Fair Partitions

A convex fair partition of a convex polygonal region is defined as a partition on which all regions are convex and have equal area and equal perimeter. In this article we describe an algorithm that finds such fair partition.•The Fair Partitions method finds a fair partition for any given convex polygon and any given number of regions.•Our method relies on two well-known methods: Lloyd's algorithm and the Normal Flow Algorithm.•The method proposed in this article can be used in various contexts and many real-world applications.

Public schools in the United States offer tuition-free primary and secondary education to their students, and are divided into school districts funded by the local and state governments. Although the primary source of school district revenue is public money, several studies have pointed to the inequality in funding across different school districts. In this paper, we focus on the spatial geometry/distribution of such inequality, i.e., how the highly funded and lesser funded school districts are located relative to each other. Due to the major reliance on local property taxes for school funding, we find existing school district boundaries promoting financial segregation, with highly-funded school districts surrounded by lesser-funded districts and vice-versa. To counter such issues, we formally propose the Fair Partitioning problem to divide a given set of schools into k districts such that the spatial inequality in the district-level funding is minimized. However, the Fair Partitioning problem turns out to be computationally challenging, and we formally show that it is strongly -complete. We further provide a greedy algorithm to offer practical solution to Fair Partitioning, and show its effectiveness in lowering spatial inequality in school district funding across different states in the United States.

A pizza is a pair of planar convex bodies \(A\subseteq B\), where B represents the dough and A the topping of the pizza. A partition of a pizza by straight lines is a succession of double operations: a cut by a full straight line, followed by a Euclidean move of one of the resulting pieces; then the procedure is repeated. The final partition is said to be fair if each resulting slice has the same amount of A and the same amount of B. This note proves that, given an integer \(n\ge 2\), there exists a fair partition by straight lines of any pizza (A, B) into n parts if and only if n is even. The proof uses the following result: For any planar convex bodies A, B with \(A\subseteq B\), and any \(\alpha \in \,]0,\frac{1}{2}[\,\), there exists an \(\alpha \)-section of A which is a \(\beta \)-section of B for some \(\beta \ge \alpha \). (An \(\alpha \)-section of A is a straight line cutting A into two parts, one of which has area \(\alpha |A|\)). The question remains open if the word “planar” is dropped.

When a cache is shared by multiple cores, its space may be allocated either by sharing, partitioning, or both. We call the last case partition-sharing. This paper studies partition-sharing as a general solution, and presents a theory an technique for optimizing partition-sharing. We present a theory and a technique to optimize partition sharing. The theory shows that the problem of partition-sharing is reducible to the problem of partitioning. The technique uses dynamic programming to optimize partitioning for overall miss ratio, and for two different kinds of fairness. Finally, the paper evaluates the effect of optimal cache sharing and compares it with conventional solutions for thousands of 4-program co-run groups, with nearly 180 million different ways to share the cache by each co-run group. Optimal partition-sharing is on average 26% better than free-for-all sharing, and 98% better than equal partitioning. We also demonstrate the trade-off between optimal partitioning and fair partitioning.

An efficient aqueous two phase systems using dual inorganic electrolytes to separate 1,3-propanediol from the fermented broth

This paper presents a detailed study of fairness in cache sharing between threads in a chip multiprocessor (CMP) architecture. Prior work in CMP architectures has only studied throughput optimization techniques for a shared cache. The issue of fairness in cache sharing, and its relation to throughput, has not been studied. Fairness is a critical issue because the operating system (OS) thread scheduler's effectiveness depends on the hardware to provide fair cache sharing to co-scheduled threads. Without such hardware, serious problems, such as thread starvation and priority inversion, can arise and render the OS scheduler ineffective. This paper makes several contributions. First, it proposes and evaluates five cache fairness metrics that measure the degree of fairness in cache sharing, and shows that two of them correlate very strongly with the execution-time fairness. Execution-time fairness is defined as how uniform the execution times of co-scheduled threads are changed, where each change is relative to the execution time of the same thread running alone. Secondly, using the metrics, the paper proposes static and dynamic L2 cache partitioning algorithms that optimize fairness. The dynamic partitioning algorithm is easy to implement, requires little or no profiling, has low overhead, and does not restrict the cache replacement algorithm to LRU. The static algorithm, although requiring the cache to maintain LRU stack information, can help the OS thread scheduler to avoid cache thrashing. Finally, this paper studies the relationship between fairness and throughput in detail. We found that optimizing fairness usually increases throughput, while maximizing throughput does not necessarily improve fairness. Using a set of co-scheduled pairs of benchmarks, on average our algorithms improve fairness by a factor of 4/spl times/, while increasing the throughput by 15%, compared to a nonpartitioned shared cache.

In 3GPP LTE-Advanced networks deployed with type I relay nodes (RNs), resource partition is required to support in-band relaying. This paper focuses on how to partition system resources in order to attain improved fairness and efficiency. We first formulate the generalized proportional fair (GPF) resource allocation problem to provide fairness for all users served by the evolved node B (eNB) and its subordinate RNs. Assuming traditional proportional fair scheduling is executed independently at the eNB and each RN to achieve local fairness, we propose the proportional fair resource partition algorithm to tackle the GPF problem and ensure global fairness. Through system level simulations, the proposed algorithm is evaluated and compared with both non-relaying and relaying systems with the fixed resource partition approach. Simulation results demonstrate that the proposed algorithm can achieve a good trade-off between system throughput and fairness performance.

