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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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