UML CS
Analysis of Algorithms 91.503 (section 201)
Fall, 2004
Homework #1
Assigned: Wednesday, 9/8
Due: Wednesday, 9/15
start of lecture
This assignment covers textbook material in Chapter 15 (Dynamic Programming)
plus review material from 91.404 (undergraduate algorithms).
1. (10 points) Algorithm & Problem Bounds: Let P be a problem. Suppose that the
worst-case asymptotic running time complexity of P is in ( n2 lg(n3lgn) ) and is also
in O( n3 lg(nlgn)2 ). That is, we know that there exists an algorithm solving P whose
worst-case asymptotic running time complexity is in O(n3 lg(nlgn)2) and we know
that time proportional to n2 lg(n3lgn) is a lower bound on the worst-case asymptotic
running time complexity of any algorithm solving P. Now, let A be an algorithm that
solves P. For each of the following statements about A below [(1)-(4)] determine
whether or not the statement is consistent with the information provided about the
worst-case asymptotic time complexity of P. For example, for (1), determine
whether or not it is possible for A to have worst-case time complexity in ( n2 lglgn)
given that P’s worst-case time complexity is in O(n3 lg(nlgn)2) and is also in ( n2
lg(n3lgn)). Note that when working on (1) you can ignore statements [(2)-(4)].
1) A has worst-case time complexity in (n2 lglgn).
2) A has worst-case time complexity in (n2 lg( n3/lgn )).
3) A has worst-case time complexity in( n3 lglgn ).
4) A has worst-case time complexity in O( n2 2lgn ).
2. (10 points) Function Order of Growth:
Given the following 4 functions:
 
3n lg n n
8lg n3
lglg  2

4lg2n


5 lg n! 
List the 4 functions in order of nondecreasing asymptotic size.
Justify your answer.
3. (10 points) Recurrence:
Find tight upper and lower asymptotic bounds on the solution to the following
recurrence:
2
n
T (n)  81T    9 2 log3 n
3
You may assume that T(3) = 1 and n is a power of 3.
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Explain your answer.
UML CS
Analysis of Algorithms 91.503 (section 201)
Fall, 2004
4. (10 points) Dynamic Programming: Determine an LCS of these two strings:
<A, B, B, A, B, A, B, A>
and
<B, A, B, B, A, B, B, B, A>.
5. (60 points) Dynamic Programming: A particle moves leftwards in a straight line along
the x axis towards the origin x = 0. It starts at initial position x0 at time t = 0. The particle
moves leftwards with initial velocity v1, where moving leftwards results in positive
velocity. Its velocity remains constant until the particle reaches position x1 at time t = 1,
at which time its positive velocity becomes v2. From there its velocity again remains
constant until the particle reaches position x2 at time t = 2, at which time its positive
velocity becomes v3. From there its velocity again remains constant until the particle
reaches its final resting position at the origin x3 = 0 at time t = 3.
Solve the following problem using dynamic programming: express x1, x2, x3 and v1, v2, v3
in terms of x0 under the condition that v12 + v22 + v32 is minimized.
a. Characterize the structure of the optimal solution.
b. Recursively define the value of an optimal solution.
c. Compute the value of an optimal solution in a bottom-up fashion.
d. Construct an optimal solution from the computed information.
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