Describe an algorithm which gives initial amount , Macroeconomics

This problem involves the question of computing change for a given coin system. A coin system is defined to be a sequence of coin values v1 < v2 < . . . < vn, such that v1 = 1. For example, in the U.S. coin system we have six coins with values h1, 5, 10, 25, 50, 100i. The question is what is the best way to make change for a given integer amount A.

(a) Let c ≥ 2 be an integer constant. Suppose that you have a coin system where there are n types of coins of integer values v1 < v2 < . . . < vn, such that v1 = 1 and, for 1 < i ≤ n, vi = c · vi-1. (For example, for c = 3 and n = 4, an example would be h1, 3, 9, 27i.) Describe an algorithm which given n, c, and an initial amount A, outputs an n-element vector that indicates the minimum number of coins in this system that sums up to this amount. (Hint: Use a greedy approach.)

(b) Given an initial amount A ≥ 0, let hm1, . . . ,mni be the number of coins output by your algorithm. Prove that the algorithm is correct. In particular, prove the following:

(i) For 1 ≤ i ≤ n, mi ≥ 0

(ii) Pn

i=1mi · vi = A

(iii) The number of coins used is as small as possible

Prove that your algorithm is optimal (in the sense that of generating the minimum number of coins) for any such currency system.

(c) Give an example of a coin system (either occurring in history, or one of your own invention) for which the greedy algorithm may fail to produce the minimum number of coins for some amount.

Your coin system must have a 1-cent coin.

Hint on problem 3: Suppose that the input is n=4, c=2, A=53 (meaning four coins, growing exponentially in 2, that is, {1, 2, 4, 8}, and 53 cents as the amount). The output would be (1,0,1,6), meaning:

The input consists of n, c, and the amount A. The output is a list of the numbers of coins of each denomination that sums to A.

Posted Date: 3/28/2013 6:19:14 AM | Location : United States







Related Discussions:- Describe an algorithm which gives initial amount , Assignment Help, Ask Question on Describe an algorithm which gives initial amount , Get Answer, Expert's Help, Describe an algorithm which gives initial amount Discussions

Write discussion on Describe an algorithm which gives initial amount
Your posts are moderated
Related Questions
Balance of T rade A country's present account reflects a money drain when exports exceed imports. The net distinction in-between the dollar value of a world imports an

Explain the impact of Wal-Mart's supply chain management on its total product, marginal product, and average product curves. What has been the effect on its retail prices?

Collateral Management is a function to handle collateral effectively. It gives interface to enter collateral data, and it has a master data of collateral descriptions and types. It

A person chooses between leisure and consumption. All of their consumption comes from current income. The utility derived from any combination of leisure and consumption is given b

if govtment face cost push inflation which policy govtment should take to control inflatoin?

Calculate the equilibrium price and quantity?

Which is better meausute of welfare?

Are there any current subsidy or welfare issues that are being discussed or addressed in parliament or in municipalities

Answer the following questions for a hypothetical economy whose situation in year 1 was as follows: M = $800 billion; long-term annual growth of real GDP = 3%; V = 4. The bankin

What are the four managerial factors that lead to diseconomies of scale