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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online