Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 15.3, Problem 6E
Program Plan Intro
To explain the optimality of substructure of the finding the best sequence of exchanges having sequence
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We examine a problem in which we are handed a collection of coins and are tasked with forming a sum of money n out of the coins. The currency numbers are coins = c1, c2,..., ck, and each coin can be used as many times as we want. What is the bare amount of money required?If the coins are the euro coins (in euros) 1,2,5,10,20,50,100,200 and n = 520, we need at least four coins. The best option is to choose coins with sums of 200+200+100+20.
Suppose the economies of the world use a set of currencies C1, . . . , Cn; think of these as dollars, pounds, Bitcoin, etc. Your bank allows you to trade each currency Ci for any other currency Cj, and finds some way to charge you for this service. Suppose that for each ordered pair of currencies (Ci, Cj ), the bank charges a flat fee of fij > 0 dollars to exchange Ci for Cj (regardless of the quantity of currency being exchanged).
Describe an algorithm which, given a starting currency Cs, a target currency Ct, and a list of fees fij for all i, j ∈ {1, . . . , n}, computes the cheapest way (that is, incurring the least in fees) to exchange all of our currency in Cs into currency Ct. Also, justify the its runtime.
[A description or pseudocode (either is OK) of the algorithm, as well as a brief justification of its runtime.]
Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies.
Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you fill in the values is the correct one. Notice how it is a lot easier to analyze the running time of…
Chapter 15 Solutions
Introduction to Algorithms
Ch. 15.1 - Prob. 1ECh. 15.1 - Prob. 2ECh. 15.1 - Prob. 3ECh. 15.1 - Prob. 4ECh. 15.1 - Prob. 5ECh. 15.2 - Prob. 1ECh. 15.2 - Prob. 2ECh. 15.2 - Prob. 3ECh. 15.2 - Prob. 4ECh. 15.2 - Prob. 5E
Ch. 15.2 - Prob. 6ECh. 15.3 - Prob. 1ECh. 15.3 - Prob. 2ECh. 15.3 - Prob. 3ECh. 15.3 - Prob. 4ECh. 15.3 - Prob. 5ECh. 15.3 - Prob. 6ECh. 15.4 - Prob. 1ECh. 15.4 - Prob. 2ECh. 15.4 - Prob. 3ECh. 15.4 - Prob. 4ECh. 15.4 - Prob. 5ECh. 15.4 - Prob. 6ECh. 15.5 - Prob. 1ECh. 15.5 - Prob. 2ECh. 15.5 - Prob. 3ECh. 15.5 - Prob. 4ECh. 15 - Prob. 1PCh. 15 - Prob. 2PCh. 15 - Prob. 3PCh. 15 - Prob. 4PCh. 15 - Prob. 5PCh. 15 - Prob. 6PCh. 15 - Prob. 7PCh. 15 - Prob. 8PCh. 15 - Prob. 9PCh. 15 - Prob. 10PCh. 15 - Prob. 11PCh. 15 - Prob. 12P
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