CS 1510 Midterm 1
Fall 2005
Note that the only difference in the first two problems is the definition of total penalty.
1. The input to this problem consists of an ordered list of nwords. The length of the ith word
is wi, that is the ith word takes up wispaces. (For simplicity assume that there are no spaces
between words.) The goal is to break this ordered list of words into lines, this is called a
layout. Note that you can not reorder the words. The length of a line is the sum of the
lengths of the words on that line. The ideal line length is L. No line may be longer than L,
although it may be shorter. The penalty for having a line of length Kis L−K.The total
penalty is the sum of the line penalties. The problem is to find a layout that minimizes the
total penalty.
Prove of disprove that the following greedy algorithm correctly solves this problem. A proof
of correctness must use an exchange argument.
For i= 1 to n
Place the ith word on the current line if it fits
else place the ith word on a new line
2. The input to this problem consists of an ordered list of nwords. The length of the ith word
is wi, that is the ith word takes up wispaces. (For simplicity assume that there are no spaces
between words.) The goal is to break this ordered list of words into lines, this is called a
layout. Note that you can not reorder the words. The length of a line is the sum of the
lengths of the words on that line. The ideal line length is L. No line may be longer than
L, although it may be shorter. The penalty for having a line of length Kis L−K. The
total penalty is the maximum of the line penalties. The problem is to find a layout that
minimizes the total penalty.
Prove of disprove that the following greedy algorithm correctly solves this problem. A proof
of correctness must use an exchange argument.
For i= 1 to n
Place the ith word on the current line if it fits
else place the ith word on a new line
3. We consider the problem of computing the longest increasing subsequence of a sequence
x1,...,xnof integers. We considered this problem several times during class.
(a) (10 points) Give C-like pseudo-code to compute the length of the longest increasing
subsequence. The running time of your code should be O(n2). Include some explanation
of your variables and data structures, and of your code in general. I suggest you read
the following items in this question before you answer this one.
(b) (5 points) Explain how to in O(n) time compute the actual subsequence, that is the
longest increasing subsequence, from the data structure constructed by your code in
part a.
