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## Notes on Exercises

Classifying exercises is an activity fraught with peril because readers of a tutorial such as this come to the material with various levels of knowledge and experience. Nonetheless, guidance is appropriate, so many of the exercises carry one of four annotations to help you decide how to approach them. Exercises that test your understanding of the material are marked with an open triangle, as follows:

18.34 Consider the graph

Draw its DFS tree and use the tree to find the graph's bridges and edge-connected components.

Most often, such exercises relate directly to examples in the text. They should present no special difficulty, but working them might teach you a fact or concept that may have eluded you when you read the text. Exercises that add new and thought-provoking information to the material are marked with an open circle, as follows:

19.106 Write a program that counts the number of different possible results of topologically sorting a given DAG.

Such exercises encourage you to think about an important concept that is related to the material in the text, or to answer a question that may have occurred to you when you read the text. You may find it worthwhile to read these exercises, even if you do not have the time to work them through. Exercises that are intended to challenge you are marked with a black dot, as follows:

• 20.73 Describe how you would find the MST of a graph so large that only V edges can fit into main memory at once.

Such exercises may require a substantial amount of time to complete, depending on your experience. Generally, the most productive approach is to work on them in a few different sittings. A few exercises that are extremely difficult (by comparison with most others) are marked with two black dots, as follows:

•• 20.37 Develop a reasonable generator for random graphs with V vertices and E edges such that the running time of the heap-based PFS implementation of Dijkstra's algorithm is superlinear.

These exercises are similar to questions that might be addressed in the research literature, but the material in the tutorial may prepare you to enjoy trying to solve them (and perhaps succeeding). The annotations are intended to be neutral with respect to your coding and mathematical ability. Those exercises that require expertise in coding or in mathematical analysis are self-evident. All readers are encouraged to test their understanding of the algorithms by implementing them. Still, an exercise such as this one is straightforward for a practicing programmer or a student in a coding course, but may require substantial work for someone who has not recently programmed:

• 17.74 Write a program that generates V random points in the plane, then builds a network with edges (in both directions) connecting all pairs of points within a given distance d of one another (see Program 3.20), setting each edge's weight to the distance between the two points that it connects. Determine how to set d so that the expected number of edges is E.

In a similar vein, all readers are encouraged to strive to appreciate the analytic underpinnings of our knowledge about properties of algorithms. Still, an exercise such as this one is straightforward for a scientist or a student in a discrete mathematics course, but may require substantial work for someone who has not recently done mathematical analysis:

19.5 How many digraphs correspond to each undirected graph with V vertices and E edges?

There are far too many exercises for you to read and assimilate them all; my hope is that there are enough exercises here to stimulate you to strive to come to a broader understanding on the topics that interest you than you can glean by simply reading the text.

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