CS 573: Graduate Algorithms, Fall 2008
Homework 2
Due at 11:59:59pm, Wednesday, October 1, 2008
•
For this and all future homeworks, groups of up to three students may submit a single, common
solution. Please neatly print (or typeset) the full name, NetID, and alias (if any) of every group
member on the first page of your submission.
•
We will use the following point breakdown to grade dynamic programming algorithms: 60% for a
correct recurrence (including base cases), 20% for correct running time analysis of the memoized
recurrence, 10% for correctly transforming the memoized recursion into an iterative algorithm.
•A greedy algorithm must be accompanied by a proof of correctness in order to receive any credit.
1. (a)
Let
X
[1
.. m
]and
Y
[1
.. n
]be two arbitrary arrays of numbers. A
common supersequence
of
X
and
Y
is another sequence that contains both
X
and
Y
as subsequences. Describe and
analyze an efficient algorithm to compute the function
scs
(
X,Y
), which gives the length of
the shortest common supersequence of Xand Y.
(b)
Call a sequence
X
[1
.. n
]of numbers
oscillating
if
X
[
i
]
<X
[
i
+1]for all even
i
, and
X
[
i
]
>X
[
i
+1]for all odd
i
. Describe and analyze an efficient algorithm to compute the
function
los
(
X
), which gives the length of the longest oscillating subsequence of an arbitrary
array Xof integers.
(c)
Call a sequence
X
[1
.. n
]of numbers
accelerating
if 2
·X
[
i
]
<X
[
i−
1] +
X
[
i
+1]for all
i
.
Describe and analyze an efficient algorithm to compute the function
lxs
(
X
), which gives the
length of the longest accelerating subsequence of an arbitrary array Xof integers.
2.
Apalindrome is a string that reads the same forwards and backwards, like x,pop,noon,redivider,
or amanaplanacatahamayakayamahatacanalpanama. Any string can be broken into sequence of
palindromes. For example, the string bubbaseesabanana (‘Bubba sees a banana.’) can be broken
into palindromes in several different ways; for example:
bub + baseesab + anana
b+u+bb+a+sees+aba+nan+a
b+u+bb+a+sees+a+b+anana
b+u+b+b+a+s+e+e+s+a+b+a+n+a+n+a
Describe and analyze an efficient algorithm to find the smallest number of palindromes that make
up a given input string. For example, given the input string bubbaseesabanana, your algorithm
would return the integer 3.
3.
Describe and analyze an algorithm to solve the traveling salesman problem in
O
(2
npoly
(
n
)) time.
Given an undirected
n
-vertex graph
G
with weighted edges, your algorithm should return the
weight of the lightest Hamiltonian cycle in
G
, or
∞
if
G
has no Hamiltonian cycles. [Hint: The
obvious recursive algorithm takes O(n!)time.]
