Generic Traverse. CS 362, Lecture 19. DFS and BFS. Today s Outline

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1 Generic Travere CS 62, Lecture 9 Jared Saia Univerity of New Mexico Travere(){ put (nil,) in bag; while (the bag i not empty){ take ome edge (p,v) from the bag if (v i unmarked) mark v; parent(v) = p; for each edge (v,w) incident to v{ put (v,w) into the bag; 2 Today Outline DFS and BFS BFS and DFS Wrapup Shortet Path If we implement the bag by uing a tack, we have Depth Firt Search If we implement the bag by uing a queue, we have Breadth Firt Search

2 Analyi Final Note Note that if we ue adjacency lit for the graph, the overhead for the for loop i only a contant per edge (no matter how we implement the bag) If we implement the bag uing either tack or queue, each operation on the bag take contant time Hence the overall runtime i O( V + E ) = O( E ) Now aume the edge are weighted If we implement the bag uing a priority queue, alway extracting the minimum weight edge from the bag, then we have a verion of Prim algorithm Each extraction from the bag now take O( E ) time o the total running time i O( V + E log E ) 6 DFS v BFS Example Note that DFS tree tend to be long and kinny while BFS tree are hort and fat In addition, the BFS tree contain hortet path from the tart vertex to every other vertex in it connected component. (here we define the length of a path to be the number of edge in the path) a b c d e f A depth-firt panning tree and a breadth-firt panning tree of one component of the example graph, with tart vertex a. a b c d e f 5

3 Searching Diconnected Graph DFS in Directed Graph If the graph i diconnected, then Travere only viit node in the connected component of the tart vertex. If we want to viit all vertice, we can ue the following wrapper around Travere TravereAll(){ for all vertice v{ if (v i unmarked){ Travere(v); Tree edge are edge that are in the tree itelf Back edge are thoe edge (u, v) connecting a vertex u to an ancetor v in the DFS tree Forward edge are nontree edge (u, v) that connect a vertex u to a decendant in a DFS tree Cro edge are all other edge. They go between two vertice where neither vertex i a decendant of the other 0 DFS and BFS Acyclic graph Note that we can do DFS and BFS equally well on undirected and directed graph If the graph i undirected, there are two type of edge in G: edge that are in the DFS or BFS tree and edge that are not in thi tree If the graph i directed, there are everal type of edge Ueful Fact: A directed graph G i acyclic if and only if a DFS of G yeild no back edge Challenge: Try to prove thi fact. 9

4 Take Away Example BFS and DFS are two ueful algorithm for exploring graph Each of thee algorithm i an intantiation of the Travere algorithm. BFS ue a queue to hold the edge and DFS ue a tack Each of thee algorithm contruct a panning tree of all the node which are reachable from the tart node Imagine we want to find the fatet way to drive from Albuquerque,NM to Seattle,WA We might ue a graph whoe vertice are citie, edge are road, weight are driving time, i Albuquerque and t i Seattle The graph i directed ince driving time along the ame road might be different in different direction (e.g. becaue of contruction, peed trap, etc) Shortet Path Problem SSSP Another intereting problem for graph i that of finding hortet path Aume we are given a weighted directed graph G = (V, E) with two pecial vertice, a ource and a target t We want to find the hortet directed path from to t In other word, we want to find the path p tarting at and ending at t minimizing the function w(p) = e p w(e) Every algorithm known for olving thi problem actually olve the following more general ingle ource hortet path or SSSP problem: Find the hortet path from the ource vertex to every other vertex in the graph Thi problem i uually olved by finding a hortet path tree rooted at that contain all the deired hortet path 5

5 Shortet Path Tree MST v SPT It not hard to ee that if the hortet path are unique, then they form a tree To prove thi, we need only oberve that the ub-path of hortet path are themelve hortet path If there are multiple hotet path to the ame vertex, we can alway chooe jut one of them, o that the union of the path i a tree If there are hortet path to two vertice u and v which diverge, then meet, then diverge again, we can modify one of the path o that the two path diverge once only. Note that the minimum panning tree and hortet path tree can be different For one thing there may be only one MST but there can be multiple hortet path tree (one for every ource vertex) 6 Example Example a b x If a b c d v and a x y d u are both hortet path, then a b c d u i alo a hortet path. c y d u v A minimum panning tree (left) and a hortet path tree rooted at the topmot vertex (right). 9

