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

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 wi wi+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′ (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, otherwise the last line is not ugly at all.

a. Give a dynamic programming algorithm to find the least ugly setting of w0, w2, .. ., wN into lines of length L. (Hint: For i = N, N − 1, .. ., 1, compute the best way to set wi, wi+1, .. ., wN.)

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

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.