final2f03

# final2f03 - asymptotic upper bounds on their time...

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Final 2 : COT5405 Fall ’03 Time: 120 minutes Open Book; Open Notes 1. (25 pts) Prove or disprove (Without using Sterling’s formulae): 2. (15 pts) Suppose that is the minimum-weight edge of an undirected weighted graph and all other edges have strictly larger weights. Prove that every minimum spanning tree of includes the edge . (Note: Any reference to minimum spanning tree algorithms such as Kruskal’s or Prim’s will get you no points.) 3. (15+15 pts) Consider a connected, unweighted, undirected graph . The “Height” of a rooted spanning tree is the maximum distance from root to any leaf (over all leaves) of that spanning tree. For each of the two questions below either give an efﬁcient (polynomial time in vertices/edges) algorithm or formally argue via reduction why one might not exist. a. Is there a rooted spanning tree of “height” at most . b. Is there a rooted spanning tree of “height” at least . If you do provide algorithms, prove that they give the correct answer. Also, analyze and give
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Unformatted text preview: asymptotic upper bounds on their time complexity. If not, show the polynomial reduction from a NP-hard problem. 4. (5+25 pts) The Set-Partition Problem is deﬁned as follows. As input we are given a ﬁnite set of positive integers. The question is whether there is a subset such that the sum of the elements in equals the sum of the elements in . The Ruler-Folding Problem is deﬁned as follows. As input we are given a sequence of hinged ruler segments with integer lengths , and a pocket size . The question is whether the ruler can be folded (that is each hinge folded to either or degrees) to ﬁt into the pocket (See Figure below). k 5 4 3 2 1 l l l l l a. Prove that the Set-Partition Problem and the Ruler-Folding Problem are in NP. b. Prove that the Ruler-Folding Problem is NP-Complete assuming that the Set-Partition Problem is NP-Complete. 1...
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## This note was uploaded on 05/27/2011 for the course COT 5405 taught by Professor Ungor during the Fall '08 term at University of Florida.

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