Applications of BFS and DFS

Size: px

Start display at page:

Download "Applications of BFS and DFS"

Ashlynn Randall
5 years ago
Views:

1 pplications of S and S S all // : PM Some pplications of S and S S o find the shortest path from a vertex s to a vertex v in an unweighted graph o find the length of such a path o construct a S tree/forest from a graph o find out if a strongly connected directed graph contains cycles S o find a path from a vertex s to a vertex v. o find the length of such a path. o construct a S tree/forest from a graph.

2 inding Shortest Paths Using S inding Shortest Paths he S code we have seen find outs if there exists a path from a vertex s to a vertex v prints the vertices of a graph (connected/strongly connected). What if we want to find the shortest path from s to a vertex v (or to every other vertex)? the length of the shortest path from s to a vertex v? In addition to array flag[ ], use an array named prev[ ], one element per vertex. prev[w] = v means that vertex w was visited right after v

3 xample djacency List Visited able (/) prev[ ] prev[ ] now can be traced backward to report the path! S and inding Shortest Path initialize all pred[v] to already got shortest path from s to v record where you came from

4 Shortest Path lgorithm for each w adjacent to v if flag[w] = false { flag[w] = true; prev[w] = v; // visited w right after v enqueue(w); } o print the shortest path from s to a vertex u, start with prev[u] and backtrack until reaching the s. Running time of backtracking =? o find the length of the shortest path from s to u, start with prev[u], backtrack and increment a counter until reaching s. Running time =? xample Q = { } Initialize Q to be empty djacency List Visited able (/) Initialize visited table (all false) Initialize prev[ ] to prev[ ]

5 djacency List Visited able (/) prev Q = { } Place on the queue. lag that has been visited. Q = {} {,, } Neighbors djacency List equeue. Place all unvisited neighbors of on the queue Visited able (/) Mark neighbors as visited. prev Record in prev that we came from.

6 djacency List Visited able (/) Mark Q = {,, } {,,, } Neighbors prev new visited Neighbors. Record in prev equeue. that we came Place all unvisited neighbors of on the queue. from. Notice that is not placed on the queue again, it has been visited! Q = {,,, } {,,,, } Neighbors djacency List equeue. Place all unvisited neighbors of on the queue. Only nodes and haven t been visited yet. Visited able (/) prev Mark new visited Neighbors. Record in prev that we came from.

7 djacency List Visited able (/) Neighbors prev Q = {,,,, } {,,, } equeue. has no unvisited neighbors! djacency List Visited able (/) Neighbors prev Q = {,,, } {,, } equeue. has no unvisited neighbors!

8 djacency List Visited able (/) Neighbors prev Q = {,, } {, } equeue. has no unvisited neighbors! djacency List Visited able (/) Q = {, } {, } equeue. place neighbor on the queue. Neighbors prev Mark new visited Vertex. Record in prev that we came from.

9 Q = {, } {, } Neighbors equeue. place neighbor on the queue. djacency List Visited able (/) prev Mark new visited Vertex. Record in prev that we came from. djacency List Visited able (/) Neighbors prev Q = {, } { } equeue. no unvisited neighbors of.

10 djacency List Visited able (/) Neighbors prev Q = { } { } equeue. no unvisited neighbors of. S inished djacency List Visited able (/) prev[ ] Q = { } SOP!!! Q is empty!!! prev[ ] now can be traced backward to report the path!

11 xample of Path Reporting nodes visited from ry some examples; report path from s to v: Path() Path() Path() Path Reporting Given a vertex w, report the shortest path from s to w currentv = w; while (prev[currentv] ) { output currentv; // or add to a list currentv = prev[currentv]; } output s; // or add to a list he above code prints the path in reverse order.

12 Path Reporting () o output the path in the right order, Print the list in reverse order. Use a stack instead of a list. Use a recursive method (implicit use of a stack). printpath (w) { if (prev[w] ) printpath (prev[w]); output w; } inding Shortest Path Length o find the length of the shortest path from s to u, start with prev[u], backtrack and increment a counter until reaching the s. Running time of backtracking =? ollowing is a faster way to find the length of the shortest path from s to u (at the cost of using more space) llocate an array d[ ], one element per vertex. When S algorithm ends, d[u] records the length of the shortest path from s to u. Running time of finding path length =?

