Today s Outline. CS 362, Lecture 19. DFS and BFS. Generic Traverse. BFS and DFS Wrapup Shortest Paths. Jared Saia University of New Mexico

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1 Today Outline CS 362, Lecture 9 Jared Saia Univerity of New Mexico BFS and DFS Wrapup Shortet Path Generic Travere DFS and BFS 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; 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 3

2 Analyi Final DFS v Note BFS Note Notethat thatif ifwe weue ueadjacency lit litfor forthe thegraph, the theoverhead for forthe the for for loop loopi ionly onlya acontant per peredge (no (nomatter how we weimplement the thebag) If Ifwe weimplement the thebag baguing either tack or orqueue, each operation on onthe thebag bagtake contant time Hence the theoverall runtime i io( V + E ) = O( E ) Now Noteaume that DFS thetree edgetend are to weighted be long and kinny while BFS If tree we are implement hort and the fat bag uing a priority queue, alway extracting In addition, the the minimum BFS tree weight contain edgehortet from the path bag, from thenthe we have tartavertex verion to ofevery Prim other algorithm vertex in it connected component. extraction (here we define from the length bag of now a path take to O( E ) be the number time o Each the of edge total running the path) time i O( V + E log E ) 56 DFS Finalv Note BFS Note Nowthat aume DFSthe tree edge tend aretoweighted be long and kinny while BFS tree If weare implement hort andthe fat bag uing a priority queue, alway Inextracting addition, the minimum BFS treeweight contain edge hortet from the path bag, from thenthe we tart havevertex a verion toof every Prim other algorithm vertex in it connected component. Each(here extraction we define fromthe the length bag ofnow a path take to be O( E ) the number time o ofthe edge total inrunning the path) time i O( V + E log E ) b d b d a f a f c e c e depth-firt panning tree and breadth-firt panning tree A depth-firt panning tree and a breadth-firt panning tree of one component of the example graph, with tart vertex a. of one component of the example graph, with tart vertex a. 56

3 Searching Diconnected Graph DFS and BFS 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); 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 9 DFS in Directed Graph Acyclic graph 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 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. 0

4 Take Away Shortet Path Problem 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 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) 3 SSSP 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) 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 It not hard to ee that if the hortet path are unique, then they form a tree Shortet Path 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 hortet path are themelve hortet path diverge, then meet, then diverge again, we can modify one of the path o that the two path diverge once only. 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 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. 6 MST v SPT a b x If If a a b b c c d d v vand a x y d u are areboth hortet path, Note that the minimum panning tree and hortet path tree then then a a b b c c d u i alo a hortet path. 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) c y d u v MST v SPT 6 b c u a d Note that the minimum panning tree and hortet path tree can be different x y v For one thing there may be only one MST but there can be hortet path, multiple hortet path tree (one for every ource vertex) If a b c d v and a x y d u are both then a b c d u i alo a hortet path A minimum panning tree (left) and a hortet path tree rooted at the topmot vertex (right). A minimum panning tree (left) and a hortet path tree rooted at the topmot vertex (right). 9 9

6 Negative Weight e ll actually allow negative weight on edge he preence of a negative cycle might mean that there i We ll actually allow negative weight on edge hortet path The preence of a negative cycle might mean that there i hortet no path hortetfrom path to t exit if and only if there i at at one Apath hortetfrom path from to tot tbut exitno if and path only from if there i at to t that leat one path from to t no path from to t that uche atouche negative a negative cycle cycle the following In the following example, there i no hortet path path from to from t to t t SSSP Algorithm Each vertex v in the graph will tore t a tentative hortet path from to v 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 dit(v) i the length of the tentati and v pred(v) i the predeceor of v in th The predeceor pointer automa hortet path tree 20 2 SSSP Algorithm 20 Defn SSSP Algorithm 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 e ll now The gopredeceor over ome pointer algorithm automaticallyfor define SSSP a tentative on directed aph. hortet path tree hee algorithm will work for undirected graph with light odification particular, we mut pecifically prohibit alternating back 22 d forth acro the ame undirected negative-weight edge Initially we et: Defn dit() = 0, pred() = NULL For every vertex v, dit(v) = and pred(v) = NULL Initially we et: dit() = 0, pred() = NULL 23 For every vertex v, dit(v) =

7 Relaxation Relax 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 Relax(u,v){ dit(v) = dit(u) + w(u,v); pred(v) = u; 2 25 Correctne Claim 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 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) 26 2

8 Claim 2 Claim 3 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) 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 Generic SSSP InitSSSP 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 InitSSSP(){ dit() = 0; pred() = NULL; for all vertice v = { dit(v) = infinity; pred(v) = NULL; 30 3

9 GenericSSSP Generic SSSP 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; 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 Diktra Algorithm Dijktra Algorithm 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 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 ) 3 35

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) Four phae of Dijktra algorithm run on graph with no negative edge. 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 At each phae, the haded vertice are in the heap, and the bold vertex ha jut been canned. The bold edge decribe jut been the canned. evolving hortet path tree. The bold edge decribe the evolving hortet path tree