Ex. 15-10 : CLRS Algorithm Book
Your knowledge of algorithms helps you obtain an exciting job with the Acme
Computer Company, along with a $10,000 signing bonus. You decide to invest
this money with the goal of maximizing your return at the end of 10 years. You
decide to use the Amalgamated Investment Company to manage your investments.
Amalgamated Investments requires you to observe the following rules. It offers n
different investments, numbered 1 through n. In each year j , investment i provides
a return rate of rij . In other words, if you invest d dollars in investment i in year j ,
then at the end of year j , you have drij dollars. The return rates are guaranteed,
that is, you are given all the return rates for the next 10 years for each investment.
You make investment decisions only once per year. At the end of each year, you
can leave the money made in the previous year in the same investments, or you
can shift money to other investments, by either shifting money between existing
investments or moving money to a new investment. If you do not move your
money between two consecutive years, you pay a fee of f1 dollars, whereas if you
switch your money, you pay a fee of f2 dollars, where f2 > f1.
a. The problem, as stated, allows you to invest your money inmultiple investments
in each year. Prove that there exists an optimal investment strategy that, in
each year, puts all the money into a single investment. (Recall that an optimal
investment strategy maximizes the amount of money after 10 years and is not
concerned with any other objectives, such as minimizing risk.)
b. Prove that the problem of planning your optimal investment strategy exhibits
c. Design an algorithm that plans your optimal investment strategy. What is the
running time of your algorithm?
d. Suppose that Amalgamated Investments imposed the additional restriction that,
at any point, you can have no more than $15,000 in any one investment. Show
that the problem of maximizing your income at the end of 10 years no longer
exhibits optimal substructure.
This question was answered on: May 23, 2022
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