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ex2-sol07 - AMS 301.3(Fall 2007 Estie Arkin Exam 2 Solution...

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AMS 301.3 (Fall, 2007) Estie Arkin Exam 2 – Solution sketch Mean 76.27, median 80, high 97, low 36. 1. (12 points) Let T be a balanced m-ary tree with 201 nodes, 161 of which are leaves. Show your work! (A correct guess with no work shown will recieve very partial credit.) (a). What is m equal to? n = mi + 1, n = 201, l = 161, i = n - l = 201 - 161 = 40, so 201 = 40 m + 1, m = 200 / 40 = 5. (b). How many edges does the tree have? 201 - 1 = 200 (c). What is the height of T ? d log 5 161 e = 4 2. (10 points) Consider the following graph. (a). Highlight the edges of a minimum spanning tree of the graph. (A,E) (B,F) (C,G) (D,H) (A,B) (D,E) (B,C). (b). Edge (A,B) is currectly part of the minimum spanning tree. Suppose its cost is increased from 2 to 11. Will it still be part of the minimum spanning tree? Explain. No. Instead can put in edge(E,F) of lower cost. 3. (8 points) I wish to model the word problem as a graph problem: I am at my home, but each of my 6 children is at some different location (friend’s house, soccer game, etc.). I need to collect all 6 and bring them all home for dinner. As I am short on time, I would like to get them all in a single trip. I have a minivan that can seat all 6 kids and myself. Assume travel times between locations are known. (a). What do the nodes of the graph to be constructed represent? Locations: home and 6 places where kids are. (b). What do the edgess of the graph to be constructed represent? Possible travel between location, and cost is time to travel. (c). State which graph problem it is: Shortest path, Minimum spanning tree, Traveling Salesman Problem, Breadth or Depth First Search. TSP (since we must visit all nodes before returning home).
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