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

Write a script to determine the volume of a pyramid, which is 1/3 * base * height, where the base is length * width.  On time the user to enter values for the length, width, and th

The order of a differential equation is the huge derivative there in the differential equation. Under the differential equations as listed above in equation (3) is a first order di

A two-digit number is seven times the sum of its digits.  The number formed by reversing the digits is 18 less than the  original number. Find the original number.

define even and odd function state whether given function are even odd or neither 1 f x =sin x cos x 2 f x {x}=x +x3n #Minimum 100 words accepted#

Definition 1: Given the function f (x ) then 1. f ( x ) is concave up in an interval I if all tangents to the curve on I are below the graph of f ( x ) . 2. f ( x ) is conca

Lindy has 48 chocolate chip cookies and 64 vanilla wafer cookies. How many bags can Lindy fill if she puts the chocolate chip cookies and the vanilla wafers in the same bag? She pl

Find out the determinant: Find out the determinant of the following 3 x 3 matrix, expanding about row 1. Solution: