Robert Bryan and Marshall Dodge, Bert and I and Other Stories from Down East (1961) Michelle Shocked, Arkansas Traveler (1992)

Transcription

1 Well, ya turn left by the fire tation in the village and take the old pot road by the reervoir and...no, that won t do. Bet to continue traight on by the tar road until you reach the choolhoue and then turn left on the road to Bennett Lake until...no, that won t work either. Eat Millinocket, ya ay? Come to think of it, you can t get there from here. Hey farmer! Where doe thi road go? Been livin here all my life, it ain t gone nowhere yet. Hey farmer! How do you get to Little Rock? Liten tranger, you can t get there from here. Hey farmer! You don t know very much do you? No, but I ain t lot. Robert Bryan and Marhall Dodge, Bert and I and Other Storie from Down Eat (96) Michelle Shocked, Arkana Traveler (99) CHAPTER Shortet Path Thi i the Spring 06 reviion in the new kin; it till need ignificant reviion. In particular, everal figure need to be redrawn in OmniGraffle.. Introduction Suppoe we are given a weighted directed graph G = (V, E, w) with two pecial vertice, and we want to find the hortet path from a ource vertex to a target vertex t. That i, we want to find the directed path p tarting at and ending at t that minimize the function w(p) := w(u v). u v p For example, if I want to anwer the quetion What the fatet way to drive from my old apartment in Champaign, Illinoi to my wife old apartment in Columbu, Ohio?, I might ue a graph whoe vertice are citie, edge are road, weight are driving time, Copyright 0 Jeff Erickon. Thi work i licened under a Creative Common Licene ( Free ditribution i trongly encouraged; commercial ditribution i exprely forbidden. See for the mot recent reviion.

2 . SHORTEST PATHS i Champaign, and t i Columbu.¹ The graph i directed, becaue driving time along the ame road might be different in different direction. (At one time, there wa a peed trap on I-0 jut eat of the Indiana/Ohio border, but only for eatbound traffic.) Perhap counter to intuition, we will allow the weight on the edge to be negative. Negative edge make our live complicated, becaue the preence of a negative cycle might imply that there i no hortet path. In general, a hortet path from to t exit if and only if there i at leat one path from to t, but there i no path from to t that touche a negative cycle. If any negative cycle i reachable from and can reach t, we can alway find a horter path by going around the cycle one more time. I m abuing terminology here; by definition, a path cannot repeat vertice. Negative cycle mean that there may be no hortet walk from to t. A long a t i reach able from, there i alway a hortet path; it jut NP-hard to compute. On the other hand, if there i a hortet walk from to t, that walk i in fact a path, and therefore the hortet path, from to t. Blerg. 5 t Figure.. There i no hortet path from to t. Almot every algorithm known for olving thi problem actually olve (large portion of) 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. It not hard to ee that if hortet path are unique, then they form a tree, becaue any ubpath of a hortet path i itelf a hortet path. If there are multiple hortet path to ome vertice, we can alway chooe one hortet path to each vertex o that the union of the path i a tree. If there are hortet path to two vertice u and v that diverge, then meet, then diverge again, we can modify one of the path without changing it length o that the two path only diverge once. Although they are both optimal panning tree, hortet-path tree and minimum panning tree are very different creature. Shortet-path tree are rooted and directed; minimum panning tree are unrooted and undirected. Shortet-path tree are mot naturally defined for directed graph; only undirected graph have minimum panning tree. If edge weight are ditinct, there i only one minimum panning tree, but every Wet on Church, north on Propect, eat on I-, outh on I-65, eat on Airport Expreway, north on I-65, eat on I-0, north on Grandview, eat on 5th, north on Olentangy River, eat on Dodridge, north on High, wet on Kelo, outh on Neil. Depending on traffic. We both live in Urbana now.

3 .. Warning! b c u a d x y v Figure.. If a b c d v and a x y d u are hortet path, then a b c d u i alo a hortet path. ource vertex induce a different hortet-path tree; moreover, it i poible for every hortet path tree to ue a different et of edge from the minimum panning tree Figure.. A minimum panning tree and a hortet path tree (rooted at the top vertex) of the ame undirected graph.. Warning! Throughout thi chapter, we will explicitly conider only directed graph. All of the algorithm decribed in thi lecture alo work for undirected graph with ome minor modification, but only if negative edge are prohibited. Dealing with negative edge in undirected graph i coniderably more ubtle. We cannot imply replace every undirected edge with a pair of directed edge, becaue thi would tranform any negative edge into a hort negative cycle. Subpath of an undirected hortet path that contain a negative edge are not necearily hortet path; conequently, the et of all undirected hortet path from a ingle ource vertex may not define a tree, even if hortet path are unique. u v u v u v An undirected graph where hortet path from are unique but do not define a tree. A complete treatment of undirected graph with negative edge i beyond the cope of thi chapter, or even the entire book. I will only mention, for people who want to

4 . SHORTEST PATHS follow up via Google, that a ingle hortet path in an undirected graph with negative edge can be computed in O(V E + V log V ) time, by a reduction to maximum weighted matching.. The Only SSSP Algorithm Jut like graph traveral and minimum panning tree, there are everal different SSSP algorithm, but they are all pecial cae of the a ingle generic algorithm, firt propoed by Leter Ford in 956, and independently by George Dantzig in 95.² Each vertex v in the graph tore two value, which (inductively) decribe a tentative hortet path from to v. dit(v) i the length of the tentative hortet v path, or if there i no uch path. pred(v) i the predeceor of v in the tentative hortet v path, or Null if there i no uch vertex. In fact, the predeceor pointer automatically define a tentative hortet path tree; they play exactly the ame role a the parent pointer in our generic graph traveral algorithm. At the beginning of the algorithm, we already know that dit() = 0 and pred() = Null. For every vertex v, we initially et dit(v) = and pred(v) = Null to indicate that we do not know of any path from to v. During the execution of the algorithm, we call an edge u v tene if dit(u)+w(u v) < dit(v). If u v i tene, the tentative hortet path v i clearly incorrect, becaue the path u v i horter. Our generic algorithm repeatedly find a tene edge in the graph and relaxe it: Relax(u v): dit(v) dit(u) + w(u v) pred(v) u When there are no tene edge, the algorithm halt, and we have our deired hortet path tree. The correctne of Ford generic relaxation algorithm follow from the following erie of claim:. For every vertex v, the ditance dit(v) i either or the length of ome walk from to v. Thi claim can be proved by induction on the number of relaxation.. If the graph ha no negative cycle, then dit(v) i either or the length of ome imple path from to v. Specifically, if dit(v) i the length of a walk from to v that contain a directed cycle, that cycle mut have negative weight. Thi claim implie that if G ha no negative cycle, the relaxation algorithm eventually halt, becaue there are only a finite number of imple path in G. Specifically, Dantzig howed that the hortet path problem can be phraed a a linear programming problem, and then decribed an interpretation of hi implex method in term of the original graph. Hi decription i equivalent to Ford relaxation trategy.

5 .. Bet-Firt: Dijktra Algorithm. If no edge in G i tene, then for every vertex v, the ditance dit(v) i the length of the predeceor path pred(pred(v)) pred(v) v. Specifically, if v violate thi condition but it predeceor pred(v) doe not, the edge pred(v) v i tene.. If no edge in G i tene, then for every vertex v, the path of predeceor edge pred(pred(v)) pred(v) v i a hortet path from to v. Specifically, if v violate thi condition but it predeceor u in ome hortet path doe not, the edge u v i tene. Thi claim alo implie that if the G ha a negative cycle, then ome edge i alway tene, o the generic algorithm never halt. So far I haven t aid anything about how we detect which edge can be relaxed, or in what order we relax them. To make thi eaier, we refine the relaxation algorithm lightly, into omething cloely reembling the generic graph traveral algorithm. We maintain a bag of vertice, initially containing jut the ource vertex. Whenever we take a vertex u from the bag, we can all of it outgoing edge, looking for omething to relax. Finally, whenever we uccefully relax an edge u v, we put v into the bag. Unlike our generic graph traveral algorithm, we do not mark vertice when we viit them; the ame vertex could be viited many time, and the ame edge could be relaxed many time. InitSSSP(): dit() 0 pred() Null for all vertice v dit(v) pred(v) Null 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, different bag data tructure for the give u different algorithm. There are three obviou choice to try: a tack, a queue, and a priority queue. Unfortunately, if we ue a tack, the reulting algorithm perform Θ( V ) relaxation tep in the wort cae! (Proving thi i a good homework problem.) The other two poibilitie are much more efficient.. Bet-Firt: Dijktra Algorithm If we implement the bag uing a priority queue, where the key of a vertex v i it tentative ditance dit(v), we obtain an algorithm firt publihed in 95 by a team of reearcher at the Cae Intitute of Technology, in an annual project report for the Combat Development Department of the US Army Electronic Proving Ground. The 5

6 . SHORTEST PATHS ame algorithm wa later independently redicovered and actually publicly publihed by Edger Dijktra in 959. A nearly identical algorithm wa alo decribed by George Dantzig in 95. Dijktra algorithm, a it i univerally known³, i particularly well-behaved 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); if we ue a regular binary heap, the running time i O(E log V). Thi analyi aume that no edge ha negative weight. Dijktra algorithm (in the form I m preenting here⁴) i till correct if there are negative edge, but the wort-cae running time could be exponential. (Proving thi unfortunate fact i a good homework problem.) On the other hand, in practice, Dijktra algorithm i uually quite fat even for graph with negative edge..5 Breadth-Firt: Bellman-Ford If we replace the heap in Dijktra algorithm with a FIFO queue, we obtain an algorithm firt ketched by Shimbel in 95, decribed in more detail by Moore in 95, then independently redicovered by Woodbury and Dantzig in 95 and again by Bellman in 95. In full compliance with Stigler Law, the algorithm i almot univerally known a Bellman-Ford, becaue Bellman explicitly ued Ford formulation of relaxing edge, although a few early ource refer to Bellman-Shimbel. Bellman-Ford i efficient even if there are negative edge, and it can be ued to quickly detect the preence of negative cycle. If there are no negative edge, however, Dijktra algorithm i fater. (In fact, in practice, Dijktra algorithm i often fater even for graph with negative edge.) The eaiet way to analyze the algorithm i to break the execution into phae, by introducing an imaginary token. Before we even begin, we inert the token into the queue. The current phae end when we take the token out of the queue; we begin the I will follow thi common convention, depite the hitorical inaccuracy, partly becaue I don t think anybody want to read about the Leyzorek-Gray-Johnon-Ladew-Meaker-Petry-Seitz algorithm, and partly becaue paper that aren t actually publically publihed don t count. Mot algorithm textbook, Wikipedia, and even Dijktra original paper preent a verion of Dijktra algorithm that give incorrect reult for graph with negative edge, becaue it never viit the ame vertex more than once. Thi i alo the verion of Dijktra algorithm that I decribed a bet-firt earch in Chapter 5. Here I ve taken the liberty of correcting Dijktra mitake. Even Dijktra would agree that a correct algorithm that i ometime low (and in practice, rarely low) i better than a fat algorithm that doen t alway work. 6

7 .5. Breadth-Firt: Bellman-Ford Figure.. 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. next phae by reinerting the token into the queue. The 0th phae conit entirely of canning the ource vertex. The algorithm end when the queue contain only the token. A imple inductive argument (hint, hint) implie the following invariant for every integer i and vertex v: After i phae of the algorithm, dit(v) i at mot the length of the hortet walk from to v coniting of at mot i edge. Include the induction proof. Since a hortet path can only pa through each vertex once, either the algorithm halt before the V th phae, or the graph contain a negative cycle. In each phae, we can each vertex at mot once, o we relax each edge at mot once, o the running time of a ingle phae i O(E). Thu, the overall running time of Bellman-Ford i O(VE). Once we undertand how the phae of Bellman-Ford behave, we can implify the algorithm coniderably by producing the ame behavior on purpoe (following the ame deign pattern that we advocated for dynamic programming in Chapter ). Intead of performing a partial breadth-firt earch of the graph in each phae, we can imply can through the adjacency lit directly, relaxing every tene edge we find in the graph.

8 . SHORTEST PATHS d e b f d e b f d e b f a 0 d 6 c a e 9 b f 0 6 c a d f e 9 b 0 c a 0 c a 0 c Figure.5. Four phae of Bellman-Ford run on a directed graph with negative edge. Node are taken from the queue in the order a b c d f b a e d d a, where i the end-of-phae token. Shaded vertice are in the queue at the end of each phae. The bold edge decribe the evolving hortet path tree. Shimbel-Moore-Woodbury-Dantzig-Bellman-Ford-Broh: Relax ALL the tene edge and recure. BellmanFordSSSP() InitSSSP() repeat V time: for every edge u v if u v i tene Relax(u v) for every edge u v if u v i tene return Negative cycle! Thi i how mot textbook preent the Bellman-Ford algorithm.⁵ The O(V E) running 5 In fact, thi i eentially the formulation propoed by both Shimbel and Bellman. Bob Tarjan

9 .6. Dynamic Programming: Bellman-Ford Again time of thi formulation of the algorithm hould be obviou, but it may be le clear that the algorithm i till correct. In fact, correctne follow from exactly the ame invariant a before: After i phae of the algorithm, dit(v) i at mot the length of the hortet walk from to v coniting of at mot i edge. A before, it i traightforward to prove by induction (hint, hint) that thi invariant hold for every integer i and vertex v..6 Dynamic Programming: Bellman-Ford Again Like everything ele with Richard Bellman name on it, Bellman-Ford can alo be recat a a dynamic programming algorithm. Let dit i (v) denote the length of the hortet path v coniting of at mot i edge. It not hard to ee that thi function obey the following Bellman equation recurrence: 0 if i = 0 and v = if i = 0 and v dit i (v) = dit i (v), min min (dit otherwie i (u) + w(u v)) u v E For the moment, let aume the graph ha no negative cycle; our goal i to compute dit V (t). A traightforward dynamic-programming evaluation of thi recurrence look like thi: BellmanFordDP() dit[0, ] 0 for every vertex v dit[0, v] for i to V for every vertex v dit[i, v] dit[i, v] for every edge u v if dit[i, v] > dit[i, u] + w(u v) dit[i, v] dit[i, u] + w(u v) Now let u make two minor change to thi algorithm. Firt, we remove one level of indentation from the lat three line. Thi may change the order in which we examine edge, but the modified algorithm till compute dit i (v) for all i and v. Second, we recognized in the early 90 that Bellman-Ford i equivalent to Dijktra algorithm with a queue intead of a heap. 9

10 . SHORTEST PATHS change the indice in the lat two line from i to i. Thi change may caue the ditance dit[i, v] to approach the true hortet-path ditance more quickly than before, but the algorithm i till correct. BellmanFordDP() dit[0, ] 0 for every vertex v dit[0, v] for i to V for every vertex v dit[i, v] dit[i, v] for every edge u v if dit[i, v] > dit[i, u] + w(u v) dit[i, v] dit[i, u] + w(u v) Now notice that the iteration index i i completely redundant! We really only need to keep a one-dimenional array of ditance, which mean we don t need to can the vertice in each iteration of the main loop. BellmanFordDP() dit[] 0 for every vertex v dit[v] for i to V for every edge u v if dit[v] > dit[u] + w(u v) dit[v] dit[u] + w(u v) The reulting algorithm i almot identical to our earlier algorithm BellmanFordSSSP! The firt three line initialize the hortet path ditance, and the lat two line check whether an edge i tene, and if o, relaxe it. The only feature miing from the new algorithm i explicit maintenance of predeceor, but that eay to add. Exercie Need more!. Prove that the following invariant hold for every integer i and every vertex v: After i phae of Bellman-Ford (in either formulation), dit(v) i at mot the length of the hortet path v coniting of at mot i edge.. Let G be a directed graph with edge weight (which may be poitive, negative, or zero), and let be an arbitrary vertex of G. 0

11 Exercie (a) Suppoe every vertex v tore a number dit(v). Decribe and analyze an algorithm to determine whether dit(v) i the hortet-path ditance from to v, for every vertex v. (b) Suppoe intead that every vertex v tore a pointer pred(v) to another vertex in G. Decribe and analyze an algorithm to determine whether thee predeceor pointer define a ingle-ource hortet path tree rooted at. Do not aume that G contain no negative cycle.. A looped tree i a weighted, directed graph built from a binary tree by adding an edge from every leaf back to the root. Every edge ha a non-negative weight A looped tree. (a) How much time would Dijktra algorithm require to compute the hortet path between two vertice u and v in a looped tree with n node? (b) Decribe and analyze a fater algorithm.. Suppoe we are given a directed graph G with weighted edge and two vertice and t. (a) Decribe and analyze an algorithm to find the hortet path from to t when exactly one edge in G ha negative weight. (b) Decribe and analyze an algorithm to find the hortet path from to t when exactly k edge in G have negative weight. How doe the running time of your algorithm depend on k? 5. Suppoe we are given an undirected graph G in which every vertex ha a poitive weight. (a) Decribe and analyze an algorithm to find a panning tree of G with minimum total weight. (The total weight of a panning tree i the um of the weight of it vertice.) (b) Decribe and analyze an algorithm to find a path in G from one given vertex to another given vertex t with minimum total weight. (The total weight of a path i the um of the weight of it vertice.)

12 . SHORTEST PATHS [Hint: One of thee problem i trivial.] 6. For any edge e in any graph G, let G \ e denote the graph obtained by deleting e from G. (a) Suppoe we are given a directed graph G in which the hortet path σ from vertex to vertex t pae through every vertex of G. Decribe an algorithm to compute the hortet-path ditance from to t in G \ e, for every edge e of G, in O(E log V ) time. Your algorithm hould output a et of E hortet-path ditance, one for each edge of the input graph. You may aume that all edge weight are non-negative. [Hint: If we delete an edge of the original hortet path, how do the old and new hortet path overlap?] (b) Let and t be arbitrary vertice in an arbitrary undirected graph G. Decribe an algorithm to compute the hortet-path ditance from to t in G \ e, for every edge e of G, in O(E log V ) time. Again, you may aume that all edge weight are non-negative.. Let G = (V, E) be a connected directed graph with non-negative edge weight, let and t be vertice of G, and let H be a ubgraph of G obtained by deleting ome edge. Suppoe we want to reinert exactly one edge from G back into H, o that the hortet path from to t in the reulting graph i a hort a poible. Decribe and analyze an algorithm that chooe the bet edge to reinert, in O(E log V ) time.. When there i more than one hortet path from one node to another node t, it i often convenient to chooe a hortet path with the fewet edge; call thi the bet path from to t. Suppoe we are given a directed graph G with poitive edge weight and a ource vertex in G. Decribe and analyze an algorithm to compute bet path in G from to every other vertex. 9. (a) Prove that Ford generic hortet-path algorithm (while the graph contain a tene edge, relax it) can take exponential time in the wort cae when implemented with a tack intead of a priority queue (like Dijktra) or a queue (like Bellman- Ford). Specifically, for every poitive integer n, contruct a weighted directed n-vertex graph G n, uch that the tack-baed hortet-path algorithm call Relax Ω( n ) time when G n i the input graph. [Hint: Tower of Hanoi.] (b) Prove that Dijktra hortet-path algorithm can require exponential time in the wort cae when edge are allowed to have negative weight. Specifically, for every poitive integer n, contruct a weighted directed n-vertex graph G n, uch that Dijktra algorithm call Relax Ω( n ) time when G n i the input graph. [Hint: Thi i relatively eay if you ve already olved part (a).] 0. (a) Decribe and analyze a modification of the Bellman-Ford hortet-path algorithm that actually return a negative cycle if any uch cycle i reachable from, or a

