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
Treatment of Na2PdCI4 with 3-chloro-2-methyl-1-propene under an atmosphere of CO yields the dimer [(11 3 - C 4 H 7 )PdClb (A). The 1H NMR spectrum of A at 298 K shows 3 signals: a

Suppose that Ana is buying only 2 goods: good 1 and 2. If the price of good 1 doubles and the price of good 2 drops by one third, then what happens with the budget constraint? (Ass

difference between gdp at market price and nnp at factor cost

Q. Monetary base and the supply of money? It isn't possible for central bank to print and distribute money -which would increase their debt without increasing their assets. Rat

Compare Money with wealth and income Money isn't the same as wealth. An individual may be very wealthy however have no money (for instance by owning stocks and real estate). An

Suppose a company is considering two independent projects, Project A and Project B.  The cash outlay for Project A is $14,000. The cash outlay for Project B is $20,000. The company

Suppose the supply function for product X is given by Qsx = -50 + 0.5Px - 5Pz. A. How much of product X is produced when Px = $500 and Pz = $30? B. How much of product X is p

outline two main restrictions by indian government applied to import. Using the data from your case study analyse and explain who would benefit directly and who would lose directly

what is static and dynamic multiplier in keynesian theory?

Diagramatic explanation of pareto optimality