Computing change for a given coin system, Mathematics

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.

Posted Date: 3/23/2013 1:41:28 AM | Location : United States

Related Discussions:- Computing change for a given coin system, Assignment Help, Ask Question on Computing change for a given coin system, Get Answer, Expert's Help, Computing change for a given coin system Discussions

Write discussion on Computing change for a given coin system
Your posts are moderated
Related Questions
At the school bookstore and two binders and three pens cost $12.50. Three binders and five pens cost $19.50. What is the approximate cost of 1 binder and 1 pen? Let x = the cos

It’s been a busy weekend for Larry. Five people in his neighborhood left on vacation Saturday morning and each of them left a pet for Larry to care for until they return. It’s a go

darien agrees to sponsor her sister $8 plus $1 for every mile she walks.Write an expression to show her total money

Consider the equation x 2 y′′+ xy′- y = 4x ln x (a) Verify that x is a solution to the homogeneous equation. (b) Use the method of reduction of order to derive the second

How do you calculate compound interest

Describe Subtracting Negative Fractions? Subtracting two fractions, whether one is positive and one is negative, or whether they are both negative, is almost the same process a

PROOF OF VARIOUS INTEGRAL FACTS/FORMULAS/PROPERTIES In this section we've found the proof of several of the properties we saw in the Integrals section and also a couple from t

The students at Norton School were asked to name their favorite type of pet. Of the 430 students surveyed, 258 said in that their favorite type of pet was a dog. Assume that only 1

Measures of Central Tendency Measures of Central Tendency are statistical values which tend to happen at the centre of any well ordered set of data. When these measures happen

Computation method           Whereas L = Lower class boundary of the class having the mode             f 0 = Frequency of the class below the modal class