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Shortest Path Algorithms Give two more shortest path trees for the fol lowing graph 6 u v 3 1 2 4 s 2 7 5 3 x 6 y Run through Dijkstras algorithmand see where there are ties which can b e arbitrarily selected There are twochoices forhowtoget to the third vertex xboth of which cost There are twochoices forhowto get to vertex v both of which cost Lessons from the Backtracking contest As predicted the sp eed dierence between the fastest programs and average program dwarfed the dierence b etween a sup ercomputer and a micro computer Algorithms have a bigger impact on performance than hardware Dierent algorithmsperform dierently on dier ent data Thus even hard problemsmaybe tractable on the kind of data you might b e interested in None of the programs could eciently handle all instances for n We will nd out why after the midtermwhenwe discuss NPcompleteness Many of the fastest programswere very short and simple KISS My bet is that many of the en hancements students built into them actually showed them down This is where proling can comein handy The fast programswere often recursive Winning Optimizations Finding a good initial solution via randomization orheuristic improvement help ed byestablishing a go o d upp er b ound to constrict search Using half the largest vertex degree as a lower b ound similarly constricted search Pruning a partial p ermutation the instant an edge was the target made the dierence in going from say to Positioning the partial p ermutation vertices sepa rated by b instead of meant signicantly earlier cutos since any edge do es the job Mirrorsymmetry can only save a factorof but perhaps more could follow from partitioning the vertices into equivalence classes by the same neigh borho o d Shortest Paths Finding the shortest path b etween twonodes in a graph arises in many dierent applications Transp ortation problems nding the cheap est waytotravel b etween twolocations Motion planning what is the most natural way foracarto on character to move ab out a simulated environment Communications problems howlookwillittake foramessage to get b etween twoplaces Which two lo cations are furthest apart ie what is the diameter of the network Shortest Paths and Sentence Disambiguation In our work on reconstructing text typ ed on an over loaded telephone keypad we had to select which of many p ossible interpretations was most likely INPUT Blank Recognition Token Token Token Token Candidate Construction Token Token Token Token give a ping of me ring hive sing Sentence Disambiguating a give ping of P P P P Pq Pq me ring hive sing GIVE ME A RING OUTPUT We constructed a graph where the vertices were the p ossible wordsp ositions in the sentence with an edge between p ossible neighb oring words Code C 1 Code C 2 Code C 3 P(W 1/C ) 2 P(W 3/C ) 1 1 P(W1 /C 2) 1 3 4 P(#/W 1) 1 P(W1 /#) 2 1 1 P(W /W ) 3 P(W2 /C 1) 2 1 P(W2 /C 3) 1 P(W2 /#) 2 P(W /C 2) # 1 2 # P(W3 /#) 1 3 P(W3 /C 1) P(W3 /C 3) 1 P(W4 /#) 1 3 P(W4 /C 1) P(W4 /C 3) The weight of each edge is a function of the probability that these two words will be next to each other in a sentence hive me would be less than give me for example The nal system worked extremely well identifying over of characters correctly based on grammatical and statistical constraints Dynamic programming the Viterbi algorithm can b e used on the sentences to obtain the sameresults by nding the shortest paths in the underlying DAG Finding Shortest Paths In an unweighted graph the cost of a path is just the number of edges on the shortest path which can be found in O n m time via breadthrst search Inaweighted graph the weight of a path b etween two vertices is the sum of the weights of the edges on a path BFS will not work on weighted graphs b ecause some times visiting more edges can lead to shorter distance ie Note that there can b e an exp onential numb er of short est paths b etween twonodessowe cannot rep ort all shortest paths eciently Note that negative cost cycles render the problem of nding the shortest path meaningless since you can always lo op around the negative cost cycle more to reduce the cost of the path Thus in our discussions we will assumethat all edge weights are p ositive Other algorithmsdeal correctly with negative cost edges Minimum spanning trees are uneected by negative cost edges Dijkstras Algorithm We can use Dijkstras algorithm to nd the shortest path b etween any twovertices s and t in G The correctness of Dijkstras algorithm is based on this If s xt is the shortest path from s to t then s x had b etter b e the shortest path from s to x orelsewecould makeitshorter The distance from s to x