tailieunhanh - Báo cáo toán học: "Totally Greedy Coin Sets and Greedy Obstructions"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Totally Greedy Coin Sets and Greedy Obstructions. | Totally Greedy Coin Sets and Greedy Obstructions . Cowen Robert Cowen Arthur Steinberg Department of Mathematics Queens College CUNY Submitted May 24 2008 Accepted Jul 5 2008 Published Jul 14 2008 Mathematics Subject Classifications 05A99 68R05 90C27 Abstract A coin set is a strictly increasing list of positive integers that always begins with 1. A coin set is called greedy when the simple greedy change-making algorithm always produces the fewest number of coins in change. Here the greedy changemaking algorithm repeatedly selects the largest denomination coin less than the remaining amount until it has assembled the correct change. Pearson has provided an efficient algorithm for determining whether a coin set is greedy. We study a stricter property on coin sets called total greediness which requires that all initial subsequences of the coin set also be greedy and a simple property makes it easy to test if a coin set is totally greedy. We begin to explore the theory of greedy obstructions- those coin sets that cannot be extended to greedy coin sets by the addition of coins in larger denominations. 1 Introduction One well-known property of US coin denominations comes from the famous change making problem. In particular suppose a cashier needs to return a value of c cents change to a customer. The cashier has pennies worth 1 cent nickels worth 5 cents dimes worth ten cents and quarters worth 25 cents . There may be several different ways to represent c cents using this coin set for example eleven cents could be made using eleven pennies or two nickels and a penny or a dime and a penny. We call a representation of c cents optimal if it uses the fewest number of coins possible so the dime and penny is the optimal representation of eleven cents in the US coin set. In general for a coin set C with denominations a1 a2 a3 . ak we denote by N C y the number of coins in an optimal representation of y using coin set C or just N y when C is clear from context. In what .