*Consider the problem of right-justifying a paragraph. The paragraph contains a sequence o...

*Consider the problem of right-justifying a paragraph. The paragraph contains a sequence of words w1, w2,…, wN of length a1, a2,…, aN, which we wish to break into lines of length L. Words are separated by blanks whose ideal length is b (millimeters), but blanks can stretch or shrink as necessary (but must be >0), so that a line wiwi+1…wj has length exactly L. However, for each blank b′ we charge |b′ − b| ugliness points. The exception to this is the last line, for which we charge only if b′ < b (in other words, we charge only for shrinking), since the last line does not need to be justified. Thus, if bi is the length of the blank between ai and ai+1, then the ugliness of setting any line (but the last) wiwi+1…wj for j > i is , where b′ is the average size of a blank on this line. This is true of the last line only if b′ < b, otherwise the last line is not ugly at all.

a. Give a dynamic programming algorithm to find the least ugly setting of w1, w2,…, wN into lines of length L. (Hint: For i = N, N − 1,…, 1, compute the best way to set wi, wi+1, . . . , wN.)

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b. Give the time and space complexities for your algorithm (as a function of the number of words, N).

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c. Consider the special case where we are using a fixed-width font, and assume the optimal value of b is 1 (space). In this case, no shrinking of blanks is allowed, since the next smallest blank space would be 0. Give a linear-time algorithm to generate the least ugly setting for this case.