFigure 2: Breadth-First Search Breadth-first Search (BFS): See Figure 2 Explore graph level by level from S • level φ = {s} • level i = vertices reachable by path of i edges but not fewer • build level i> 0 from level i − 1 by trying all outgoing edges, but ignoring vertices from previous levels BFS (V,Adj,s): level = { s: φ }parent = {s : None }i =1 frontier =[s] � previous level, i − 1 while frontier: next = [ ] � next level, i for u in frontier: for v in Adj [u]: if v not in level: � not yet seen level[v]= i� = level[u]+1 parent[v]= u next.append(v) frontier = next i +=1 2 � � Lecture 13 Searching II 6.006 Spring 2008Example:asdfvcxz1Ø233221level Ølevel 1level 2level 3frontierØ = {s}frontier1 = {a, x}frontier2 = {z, d, c}frontier3 = {f, v}(not x, c, d)Figure 3: Breadth-First Search Frontier Analysis: • vertex V enters next (& then frontier)only once (because level[v] then set)base case: v = s = Adj[v] looped through only once •⇒ time = � Adj[V ]= | E | for directed graphs v�V ||2 | E | for undirected graphs • O(E) time -O(V + E) to also list vertices unreachable from v (those still not assigned level) “LINEAR TIME” Shortest Paths: • for every vertex v, fewest edges to get from s to v is level[v] if v assigned level ∞ else (no path) • parent pointers form shortest-path tree = union of such a shortest path for each v = to find shortest path, take v, parent[v], parent[parent[v]], etc., until s (or None) ⇒ 3Lecture 13 Searching II 6.006 Spring 2008Depth-First Search (DFS): This is like exploring a maze. sFigure 4: Depth-First Search Frontier • follow path until you get stuck • backtrack along breadcrumbs until reach unexplored neighbor • recursively explore parent = {s: None}DFS-visit (V, Adj, s): for v in Adj [s]: if v not in parent: parent [v] = s DFS-visit (V, Adj, v) DFS (V, Adj) parent = { } for s in V: if s not in parent: parent [s] = None DFS-visit (V, Adj, s)}}search from start vertex s(only see stuff reachable from s)explore entire graph(could do same to extend BFS)Figure 5: Depth-First Search Algorithm 4 • � Lecture 13 Searching II 6.006 Spring 2008Example:1326back edge7548back edgeforward edgecross edgeabcdefS1S2Figure 6: Depth-First Traversal Edge Classification: back edge: to ancestorforward edge: to descendantcross edge (to another subtree)tree edges (formed by parent)nontree edgesFigure 7: Edge Classification To compute this classification, keep global time counter and store time interval during which each vertex is on recursion stack. Analysis: DFS-visit gets called with a vertex s only once (because then parent[s] set) = ⇒ time in DFS-visit = | Adj[s] |= O(E) s�V • DFS outer loop adds just O(V )= O(V + E) time (linear time)⇒ 5