I only need part B in the image, I have already completed part A. How do I formally prove that by using consecutive powers for the values of coins, that it will give me the optimal solution? (using induction preferably unless an easier formal proof is available) 14. Suppose you are a simple shopkeeper living in a country with n differenttypes of coins, with values 1 = c[1] < c[2] < recursively give change for the remainder. El Generalissimo does notapprove of American capitalist greed. Show that there is a set of coinvalues for which the greedy algorithm does not always give the smallestpossible of coins.(b) Now suppose El Generalissimo decides to impose a currency systemwhere the coin denominations are consecutive powers bº, b¹,b²,..., bkof some integer b≥ 2. Prove that despite El Generalissimo's disapproval,the greedy algorithm described in part (a) does make optimal change inthis currency system.

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Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

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I only need part B in the image, I have already completed part A. How do I formally prove that by using consecutive powers for the values of coins, that it will give me the optimal solution? (using induction preferably unless an easier formal proof is available)

14. Suppose you are a simple shopkeeper living in a country with n different
types of coins, with values 1 = c[1] < c[2] < <c[n]. (In the U.S.,
for example, n = 6 and the values are 1, 5, 10, 25, 50 and 100 cents.)
Your beloved and benevolent dictator, El Generalissimo, has decreed that
whenever you give a customer change, you must use the smallest possible
number of coins, so as not to wear out the image of El Generalissimo lovingly
engraved on each coin by servants of the Royal Treasury.
(a) In the United States, there is a simple greedy algorithm that always
results in the smallest number of coins: subtract the largest coin and

recursively give change for the remainder. El Generalissimo does not
approve of American capitalist greed. Show that there is a set of coin
values for which the greedy algorithm does not always give the smallest
possible of coins.
(b) Now suppose El Generalissimo decides to impose a currency system
where the coin denominations are consecutive powers bº, b¹,b²,..., bk
of some integer b≥ 2. Prove that despite El Generalissimo's disapproval,
the greedy algorithm described in part (a) does make optimal change in
this currency system.

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