Making Change in 2048

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

Diseño del encofrado para muros usando encofrados modulares

Concrete is one of the most versatile materials used in construction. At present, it is extensively employed in the industry as it can be molded in almost any shape through rigid molds called formwork. The arrangement of the different elements that compose the formwork and its associated restrictions pose interesting computational challenges that directly impact the construction industry. In this article, we propose the implementation of a model with constraints that generates a formwork layout for rectilinear walls using modular forms. This approach is based on the decomposition of walls and their representation as instances of the change-making problem. For that purpose, we used the concept of oriented rectangular decomposition to simplify the problem input and address the solution as a Constraint Optimization Problem (COP). The experiments with test inputs provided solutions in less than 200 seconds, thus offering a practical and efficient approach to the complex task of formwork design.  El concreto u hormigón es uno de los materiales más versátiles usados en la construcción.  En la actualidad, se usa extensivamente en la industria, ya que puede ser moldeado de formas casi arbitrarias a través de moldes rígidos denominados encofrados. La manera en que se disponen los distintos elementos para formar el encofrado y las restricciones asociadas, genera retos computacionales interesantes que impactan directamente la industria de la construcción. En este artículo se propone la implementación de un modelo de restricciones que genera encofrados para muros rectilíneos usando formaletas modulares. Este enfoque se basa en la descomposición de muros y su representación en instancias del problema de cambio de monedas. Para este propósito, se utiliza el concepto de descomposición rectangular orientada para simplificar la entrada del problema y abordar la solución como un problema de optimización por restricciones (COP). Los experimentos con entradas de prueba ofrecen soluciones en tiempos inferiores a 200 segundos, cuyo resultado es un enfoque práctico y eficiente para la compleja tarea del diseño de encofrados

Enabling sustainable power distribution networks by using smart grid communications

Smart grid modernization enables integration of computing, information and communications capabilities into the legacy electric power grid system, especially the low voltage distribution networks where various consumers are located. The evolutionary paradigm has initiated worldwide deployment of an enormous number of smart meters as well as renewable energy sources at end-user levels. The future distribution networks as part of advanced metering infrastructure (AMI) will involve decentralized power control operations under associated smart grid communications networks. This dissertation addresses three potential problems anticipated in the future distribution networks of smart grid: 1) local power congestion due to power surpluses produced by PV solar units in a neighborhood that demands disconnection/reconnection mechanisms to alleviate power overflow, 2) power balance associated with renewable energy utilization as well as data traffic across a multi-layered distribution network that requires decentralized designs to facilitate power control as well as communications, and 3) a breach of data integrity attributed to a typical false data injection attack in a smart metering network that calls for a hybrid intrusion detection system to detect anomalous/malicious activities. In the first problem, a model for the disconnection process via smart metering communications between smart meters and the utility control center is proposed. By modeling the power surplus congestion issue as a knapsack problem, greedy solutions for solving such problem are proposed. Simulation results and analysis show that computation time and data traffic under a disconnection stage in the network can be reduced. In the second problem, autonomous distribution networks are designed that take scalability into account by dividing the legacy distribution network into a set of subnetworks. A power-control method is proposed to tackle the power flow and power balance issues. Meanwhile, an overlay multi-tier communications infrastructure for the underlying power network is proposed to analyze the traffic of data information and control messages required for the associated power flow operations. Simulation results and analysis show that utilization of renewable energy production can be improved, and at the same time data traffic reduction under decentralized operations can be achieved as compared to legacy centralized management. In the third problem, an attack model is proposed that aims to minimize the number of compromised meters subject to the equality of an aggregated power load in order to bypass detection under the conventionally radial tree-like distribution network. A hybrid anomaly detection framework is developed, which incorporates the proposed grid sensor placement algorithm with the observability attribute. Simulation results and analysis show that the network observability as well as detection accuracy can be improved by utilizing grid-placed sensors. Conclusively, a number of future works have also been identified to furthering the associated problems and proposed solutions

A Polynomial-time Algorithm for the Change-Making Problem

The change-making problem is the problem of representing a given value with the fewest coins possible from a given set of coin denominations. To solve this problem for arbitrary coin systems is NP-hard [L]. We investigate the problem of determining whether the greedy algorithm always produces the optimal result for a given coin system. Chang and Gill [CG] show that this can be solved in time polynomial in the size of the largest coin and in the number of coins. Kozen and Zaks [KZ] give a more efficient algorithm, and pose as an open problem whether there is an algorithm to solve this problem which is polynomial in the size of the input. In this paper, we will derive such an algorithm. We first obtain a characterization of the smallest counterexample (if there is one) for which the greedy algorithm is not optimal. We then derive a set of O(n 2 ) possible values (where n is the number of coins) which must contain the smallest counterexample. Each can be tested with O(n) arithmetic ope..