6 Negative Weight SSSP Algorithm We ll actually allow negative weight on edge The preence of a negative cycle might mean that there i no hortet path A hortet path from to t exit if and only if there i at leat one path from to t but no path from to t that touche a negative cycle In the following example, there i no hortet path from to t 5 2 t Each vertex v in the graph will tore two value which decribe a tentative hortet path from to v dit(v) i the length of the tentative hortet path between and v pred(v) i the predeceor of v in thi tentative hortet path The predeceor pointer automatically define a tentative hortet path tree SSSP Algorithm Defn We ll now go over ome algorithm for SSSP on directed graph. Thee algorithm will work for undirected graph with light modification In particular, we mut pecifically prohibit alternating back and forth acro the ame undirected negative-weight edge Like for graph traveral, all the SSSP algorithm will be pecial cae of a ingle generic algorithm Initially we et: dit() = 0, pred() = NULL For every vertex v, dit(v) = and pred(v) = NULL 2 2

7 Relaxation Correctne We call an edge (u, v) tene if dit(u) + w(u, v) < dit(v) If (u, v) i tene, then the tentative hortet path from to v i incorrect ince the path to u and then (u, v) i horter Our generic algorithm repeatedly find a tene edge in the graph and relaxe it If there are no tene edge, our algorithm i finihed and we have our deired hortet path tree The correctne of the relaxation algorithm follow directly from three imple claim The run time of the algorithm will depend on the way that we make choice about which edge to relax 2 26 Relax Claim Relax(u,v){ dit(v) = dit(u) + w(u,v); pred(v) = u; If dit(v), then dit(v) i the total weight of the predeceor chain ending at v: pred(pred(v)) pred(v) v. Thi i eay to prove by induction on the number of edge in the path from to v. (left a an exercie) 25 2

8 Claim 2 Generic SSSP If the algorithm halt, then dit(v) w( v) for any path v. Thi i eay to prove by induction on the number of edge in the path v. (which you will do in the hw) We haven t yet aid how to detect which edge can be relaxed or what order to relax them in The following Generic SSSP algorithm anwer thee quetion We will maintain a bag of vertice initially containing jut the ource vertex Whenever we take a vertex u out of the bag, we can all of it outgoing edge, looking for omething to relax Whenever we uccefully relax an edge (u, v), we put v in the bag 2 0 Claim InitSSSP The algorithm halt if and only if there i no negative cycle reachable from. The only if direction i eay if there i a reachable negative cycle, then after the firt edge in the cycle i relaxed, the cycle alway ha at leat one tene edge. The if direction follow from the fact that every relaxation tep reduce either the number of vertice with dit(v) = by or reduce the um of the finite hortet path length by ome poitive amount. InitSSSP(){ dit() = 0; pred() = NULL; for all vertice v!= { dit(v) = infinity; pred(v) = NULL; 29

9 GenericSSSP Diktra Algorithm GenericSSSP(){ InitSSSP(); put in the bag; while the bag i not empty{ take u from the bag; for all edge (u,v){ if (u,v) i tene{ Relax(u,v); put v in the bag; If we implement the bag a a heap, where the key of a vertex v i dit(v), we obtain Dijktra algorithm Dijktra algorithm doe particularly well if the graph ha no negative-weight edge In thi cae, it not hard to how (by induction, of coure) that the vertice are canned in increaing order of their hortet-path ditance from It follow that each vertex i canned at mot once, and thu that each edge i relaxed at mot once 2 Generic SSSP Dijktra Algorithm Jut a with graph traveral, uing different data tructure for the bag give u different algorithm Some obviou choice are: a tack, a queue and a heap Unfortunately if we ue a tack, we need to perform Θ(2 E ) relaxation tep in the wort cae (an exercie for the diligent tudent) The other poibilitie are more efficient Since the key of each vertex in the heap i it tentative ditance from, the algorithm perform a DecreaeKey operation every time an edge i relaxed Thu the algorithm perform at mot E DecreaeKey Similarly, there are at mot V Inert and ExtractMin operation Thu if we tore the vertice in a Fibonacci heap, the total running time of Dijktra algorithm i O( E + V log V ) 5

10 Negative Edge Thi analyi aume that no edge ha negative weight The algorithm given here i till correct if there are negative weight edge but the wort-cae run time could be exponential The algorithm in our text book give incorrect reult for graph with negative edge (which they make clear) 6 Example Four phae of Dijktra algorithm run on a graph with no negative edge. At each phae, the haded vertice are in the heap, and the bold vertex ha jut been canned. The bold edge decribe the evolving hortet path tree.