13 Exercie hortet-path tree if there i no uch cycle. The modified algorithm hould till run in O(V E) time. (b) Decribe and analyze a modification of Bellman-Ford that compute the correct hortet path ditance from to every other vertex of the input graph, even if the graph contain negative cycle. Specifically, if any walk from to v contain a negative cycle, your algorithm hould end with dit(v) = ; otherwie, dit(v) hould contain the length of the hortet path from to v. The modified algorithm hould till run in O(V E) time. (c) Repeat part (a) and (b), but for Ford generic hortet-path algorithm. You may aume that the unmodified algorithm halt in O( V ) tep if there i no negative cycle; your modified algorithm hould alo run in O( V ) time.. Although we typically peak of the hortet path between two node, a ingle graph could contain everal minimum-length path with the ame endpoint Four (of many) equal-length hortet path. Decribe and analyze an algorithm to determine the number of hortet path from a ource vertex to a target vertex t in an arbitrary directed graph G with weighted edge. You may aume that all edge weight are poitive and that all neceary arithmetic operation can be performed in O() time. [Hint: Compute hortet path ditance from to every other vertex. Throw away all edge that cannot be part of a hortet path from to another vertex. What left?]. You jut dicovered your bet friend from elementary chool on Twitbook. You both want to meet a oon a poible, but you live in two different cite that are far apart. To minimize travel time, you agree to meet at an intermediate city, and then you imultaneouly hop in your car and tart driving toward each other. But where exactly hould you meet? You are given a weighted graph G = (V, E), where the vertice V repreent citie and the edge E repreent road that directly connect citie. Each edge e ha a weight w(e) equal to the time required to travel between the two citie. You are alo given a vertex p, repreenting your tarting location, and a vertex q, repreenting your friend tarting location. Decribe and analyze an algorithm to find the target vertex t that allow you and your friend to meet a quickly a poible.

14 . SHORTEST PATHS. After moving to a new city, you decide to chooe a walking route from your home to your new office. To get a good daily workout, your route mut conit of an uphill path (for exercie) followed by a downhill path (to cool down), or jut an uphill path, or jut a downhill path. (You ll walk the ame path home, o you ll get exercie one way or the other.) But you alo want the hortet path that atifie thee condition, o that you actually get to work on time. Your input conit of an undirected graph G, whoe vertice repreent interection and whoe edge repreent road egment, along with a tart vertex and a target vertex t. Every vertex v ha a value h(v), which i the height of that interection above ea level, and each edge uv ha a value l(uv), which i the length of that road egment. (a) Decribe and analyze an algorithm to find the hortet uphill downhill walk from to t. Aume all vertex height are ditinct. (b) Now uppoe we allow ome or all vertex height to be equal. Decribe and analyze an algorithm to find the hortet uphill downhill walk from to t; you may ue flat edge in both the uphill and downhill portion of your walk. (c) Finally, uppoe you dicover that there i no path from to t with the tructure you want. Decribe an algorithm to find a path from to t that alternate between uphill and downhill ubpath a few time a poible, and ha minimum length among all uch path.. After a grueling algorithm midterm, you decide to take the bu home. Since you planned ahead, you have a chedule that lit the time and location of every top of every bu in Champaign-Urbana. Unfortunately, there in t a ingle bu that viit both your exam building and your home; you mut tranfer between bu line at leat once. Decribe and analyze an algorithm to determine the equence of bu ride that will get you home a early a poible, auming there are b different bu line, and each bu top n time per day. Your goal i to minimize your arrival time, not the time you actually pend traveling. Aume that the bue run exactly on chedule, that you have an accurate watch, and that you are too tired to walk between bu top. 5. After graduating you accept a job with Aerophobe- -U, the leading traveling agency for people who hate to fly. Your job i to build a ytem to help cutomer plan airplane trip from one city to another. All of your cutomer are afraid of flying (and by extenion, airport), o any trip you plan need to be a hort a poible. You know all the departure and arrival time of all the flight on the planet. Suppoe one of your cutomer want to fly from city X to city Y. Decribe an algorithm to find a equence of flight that minimize the total time in tranit the length of time from the initial departure to the final arrival, including time at R

