The Folds and Cuts Problem - Computer Science Assignment Help

Assignment Task

Recall the FOLDS AND CUTS problem from Assignment 2, where you are given a rectangular strip of paper P of length t with n > 0 vertical folds, located at distances h f2. fn from the left end of P. Let fn+i = 1. In the "dual" version of the problem, you are also given a nonnegative integer k. You want to cut P along exactly k < n folds, such that the maximum length of the k + 1 resulting strip; is as short as passible. We'll denote this length by L(n. k). More generally, for any i in the range [01..4 let Pi denote the strip of paper of length 11+1 that has folds at distances fi, f2,..., If we cut P1 along exactly j folds, where 0 < j < i, such that the maximum length of the j + 1 resulting strips is as short as possible, we denote the resulting maximum length by L(i, j). For example, if 11,12, 13 are 1,3,7, respectively, f4 = t = 10, and k = 2 then cutting at 3 and 7 is optimal, resulting in a max length L(3, 2) = 4. Also for this example, L(2,1) = L(2, 2) = 4, while L(2,0) = 7. 
 

1. When k < n, explain why L(n, k) < t.

2. When k < n, explain why L(n, k) > rf.r.

3. Write a recursive definition of L(i, j) when 0 < j < i. You do not need to provide a justification for your recurrence.

4. Using your recurrence, give a memoized algorithm for computing L(n, k).

5. Explain the runtime of your algorithm. 
 

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