George Washington University
Department of Computer Science
Csci 212 - Homework 3
Due date: April 23, 2009
Email to bowuzh@gwmail.gwu.edu by 11:59PM or submit to CS Dept by 5PM.
1. Given a sequence of nreal numbers A(1),· · · , A(n), give a dynamic programming algorithm
to determine a contiguous subsequence A(i),· · · , A(j) for which the sum of elements in the
subsequence is maximized. Analyze the time complexity of your algorithm.
2. You are given ntypes of coin denominations of values v(1) < v(2) <· · · < v (n) (all integers).
Assume v(1) = 1, so you can always make change for any amount of money C. Give a
dynamic programming algorithm which makes change for a given amount of money Cwith
as few coins as possible. Analyze the time complexity of your algorithm.
3. Given a sequence of nreal numbers A(1),· · · , A(n), Give a dynamic programming algorithm to
determine a subsequence (not necessarily contiguous) of maximum length in which the values
in the subsequence form a strictly increasing sequence. Analyze the time complexity of your
algorithm.
4. Consider a 2-D map with a horizontal river passing through its center. There are ncities on the
southern bank with x-coordinates a(1),· · · , a(n) and ncities on the northern bank with x-
coordinates b(1),· · · , b(n). You want to connect as many north-south pairs of cities as possible
with bridges such that no two bridges cross. When connecting cities, you can only connect
city ion the northern bank to city ion the southern bank. Give a dynamic programming
algorithm to solve this problem and analyze the time complexity of your algorithm.
5. You have a set of nintegers each in the range 0,···, K. Partition these integers into two subsets
such that you minimize |S1−S2|, where S1and S2denote the sums of the elements in each of
the two subsets. Give a dynamic programming algorithm to solve this problem and analyze
the time complexity of your algorithm.
6. Given two text strings Aof length nand Bof length m, you want to transform Ainto B
with a minimum number of operations of the following types: delete a character from A,
insert a character into A, or change some character in Ainto a new character. The minimal
number of such operations required to transform Ainto Bis called the edit distance between
Aand B. Give a dynamic programming algorithm to solve this problem and analyze the time
complexity of your algorithm.
7. You are given a boolean expression consisting of a string of the symbols ’true’, ’false’, ’and’,
’or’, and ’xor’. Count the number of ways to parenthesize the expression such that it will
evaluate to true. For example, there is only 1 way to parenthesize ’true and false xor true’
such that it evaluates to true. Give a dynamic programming algorithm to solve this problem
and analyze the time complexity of your algorithm.