15 Exercie intermediate airport waiting for connecting flight. [Hint: Modify the input data and apply Dijktra algorithm.] 6. Mulder and Scully have computed, for every road in the United State, the exact probability that omeone driving on that road won t be abducted by alien. Agent Mulder need to drive from Langley, Virginia to Area 5, Nevada. What route hould he take o that he ha the leat chance of being abducted? More formally, you are given a directed graph G = (V, E), where every edge e ha an independent afety probability p(e). The afety of a path i the product of the afety probabilitie of it edge. Deign and analyze an algorithm to determine the afet path from a given tart vertex to a given target vertex t. La Vega, NV Memphi, TN Langley, VA Area 5, AZ For example, with the probabilitie hown above, if Mulder trie to drive directly from Langley to Area 5, he ha a 50% chance of getting there without being abducted. If he top in Memphi, he ha a = 6% chance of arriving afely. If he top firt in Memphi and then in La Vega, he ha a = 96.5% chance of being abducted! (That how they got Elvi, you know.) Although thi example i a dag, your algorithm mut handle arbitrary directed graph.. On an overnight camping trip in Sunnydale National Park, you are woken from a retle leep by a cream. A you crawl out of your tent to invetigate, a terrified park ranger run out of the wood, covered in blood and clutching a crumpled piece of paper to hi chet. A he reache your tent, he gap, Get out... while... you..., thrut the paper into your hand, and fall to the ground. Checking hi pule, you dicover that the ranger i tone dead. You look down at the paper and recognize a map of the park, drawn a an undirected graph, where vertice repreent landmark in the park, and edge repreent trail between thoe landmark. (Trail tart and end at landmark and do not cro.) You recognize one of the vertice a your current location; everal vertice on the boundary of the map are labeled EXIT. On cloer examination, you notice that omeone (perhap the poor dead park ranger) ha written a real number between 0 and next to each vertex and each edge. A crawled note on the back of the map indicate that a number next to an edge i the probability of encountering a vampire along the correponding trail, 5

16 . SHORTEST PATHS and a number next to a vertex i the probability of encountering a vampire at the correponding landmark. (Vampire can t tand each other company, o you ll never ee more than one vampire on the ame trail or at the ame landmark.) The note warn you that tepping off the marked trail will reult in a low and painful death. You glance down at the corpe at your feet. Ye, hi death certainly looked painful. Wait, wa that a twitch? Are hi teeth getting longer? After driving a tent take through the undead ranger heart, you wiely decide to leave the park immediately. Decribe and analyze an efficient algorithm to find a path from your current location to an arbitrary EXIT node, uch that the total expected number of vampire encountered along the path i a mall a poible. Be ure to account for both the vertex probabilitie and the edge probabilitie! Randall Munroe, xkcd ( Reproduced under a Creative Common Attribution-NonCommercial.5 Licene 6 Copyright 0 Jeff Erickon. Thi work i licened under a Creative Common Licene ( Free ditribution i trongly encouraged; commercial ditribution i exprely forbidden. See for the mot recent reviion.