13 Recording the Shortest istance d[v] = ; d[s] = ; d[v] stores shortest distance from s to v d[w] = d[v] + ; omputing Spanning rees

14 rees ree: a connected graph without cycles. Given a connected graph, remove the cycles a tree. he paths found by S(s) form a rooted tree (called a spanning tree), with the starting vertex as the root of the tree. S tree for vertex s = What would a levelorder traversal of the tree tell you? omputing a S ree Use S on a vertex S( v ) with array prev[ ] he paths from s to the other vertices form a tree

15 omputing Spanning orests omputing a S orest forest is a set of trees. connected graph gives a tree (which h is itself a forest). connected component also gives us a tree. graph with k components gives a forest of k trees.

16 xample P graph with components L N M O R Q P s G K H xample of a orest P We removed the cycles from the previous graph. L N M O R Q P s forest with trees G K H

17 omputing a S orest Use S method on a graph Search( G ), which calls S( v ) Use S( v ) with array prev[ ]. he paths originating from v form a tree. Search( G ) examines all the components to compute all the trees in the forest. esting for ycles

18 esting for ycles Method isyclic(v) returns true if a directed graph (with only one component) contains a cycle, and returns false otherwise. return false; else return true; inding ycles o output the cycle just detected, use info in prev[ ]. NO: he code above applies only to directed graphs. Homework: xplain why that code does not work for undirected graphs.

19 inding ycles in Undirected Graphs o detect/find cycles in an undirected graph, we need to classify the edges into categories during program execution: unvisited edge: never visited. discovery edge: visited for the very first time. cross edge: edge that forms a cycle. ode fragment., p.. When the S algorithm terminates, the discovery edges form a spanning tree. If there exists a cross edge, the undirected graph contains a cycle. S lgorithm (in textbook) he algorithm uses a mechanism for setting and getting labels of vertices and edges lgorithm S(G) Input graph G Output labeling of the edges and partition of the vertices of G for all u G.vertices() setlabel(u, UNXPLOR) for all e G.edges() setlabel(e, UNXPLOR) for all v G.vertices() if getlabel(v) UNXPLOR S(G, v) readthirst Search lgorithm S(G, s) L new empty sequence L.insertLast(s) setlabel(s, VISI) i while L i.ismpty() L i new empty sequence for all v L i.elements() for all e G.incidentdges(v) if getlabel(e) UNXPLOR w opposite(v,e) if getlabel(w) UNXPLOR setlabel(e, ISOVRY) setlabel(w, VISI) L i.insertlast(w) else setlabel(e, ROSS) i i

20 xample unexplored vertex L visited vertex unexplored edge L discovery edge cross edge L L L L readthirst Search xample () L L L L L L L L L L L readthirst Search

21 xample () L L L L L L L L L readthirst Search S pplications

22 pplications of S Is there a path from s to a vertex v? Is an undirected graph connected? Is a directed graph strongly connected? o output the contents (e.g., the vertices) of a graph o find the connected components of a graph o find out if a graph contains cycles and report cycles. o construct a S tree/forest from a graph S lgorithm lag all vertices as not visited lag yourself as visited or unvisited neighbors, call RS(w) recursively We can also record the paths using prev[ ]. Where do we insert the code for prev[ ]?

aptures the structure of the recursive calls when we visit a neighbor w of v, we add w as

23 S Path racking djacency List Visited able (/) S find out path too Pred ry some examples. Path() > Path() > Path() > S ree Resulting Stree. Notice it is much deeper than the S tree. aptures the structure of the recursive calls when we visit a neighbor w of v, we add w as child of v whenever S returns from a vertex v, we climb up in the tree from v to its parent

24 inding ycles Using S Similar to using S. or undirected graphs, classify the edges into categories during program execution: unvisited edge, discovery edge, and back (cross) edge. ode ragment., p.. If there exists a back edge, the undirected graph contains a cycle. pplications S vs. S What can S do and S can t? inding shortest paths (in unweighted graphs) What can S do and S can t? inding out if a connected undirected graph is biconnected connected undirected graph is biconnected if there are no vertices whose removal disconnects the rest of the graph

25 S vs. S pplications Spanning forest, connected components, paths, cycles Shortest paths iconnected components S S L L L S S inal xam Review: ecember. inal xam: ecember, PM PM Material: ll lectures notes and corresponding sections in the textbook. ssignments and. Homework and review questions.

Breadth-First Search L 1 L Goodrich, Tamassia, Goldwasser Breadth-First Search 1

Breadth-First Search L 1 L Goodrich, Tamassia, Goldwasser Breadth-First Search 1 readth-irst Search 2013 Goodrich, Tamassia, Goldwasser readth-irst Search 1 readth-irst Search readth-first search (S) is a general technique for traversing a graph S traversal of a graph G Visits all