3Soln - Computer Science C73 Scarborough Campus Solutions...

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Computer Science C73 November 7, 2007 Scarborough Campus University of Toronto Solutions for Homework Assignment #3 Answer to Question 1. a. The array of edit distances of all preFxes of the two strings computed by the algorithm is shown below. L O O P Y 0 1 2 3 4 5 0 0 1 2 3 4 5 S 1 1 1 2 3 4 5 N 2 2 2 2 3 4 5 O 3 3 3 2 2 3 4 O 4 4 4 3 2 3 4 P 5 5 5 4 3 2 3 b. The DAG of subproblems being solved with the edges appropriately labeled is shown below. A shortest (unique, in this case) path from the smallest subproblem to the largest one is identiFed with heavy edges. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 O S N O O P L O P Y 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 c. Any path through the DAG shown above consists of edges each of which goes down, to the right, or diagonally. An edge going down corresponds to deleting a letter of the Frst string; an edge going to the right corresponds to inserting a letter; and an edge labeled 1 going diagonally corresponds to changing a letter. (A diagonal edge labeled 0 corresponds to a match, and results in no editing operation.) More precisely, an edge from ( i 1 ,j ) to ( i,j ) corresponds to deleting the j -th letter of the Frst string. An edge 1
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from ( i,j 1) to ( i,j ) corresponds to inserting the j -th letter of the second string. An edge labeled 1 from ( i 1 ,j 1) to ( i,j ) corresponds to changing the i -th letter of the Frst string to the j -the letter of the second string. Answer to Question 2. The inputs are positive integers w 1 ,w 2 ,... ,w n ,W, Δ. Let S ⊆ { 1 ,... ,n } . We deFne the value of S to be i S w i . We say that a set S ⊆ { 1 ,... ,n } is w - legal if its value is at most w , and for all i,j S such that i n = j , | w i w j | ≥ Δ. ±or 0 i n and 0 w W , let S ( i,w ) be a w -legal subset of { 1 ,... ,i } of maximum value. The set we are seeking is S ( n,W ). Let M ( i,w ) be the value of S ( i,w ). We now give (and justify) a recursive formula to compute M ( i,w ). To do so, it is convenient to Frst sort the Frst n inputs so that w 1 w 2 ... w n , and to deFne, for each 1 i n , p ( i ) = max { j : j < i and w i w j Δ } , where we take max = 0. (That is, p ( i ) is the largest index preceding i of an input that is “su²ciently far apart” from w i .) Note that if i is in a w -legal set S , p ( i ) is the largest index smaller than i that can be in S . Let 0 i n and 0 w W . Case 1. i = 0. Then clearly M (0 ,w ) = 0. Case 2. i 1 and w i > w . In this case, i / S ( i,w ) (since w i by itself is not w -legal). Therefore, S ( i,w ) = S ( i 1 ,w ), and so M ( i,w ) = M ( i 1 ,w ). Case 3.
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3Soln - Computer Science C73 Scarborough Campus Solutions...

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