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Chapter 9: Graphs Breadth-First and Depth-First Search Mark Allen Weiss: Data Structures and Algorithm Analysis in Java Lydia Sinapova, Simpson College

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2 Breadth-First and Depth-First Search BFS Basic Algorithm BFS Complexity DFS Algorithm DFS Implementation Relation between BFS and DFS

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3 BFS – Basic Idea Given a graph with N vertices and a selected vertex A: for (i = 1; there are unvisited vertices ; i++) Visit all unvisited vertices at distance i (i is the length of the shortest path between A and currently processed vertices) Queue-based implementation

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4 BFS – Algorithm BFS algorithm 1. Store source vertex S in a queue and mark as processed 2. while queue is not empty Read vertex v from the queue for all neighbors w: If w is not processed Mark as processed Append in the queue Record the parent of w to be v (necessary only if we need the shortest path tree)

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5 Example 1 4 235 6 Adjacency lists 1: 2, 3, 4 2: 1, 3, 6 3: 1, 2, 4, 5, 6 4: 1, 3, 5 5: 3, 4 6: 2, 3 Breadth-first traversal: 1, 2, 3, 4, 6, 5 1: starting node 2, 3, 4 : adjacent to 1 (at distance 1 from node 1) 6 : unvisited adjacent to node 2. 5 : unvisited, adjacent to node 3 The order depends on the order of the nodes in the adjacency lists

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6 Shortest Path Tree Consider the distance table: Nodes ABCDE Distance to A01121 Parent0AACA The table defines the shortest path tree, rooted at A.

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7 BFS – Complexity Step 1 : read a node from the queue O(V) times. Step 2 : examine all neighbors, i.e. we examine all edges of the currently read node. Not oriented graph: 2*E edges to examine Hence the complexity of BFS is O(V + 2*E)

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8 Depth-First Search Procedure dfs(s) mark all vertices in the graph as not reached invoke scan(s) Procedure scan(s) mark and visit s for each neighbor w of s if the neighbor is not reached invoke scan(w)

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9 Depth-First Search with Stack Initialization: mark all vertices as unvisited, visit(s) while the stack is not empty: pop (v,w) if w is not visited add (v,w) to tree T visit(w) Procedure visit(v) mark v as visited for each edge (v,w) push (v,w) in the stack

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10 Recursive DFS DepthFirst(Vertex v) visit(v); for each neighbor w of v if (w is not visited) add edge (v,w) to tree T DepthFirst(w) Visit(v) mark v as visited

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11 Example 1 4 235 6 Adjacency lists 1: 2, 3, 4 2: 6, 3, 1 3: 1, 2, 6, 5, 4 4: 1, 3, 5 5: 3, 4 6: 2, 3 Depth first traversal: 1, 2, 6, 3, 5, 4 the particular order is dependent on the order of nodes in the adjacency lists

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12 BFS and DFS bfs(G) list L = empty tree T = empty choose a starting vertex x visit(x) while(L nonempty) remove edge (v,w) from beginning of L if w not visited add (v,w) to T visit(w) dfs(G) list L = empty tree T = empty choose a starting vertex x visit(x) while(L nonempty) remove edge (v,w) from end of L if w not visited add (v,w) to T visit(w) Visit ( vertex v ) mark v as visited for each edge (v,w) add edge (v,w) to end of L

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13 Applications of DFS Trees: preorder traversal Graphs: Connectivity Biconnectivity – articulation points Euler graphs

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