cs320-2010-t1-midterm2-solution

# cs320-2010-t1-midterm2-solution - CPSC 320 Midterm 2 Monday...

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Unformatted text preview: CPSC 320 Midterm 2 Monday, November 15th, 2010 [16] 1. Short Answers [4] a. Your boss asks you to design a divide-and-conquer algorithm to solve a problem whose input is an array of measurements that contain temperature data. What are the two main issues that you need to determine in order to design this algorithm? Solution : The two issues you need to determine are: • How to break the problem into smaller subproblems, and • How to combine the solutions to the smaller subproblems to get the solution to the initial problem. [4] b. In the Disjoint Sets data structure used by Kruskal’s minimum spanning tree algo- rithm, why does the union operation insert the root with the smaller rank as a child of the root with the larger rank, rather than the other way around? Solution : If we inserted the root with the larger rank as a child of the other root, the following sequence of n calls to unions : T ← new Node for i ← 1 to n do T ← union(T, new Node) endfor would result in a tree with height n − 1 (and hence calls to find would take O(n) time). [4] c. Using the method of your choice, give a tight bound on the function T ( n ) described by the recurrence relation T ( n ) = braceleftbigg 6 T ( n/ 3) + 4 n 2 if n ≥ 13 17 if n ≤ 12 Solution : We use the Master theorem. Because n log b a = n log 3 6 , and log 3 6 < 2 , the only case that may apply is case 3. Let us check the regularity condition. af ( n/b ) = 6 f ( n/ 3) = 6 · 4( n/ 3) 2 = 6 · 4 n 2 / 9 = (2 / 3)4 n 2 and hence the regularity condition is satisfied as long as we choose 2 / 3 < δ < 1 ....
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cs320-2010-t1-midterm2-solution - CPSC 320 Midterm 2 Monday...

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