(Solution) 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... | Snapessays.com


(Solution) 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...


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

 

optimal substructure.

 

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.

 


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This question was answered on: May 23, 2022

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