aa2-sol - Problem Set 2 Solutions Due: Feb 26 cs2me3 Winter...

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Unformatted text preview: Problem Set 2 Solutions Due: Feb 26 cs2me3 Winter 2007 Part A This problem is very similar in flavor to the segmented least squares problem. We observe that the last line ends with word n and has to start with some word j ; breaking off words j , , n we are left with a recursive sub-problem on 1 , , j- 1 . Thus, we define OPT [ i ] to be the value of the optimal solution on the set of words W i = { 1 , , i } , for any i j , let S i,j denote the slack of a line containing the words i , , j ; as a notational device, we define S i,j = if these words exceed total length L . For each fixed i , we can compute all S i,j in O ( n ) time by considering values of j in increasing order; thus we can compute all S i,j in O ( n 2 ) time. As noted above, the optimal solution must begin the last line somewhere (at word j ), and solve the sub-problem on the earlier lines optimally. We thus have the recurrence OPT [ n ] = min 1 j n S 2 j,n + OPT [ j- 1] and the line of words j , , n is used in an optimum solution if and only if the minimum is obtained using index j . Finally, we just need a loop to build up all these values: Compute all values S i,j as described above Set OPT[0]=0 For k=1, ,n OPT[k]= min 1 j k S 2 j,k +OPT[j-1] Endfor Return OPT[n] As noted above, it takes O ( n 2 ) time to compute all values S i,j . Each iteration of the loop takes time O ( n ), and there are O ( n ) iterations. Thus the total running time is O ( n 2 ). By tracing back through the array OPT, we can recover the optimal sequence of line breaks that achieve the value OPT [ n ] in O ( n ) additional time. Part B 1a Suppose by way of contradiction that T and T are two distinct MCST of G . Since T and T have the same number of edges, but are not equal, there is some edge e in T but not in T ....
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aa2-sol - Problem Set 2 Solutions Due: Feb 26 cs2me3 Winter...

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