A vertex u is reachable from vertex v iff there is a path from v to u.

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Presentation transcript:

A vertex u is reachable from vertex v iff there is a path from v to u. Graph Search Methods A vertex u is reachable from vertex v iff there is a path from v to u. 2 3 8 10 1 4 5 9 11 6 7

Graph Search Methods A search method starts at a given vertex v and visits/labels/marks every vertex that is reachable from v. 2 3 8 10 1 4 5 9 11 6 7 Visit, mark, and label used as synonyms.

Graph Search Methods Many graph problems solved using a search method. Path from one vertex to another. Is the graph connected? Find a spanning tree. Etc. Commonly used search methods: Breadth-first search. Depth-first search.

Breadth-First Search Visit start vertex and put into a FIFO queue. Repeatedly remove a vertex from the queue, visit its unvisited adjacent vertices, put newly visited vertices into the queue.

Breadth-First Search Example 2 3 8 10 1 4 5 9 11 6 7 Start search at vertex 1.

Breadth-First Search Example FIFO Queue 1 2 3 8 1 1 4 5 9 10 6 7 11 Visit/mark/label start vertex and put in a FIFO queue.

Breadth-First Search Example FIFO Queue 1 2 3 8 1 1 4 5 9 10 6 7 11 Remove 1 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 2 4 8 1 1 4 4 5 9 10 6 7 11 Remove 1 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 2 4 8 1 1 4 4 5 9 10 6 7 11 Remove 2 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 4 5 3 6 8 1 1 4 4 5 5 9 10 6 6 7 11 Remove 2 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 4 5 3 6 8 1 1 4 4 5 5 9 10 6 6 7 11 Remove 4 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 5 3 6 8 1 1 4 4 5 5 9 10 6 6 7 11 Remove 4 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 5 3 6 8 1 1 4 4 5 5 9 10 6 6 7 11 Remove 5 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 3 6 9 7 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11 Remove 5 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 3 6 9 7 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11 Remove 3 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 6 9 7 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11 Remove 3 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 6 9 7 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11 Remove 6 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 9 7 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11 Remove 6 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 9 7 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11 Remove 9 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 7 8 8 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11 Remove 9 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 7 8 8 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11 Remove 7 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 8 8 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11 Remove 7 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 8 8 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11 Remove 8 from Q; visit adjacent unvisited vertices; put in Q.

Breadth-First Search Example 2 FIFO Queue 2 3 3 8 8 1 1 4 4 5 5 9 9 10 6 6 7 7 11 Queue is empty. Search terminates.

Breadth-First Search Property All vertices reachable from the start vertex (including the start vertex) are visited.

Time Complexity Each visited vertex is put on (and so removed from) the queue exactly once. When a vertex is removed from the queue, we examine its adjacent vertices. O(n) if adjacency matrix used O(vertex degree) if adjacency lists used Total time O(mn), where m is number of vertices in the component that is searched (adjacency matrix)

Time Complexity O(n + sum of component vertex degrees) (adj. lists) = O(n + number of edges in component)

Path From Vertex v To Vertex u Start a breadth-first search at vertex v. Terminate when vertex u is visited or when Q becomes empty (whichever occurs first). Time O(n2) when adjacency matrix used O(n+e) when adjacency lists used (e is number of edges)

Is The Graph Connected? Start a breadth-first search at any vertex of the graph. Graph is connected iff all n vertices get visited. Time O(n2) when adjacency matrix used O(n+e) when adjacency lists used (e is number of edges)

Connected Components Start a breadth-first search at any as yet unvisited vertex of the graph. Newly visited vertices (plus edges between them) define a component. Repeat until all vertices are visited.

Connected Components 2 3 8 10 1 4 5 9 11 6 7

Time Complexity O(n2) when adjacency matrix used O(n+e) when adjacency lists used (e is number of edges)

Spanning Tree Breadth-first search from vertex 1. 2 2 3 3 8 8 1 1 4 4 5 5 9 9 Keep track of edges used to rreach new vertices. These edges form a spanning tree if the graph is connected. 6 6 7 7 Breadth-first search from vertex 1. Breadth-first spanning tree.

