Totally Greedy Coin Sets and Greedy Obstructions

Cowen, Lenore; Cowen, Robert; Steinberg, Arthur

doi:10.37236/814

Cited by 7 publications

(6 citation statements)

References 5 publications

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“…Theorem 4 (Magazine, Nemhauser, and Trotter Jr. [10], Hu and Lenard [8], Cowen, Cowen, and Steinberg [5]). Let C be a system (1, c 2 , .…”

Section: Related Results On Characterizationmentioning

confidence: 99%

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Characterization of canonical systems with six types of coins for the change-making problem

Suzuki¹,

Miyashiro²

2021

Preprint

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This paper analyzes a necessary and sufficient condition for the change-making problem to be solvable with a greedy algorithm. The change-making problem is to minimize the number of coins used to pay a given value in a specified currency system. This problem is NP-hard, and therefore the greedy algorithm does not always yield an optimal solution. Yet for almost all real currency systems, the greedy algorithm outputs an optimal solution. A currency system for which the greedy algorithm returns an optimal solution for any value of payment is called a canonical system. Canonical systems with at most five types of coins have been characterized in previous studies. In this paper, we give characterization of canonical systems with six types of coins, and we propose a partial generalization of characterization of canonical systems.

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“…Theorem 4 (Magazine, Nemhauser, and Trotter Jr. [10], Hu and Lenard [8], Cowen, Cowen, and Steinberg [5]). Let C be a system (1, c 2 , .…”

Section: Related Results On Characterizationmentioning

confidence: 99%

“…Some terms other than canonical are used in the literature, such as standard[8], greedy[5], and orderly[1].…”

mentioning

confidence: 99%

Characterization of canonical systems with six types of coins for the change-making problem

Suzuki¹,

Miyashiro²

2021

Preprint

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“…Optimal change-making is weakly NP-hard but has a pseudopolynomial time dynamic program that is often used as an example or an exercise in undergraduate algorithms classes [5,7]. However, although there have also been studies on sets of coins that would lead to small solutions [24] or on counting distinct ways of making change [4], much of the research on change-making has focused on a different problem: for which coinage systems is the greedy algorithm optimal [3,6,11,14,20]? This can be tested in polynomial time [20].…”

Section: Making Change In 2048mentioning

confidence: 99%

“…Alternatively, suppose we use 3-smooth tile values, the numbers whose only prime factors are two or three: 1, 2, 3,4,6,8,9,12,16,18,24,27,32,36, . .…”

Section: Introductionmentioning

confidence: 99%

Making Change in 2048

Eppstein

2018

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The 2048 game involves tiles labeled with powers of two that can be merged to form bigger powers of two; variants of the same puzzle involve similar merges of other tile values. We analyze the maximum score achievable in these games by proving a min-max theorem equating this maximum score (in an abstract generalized variation of 2048 that allows all the moves of the original game) with the minimum value that causes a greedy change-making algorithm to use a given number of coins. A widely-followed strategy in 2048 maintains tiles that represent the move number in binary notation, and a similar strategy in the Fibonacci number variant of the game (987) maintains the Zeckendorf representation of the move number as a sum of the fewest possible Fibonacci numbers; our analysis shows that the ability to follow these strategies is intimately connected with the fact that greedy change-making is optimal for binary and Fibonacci coinage. For variants of 2048 using tile values for which greedy change-making is suboptimal, it is the greedy strategy, not the optimal representation as sums of tile values, that controls the length of the game. In particular, the game will always terminate whenever the sequence of allowable tile values has arbitrarily large gaps between consecutive values.

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“…No artigo original sobre o Smoothsort, Dijkstra deixou a prova desse problema para o leitor. Apresentamos a seguir uma prova simples, baseada no Teorema 1 e no Corolário 1 desse teorema, expostos em [10].…”

Section: Heapsort E áRvores De Leonardounclassified

O Algoritmo de Ordenação Smoothsort explicado

Pfitzner¹,

Pinto

Costa

2009

Cadernos Do IME - Série Informática

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Neste trabalho apresentamos o algoritmo de ordenação Smoothsort [1][2], desenvolvido por Dijkstra em 1981 e que, inexplicavelmente, não é citado na literatura corrente sobre métodos de ordenação como um método relevante [3-5]. Sua importância vem do fato dele ter complexidade de pior caso O(n log n), restrita a O(n) para conjuntos de dados já ordenados, uma importante situação prática. Nenhum dos métodos destacados na literatura apresenta tal comportamento. Além de descrever o algoritmo, mostramos também análises alternativas sobre a complexidade do algoritmo, que não estão presentes no artigo original do método.

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Totally Greedy Coin Sets and Greedy Obstructions

Cited by 7 publications

References 5 publications

Characterization of canonical systems with six types of coins for the change-making problem

Characterization of canonical systems with six types of coins for the change-making problem

Making Change in 2048

O Algoritmo de Ordenação Smoothsort explicado

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