will b e less than than of that from s to tif all edges have p ositive weight This suggests a dynamic programminglike strategy where we store the distance from s to all nearbynodes and use them to nd the shortest path to more distant no des The shortest path from s to s ds s If all edge weights are positive the smallest edge incident to s says x denes ds x Wecanuse an arrayto store the length of the shortest path to each no de Initialize each to to start So on as weestablish the shortest path from s to a new no de xwegothrough each of its incident edges to see if there is a b etter wayfroms to other no des thru x known fsg for i to n do disti foreachedge s v distv ds v lasts while l ast t select v such that distv min disti unk now n foreachv x distxmindistx distv w v x lastv k now n known fv g INCLUDE FIGURE OF EXAMPLES OF SHORTEST PATH ALGORITHMS Complexity O n if we use adjacency lists and a Bo olean arraytomark what is known This is essentially the sameasPrims algorithm An O m lg n implementation of Dijkstras algorithm would b e faster forsparse graphs and comes from us ing a heap of the vertices ordered by distance and up dating the distance to each vertex if necessary in O lg ntimeforeachedgeout from freshly known ver tices Even b etter O n lg n mfollows from using Fib onacci heaps since they p ermit one to do a decreasekey op eration in O amortized time AllPairs Shortest Path Notice that nding the shortest path between a pair of vertices s tinworst case requires rst nding the shortest path from s to all other vertices in the graph Many applications such as nding the center or di ameter of a graph require nding the shortest path between all pairs of vertices We can run Dijkstras algorithm n times once from each possible start vertex to solve allpairs shortest path problem in O n Can wedobetter Improving the complexityisanopenquestion but there is a sup erslick dynamic programming algorithm which also runs in O n Dynamic Programming and Shortest Paths The fourstep approach to dynamic programming is Characterize the structure of an optimal solution Recursively dene the value of an optimal solution Compute this recurrence in a b ottomup fashion Extract the optimal solution from computed infor mation From the adjacency matrix wecan construct the fol lowing matrix D i j if i j and v v isnotinE i j D i j w i j if v v E i j D i j if i j This tells us the shortest path going through no inter mediate no des There are several ways to characterize the shortest path between two no des in a graph Note that the shortest path from i to j i j using at most M edges consists of the shortest path from i to k using at most M edgesW k j forsome k This suggests that we can compute allpair shortest path with an induction based on the number of edges in the optimalpath m Let di j be the length of the shortest path from i to j using at most m edges What is di j di j if i j if i j m What if weknow di j for all i j m m m di j mindi j mindi k w k j m mindi k w k j k i since w k k This gives us a recurrence which wecan evaluate in a b ottom up fashion for i to n for j to n m di j for k to n m m di j Min di k di k dk j This is an O n algorithm just likematrix multiplica tion but it only go es from m to m edges Since the shortest path between any two no des must use at most n edges unless we have negative cost cycles wemust rep eat that pro cedure n times m to nforan O n algorithm We can improve this to O n log n with the obser vation that any path using at most m edges is the function of paths using at most m edges each This is n n n just like computing a a x a So a logarithmic number of multiplications suce for exp onentiation Although this is slick observe that even O n log nis slower than running Dijkstras algorithm starting from each vertex The FloydWarshall Algorithm An alternate recurrence yields a moreecient dynamic programming formulation Number the vertices from to n k Let di j be the shortest path length from i to j us ing only vertices from k as p ossible intermediate vertices What is dj j With no intermediate vertices any path consists of at most one edge so di j w i j In general adding anew vertex k helps iapath go es through it so k di j w i j ifk k k k mindi j di k dk j ifk Although this lo oks similarto the previous recurrence it isnt The following algorithm implements it o d w for k to n for i to n for j to n k k k k di j mindi j di k dk j This obviously runs in n time which asymptoti cally is no b etter than acalls to Dijkstras algorithm However the lo ops are so tight and it is so short and simple that it runs b etter in practice by a constant factor.
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