Spanning Tree Start a breadth-first search at any vertex of the graph. If graph is connected, the n-1 edges used to get to unvisited vertices define a spanning tree (breadth-first spanning tree). Time O(n2) when adjacency matrix used O(n+e) when adjacency lists used (e is number of edges)

Depth-First Search depthFirstSearch(v) { Label vertex v as reached. for (each unreached vertex u adjacenct from v) depthFirstSearch(u); } Note that vertices adjacent from v are examined one at a time. As soon as an unreached adjacent vertex u is found, a depthFirstSearch(u) is done. Remaining vertices adjacent from v are examined after depthFirstSearch(u) completes.

Depth-First Search Example 2 3 8 10 1 4 5 9 11 6 7 2 1 Start search at vertex 1. Label vertex 1 and do a depth first search from either 2 or 4. Suppose that vertex 2 is selected.

Depth-First Search Example 2 2 2 3 8 1 1 4 5 5 9 10 6 7 11 Label vertex 2 and do a depth first search from either 3, 5, or 6. Suppose that vertex 5 is selected.

Depth-First Search Example 2 2 2 3 8 1 1 4 5 5 5 9 9 10 6 7 11 Label vertex 5 and do a depth first search from either 3, 7, or 9. Suppose that vertex 9 is selected.

Depth-First Search Example 2 2 2 3 8 8 1 1 4 5 5 5 9 9 9 10 6 7 11 Label vertex 9 and do a depth first search from either 6 or 8. Suppose that vertex 8 is selected.

Depth-First Search Example 2 2 2 3 8 8 8 1 1 4 5 5 5 9 9 9 10 6 6 7 11 Label vertex 8 and return to vertex 9. From vertex 9 do a dfs(6).

Depth-First Search Example 2 2 2 3 8 8 8 1 1 4 4 5 5 5 9 9 9 10 6 6 6 7 11 Label vertex 6 and do a depth first search from either 4 or 7. Suppose that vertex 4 is selected.

Depth-First Search Example 2 2 2 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 11 Label vertex 4 and return to 6. From vertex 6 do a dfs(7).

Depth-First Search Example 2 2 2 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 7 11 Label vertex 7 and return to 6. Return to 9.

Depth-First Search Example 2 2 2 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 7 11 Return to 5.

Depth-First Search Example 2 2 2 3 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 7 11 Do a dfs(3).

Depth-First Search Example 2 2 2 3 3 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 7 11 Label 3 and return to 5. Return to 2.

Depth-First Search Example 2 2 2 3 3 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 7 11 Return to 1.

Depth-First Search Example 2 2 2 3 3 3 8 8 8 1 1 4 4 4 5 5 5 9 9 9 10 6 6 6 7 7 7 11 Return to invoking method.

Depth-First Search Properties Same complexity as BFS. Same properties with respect to path finding, connected components, and spanning trees. Edges used to reach unlabeled vertices define a depth-first spanning tree when the graph is connected. There are problems for which bfs is better than dfs and vice versa.

Shortest Path Problems Directed weighted graph. Path length is sum of weights of edges on path. The vertex at which the path begins is the source vertex. The vertex at which the path ends is the destination vertex.

Example A path from 1 to 7. Path length is 14. 8 2 3 1 16 7 6 4 10 4 5

Example Another path from 1 to 7. Path length is 11. 8 2 3 1 16 7 6 4 5 10 4 2 4 7 5 3 14 Another path from 1 to 7. Path length is 11.

Shortest Path Problems Single source single destination. Single source all destinations. All pairs (every vertex is a source and destination).

Single Source Single Destination Possible greedy algorithm: Leave source vertex using cheapest/shortest edge. Leave new vertex using cheapest edge subject to the constraint that a new vertex is reached. Continue until destination is reached.