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Unformatted text preview: Advanced Analysis of Algorithms  Homework II (Solutions) K. Subramani LCSEE, West Virginia University, Morgantown, WV { [email protected] } 1 Problems 1. Problem 15 2 on Page 364 of [CLRS01]. Solution: Assume that the optimal solution packs the n words in r lines; this solution can be decomposed into two parts, viz., the words packed on the first line and the words packed in the remaining ( r 1) lines. Regardless of the choice for the last word on line 1 , the remaining words must be packed optimally in the lines 2 through ( r 1) . (Why?) We define s [ i,j ] to be cost if packing words w i through w j in one line. Likewise, p ij is defined as ( M j + i ∑ j t = i l t ) . Accordingly, s [ i,j ] = p 3 ij , if p ij ≥ 0 and j 6 = n = , if p ij ≥ 0 and j = n = ∞ , if p ij < Let m [ i,j ] denote the optimal cost of of packing words w i through w j . The entry of interest is m [1 ,n ] . It follows that: m [ i,j ] = min i ≤ k ≤ j ( s [ i,k ] + m [ k + 1 ,j ]) Each p ij entry can be computed in O ( n ) time and hence all the p ij entries can be computed in O ( n 3 ) time. Once the p ij values are computed, the s [ i,j ] values can be computed in O (1) time per entry for a total of O ( n 2 ) time. Finally, computing each entry in the m [ i,j ] table takes O ( n ) time and hence the table can be computed in O ( n 3 ) time. The above algorithm can be improved to run in O ( n 2 ) time; I leave this as an exercise. 2 2. Problem 15 7 on Page 369 of [CLRS01]. Solution: Without loss of generality, assume that the jobs are ordered by their deadline, so that d 1 ≤ d 2 ≤ ... ≤ d n . We define A [ i,j ] to be the maximum profit that results from scheduling the jobs { a 1 ,a 2 ,...,a i } in the time interval [0 ,j ] . Note that the entry of interest is A [ n,d n ] , since any job scheduled after its deadline and in particular, after d n results in a profit of . Since each job has a processing time of at most n , we have d n ≤ n 2 . (Why?) We need the following lemma....
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 Spring '10
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 Algorithms, Greedy algorithm, Big O notation, Analysis of algorithms, average completion time

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