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How do I do #2? This is about integer partitions, but I don't have an idea how to do upper bounds TOGETHER WITH lower bounds, and Google does not give me what I'm looking for at all.


CS/MATH111 ASSIGNMENT 4

 

due Tuesday, March 1 (5PM)

 

Problem 1: (a) Give the asymptotic value (using the ?-notation) for the number of letters that will be

 

printed by the algorithms below. In each algorithm the argument n is a positive integer. Your solution needs

 

to consist of an appropriate recurrence equation and its solution. You also need to give a brief justi?cation

 

for the recurrence (at most 10 words each).

 

(i) Algorithm PrintAs (n : integer)

 

if n < 5

 

print(?A?)

 

else

 

PrintAs( n/4 )

 

PrintAs( n/4 )

 

PrintAs( n/4 )

 

PrintAs( n/4 )

 

for i ? 1 to 5 do print(?A?)

 

(ii) Algorithm PrintBs (n : integer)

 

if n < 2

 

print(?B?)

 

else

 

for j ? 1 to 10 do PrintBs( n/2 )

 

for i ? 1 to 6n3 do print(?B?)

 

(iii) Algorithm PrintCs (n : integer)

 

if n < 4

 

print(?C?)

 

else

 

PrintCs( n/3 )

 

PrintCs( n/3 )

 

PrintCs( n/3 )

 

PrintCs( n/3 )

 

for i ? 1 to 20n2 do print(?C?)

 

(b) For each integer n ? 1 we de?ne a tree Tn , recursively, as follows. For n = 1, T1 is a single node. For

 

n > 1, Tn is obtained from four copies of T n/2 and three additional nodes, by connecting them as follows:

 


 

Tn/2

 


 

Tn/2

 


 

Tn/2

 


 

Tn/2

 


 

(In this ?gure, the subtrees are denoted Tn/2 , without rounding, to reduce clutter.) Let h(n) be the number

 

of nodes in Tn . Give a recurrence equation for h(n) and justify it. Then give the solution of this recurrence

 

using the ?() notation.

 


 

1

 


 

Problem 2: Bill is buying his wife a bouquet of carnations, daises, roses and tulips. The bouquet will have

 

26 ?owers, with

 

? between 2 and 11 carnations,

 

? at most 6 daises,

 

? at least 3 roses, and

 

? between 3 and 9 tulips.

 

How many di?erent combinations of ?owers satisfy these requirements? You need to use the counting method

 

for integer partitions and show your work.

 

Problem 3: We have three sets P , Q, R with the following properties:

 

(a) |Q| = 2|P | and |R| = 4|P |,

 

(b) |P ? Q| = 11, |P ? R| = 7, |Q ? R| = 10,

 

(c) 1 ? |P ? Q ? R| ? 11,

 

(d) |P ? Q ? R| = 121.

 

Use the inclusion-exclusion principle to determine the number of elements in P . Show your work. (Hint:

 

You may get an equation with two unknowns, but one of them has only a few possible values.)

 


 

2

 


 

 


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