Midterm2Prac

# Midterm2Prac - Solutions to Practice Problems for Midterm 2...

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Solutions to Practice Problems for Midterm 2 Peng Du University of California, San Diego 1 Exercise 5.5 1.1 Part (a) No, since every spanning tree gets the same proﬁt n - 1. 1.2 Part (b) Yes, use the same idea as problem 4.8 to give an example. 2 Exercise 5.9 2.1 Part (a) False, see Fig. 1 for a counter-example where the heaviest edge ( b,d ) is part of every MST. a b d c 1 1 10 1 Fig.1. The counter-example for exercise 5.9(a). 2.2 Part (b) True. Suppose e T for some MST T . We remove e and split T into disjoint sets of vertices S 1 and S 2 . Denote the lightest edge between S 1 and S 2 by e 0 . Since there is a cycle where e is the unique heaviest edge, we have w e 0 < w e . Therefore, T \ { e } ∪ { e 0 } is lighter than T , a contradiction.

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2 2.3 Part (c) True. Take any cut containing e , then use cut property by setting X = . 2.4 Part (d) True. Denote the lightest edge by e . Suppose some MST T doesn’t contain e . By adding e to T , we get a cycle C . Since there exists e 0 C such that w e < w e 0 , we can get a lighter tree T 0 = T \ { e 0 } ∪ { e } , a contradiction.
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Midterm2Prac - Solutions to Practice Problems for Midterm 2...

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