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algorithms and data structures presented in the text Implement only one change at a time You should run your tests on a reasonably large and somewhat sparse random graph Then do the following a When an edge is read, determine whether it is already in the graph b Implement the "dictionary" by using a sequential scan of the vertex table c Implement the queue by using the algorithm in Exercise 712 (which should affect the unweighted shortest-path algorithm) d In the unweighted shortest-path algorithm, implement the search for the minimum-cost vertex as a sequential scan of the vertex table e Implement the priority queue by using the algorithm in Exercise 714 (which should affect the weighted shortest-path algorithm) f Implement the priority queue by using the algorithm in Exercise 715 (which should affect the weighted shortest-path algorithm) g In the weighted shortest-path algorithm, implement the search for the minimum-cost vertex as a sequential scan of the vertex table h In the acyclic shortest-path algorithm, implement the search for a vertex with indegree 0 as a sequential scan of the vertex table i Implement any of the graph algorithms by using an adjacency matrix instead of adjacency lists Programming Projects

1518 A directed graph is strongly connected if there is a path from every vertex to every other vertex Do the following a Pick any vertex S Show that, if the graph is strongly connected, a shortest-path algorithm will declare that all nodes are reachable from S b Show that, if the graph is strongly connected and then the directions of all edges are reversed and a shortest-path algorithm is run from S, all nodes will be reachable from S c Show that the tests in parts (a) and (b) are sufficient to decide whether a graph is strongly connected (ie, a graph that passes both tests must be strongly connected) d Write a program that checks whether a graph is strongly connected What is the running time of your algorithm

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Explain how each of the following problems can be solved by applying a shortest-path algorithm Then design a mechanism for representing an input and write a program that solves the problem

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1519 The input is a list of league game scores (and there are no ties) If all teams have at least one win and a loss, we can generally "prove," by a silly transitivity argument, that any team is better than any other For instance, in the six-team league where everyone plays three games, suppose that we have the following results: A beat B and C; B beat C and F; C beat D; D beat E; E beat A; and F beat D and E Then we can prove that A is better than F because A beat B who in turn beat F Similarly, we can prove that F is better than A because F beat E and E beat A Given a list of game scores and two teams X and I: either find a proof (if one exists) that X is better than Y or indicate that no proof of this form can be found 1520 A word can be changed to another word by a one-character substitution Assume that a dictionary of five-letter words exists Give an algorithm to determine whether a word A can be transformed to a word B by a series of one-character substitutions, and if so, outputs the corresponding sequence of words For example, bleed converts to blood by the sequence bleed,blend,blond,blood 1521 The input is a collection of currencies and their exchange rates Is there a sequence of exchanges that makes money instantly For : instance, if the currencies are X, I and Z and the exchange rate is 1 X equals 2 Ys, 1 Y equals 2 Zs, and 1 X equals 3 Zs, then 300 Zs will buy 100 Xs, which in turn will buy 200 Ys, which in turn will buy 400 Zs We have thus made a profit of 33 percent 1522 A student needs to take a certain number of courses to graduate, and these courses have prerequisites that must be followed Assume that all courses are offered every semester and that the student can take an unlimited number of courses Given a list of courses and their prerequisites, compute a schedule that requires the minimum number of semesters 1523 The object of the Kevin Bacon Game is to link a movie actor to Kevin Bacon via shared movie roles The minimum number of links is an actor's Bacon number For instance, Tom Hanks has a Bacon number of 1 He was in Apollo 13 with Kevin Bacon Sally Field has a Bacon number of 2 because she was in Forest Gump with Tom.

Fig. 12.12 The quark diagram responsible for neutron {3decay, n -> pe - iie - The two spectator quarks which do not take part in the weak interaction are shown by dashed lines.

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