We conduct a computational analysis of fair and optimal partitions in additively separable hedonic games. We show that, for strict preferences, a Pareto optimal partition can be found in polynomial time while verifying whether a given partition is Pareto optimal is coNP-complete, even when preferences are symmetric and strict. Moreover, computing a partition with maximum egalitarian or utilitarian social welfare or one which is both Pareto optimal and individually rational is NP-hard. We also prove that checking whether there exists a partition which is both Pareto optimal and envy-free is Σ2p-complete. Even though an envy-free partition and a Nash stable partition are both guaranteed to exist for symmetric preferences, checking whether there exists a partition which is both envy-free and Nash stable is NP-complete.

Poor I/O performance can prevent an application from scaling to a large number of nodes even if the computation is parallelized appropriately. Therefore, improving I/O performance of large-scale parallel applications is very important. Caching recently and frequently accessed I/O blocks in memory is a widely used technique for improving I/O performance of these applications on high-end machines. However, simultaneous storage cache accesses of multiple applications may lead to unacceptable degradations in application performance due to interferences at the storage cache layer. As a result, efficient management of storage cache space across multiple I/O servers among competing applications is critical in order to ensure performance quality of service (QoS) to individual applications. In this paper, we propose a novel two-step approach to the management of the storage caches to provide predictable performance in multi-server storage architectures: (1)An adaptive QoS decomposition and optimization step uses max-flow algorithm to determine the best decomposition of application-level QoS to sub-QoSs such that the application performance is optimized, and (2) A storage cache allocation step uses feedback control theory to allocates hared storage cache space such that the specified QoSs are satisfied throughout the execution. Our experimental evaluation indicates that, on an average, our approach improves the I/Othroughput of applications by 48.6%, 29.2%, and 20.7%, respectively, over the uncontrolled partitioning, fair share and uniform decomposition schemes. We also observed 31.4%, 20.2%, and 44.7% improvements by our approach, in our global metric, called the fair speedup metric, against the fair share, uncontrolled partitioning and uniform decomposition schemes, respectively.

In heterogeneous networks, interference management is a crucial issue. Resource partitioning is one of the solution of enhanced intercell interference coordination (eICIC) in LTE-A. In this paper, we propose a bargaining-based resource partitioning. We model a net profit model considering service profit as well as operating expense, and analyze the Nash bargaining solution (NBS). For solving the NBS, we consider two algorithms: concurrent and sequential bargaining. In sequential bargaining, the NBS is dominated by a bargaining order. To mitigate this effect, we further study a bargaining power adaptation scheme. We verify the performance of the NBS and the bargaining power adaptation scheme by means of simulations. It is shown that proposed NBS can maintain fairness with little throughput degradation. Our results can be applied as a guideline for fair resource partitioning with economic efficiency.

Trade fairs and exhibitions have been around for centuries. They provide an important meeting point for sellers and potential buyers, be it for cars or clothes. Some fair organizers partition the exhibition space into predefined units, which may limit the flexibility. A free layout of the space allows for any shape or size of the stands and gives the exhibitor a better chance at creating something spectacular that will attract attention. Providing full flexibility, and thus a free layout, means that every new fair will be different. In addition to each individual stand having a varying size, the shape is not constrained to be rectangular, but might be, e.g., triangular or elliptic. All of this puts high demands on the production of the fairs. For example, the location and outline of every stand has to be determined and marked on the floor before the construction phase. This article gives a real world example of how robot technology can be used to automate the marking process for exhibitions and fairs in a way that both reduces the marking time and improves the working conditions. A mobile robot system equipped with a laser scanner and an ink jet printer is presented. The focus of the article is the description of the marking process and the system design. In addition, the requirements for such a system are outlined as well as results from over three years of operation and an analysis for commercialization.

Over the past few decades, increased demand of highly sophisticated real-time applications with complex functionalities has directly led to exponentially increased power consumption and significantly elevated system temperatures. These elevated temperature and thermal variations present formidable challenges towards system reliability, performance, cooling cost and leakages. This article explores the thermal management strength of two fairness based algorithms, namely Proportional Fair (PFair) and Deadline Partitioning Fair (DP-Fair). In related literature, the introduction of fairness is often considered as a tool to achieve optimality in multiprocessor scheduling algorithms. This work shows that these algorithms bring about better thermal profile when compared with the commonly used Earliest Deadline First (EDF) algorithm in similar conditions both in uniprocessor and multiprocessor environments. A simulation is conducted for periodic task set model. The obtained results are encouraging and show that use of fairness based algorithms reduces the operating temperature, peak temperature, and thermal variations.

In a multi-processing environment extra misses are endured in the last level shared cache because each process is trying to utilize the whole cache space, resulting in high interference between their working sets, and thus a degraded performance. In this paper, we propose a new cache partitioning scheme, which opposed to prior partitioning schemes in the literature, offers both high performance and fair allocation even with increasing number of simultaneously running processes and decreased cache associativity. The proposed scheme takes advantage of the broad number of sets in the cache by partitioning it dynamically set-wise. The partitioning presented is adaptive to accommodate to the changing requirements of the mix of running processes without the need for any offline information and with an eye on fair allocation of cache resources. Performance is evaluated by experimentation on different cache sizes with different associativity and comparison with the Least Recently Used baseline cache and partitioning schemes. We experimented on mixes from the Splash Benchmark Suite. The results shows a performance speedup up to 44% with an average speedup of 15% over all mixes, and also fairness improvement up to 46% in terms of the weighted harmonic mean, with an average of 14% over all mixes.

The Citizens Broadband Radio Service (CBRS) band has three tiers of users. The General Authorized Access (GAA) has the lowest priority and operates without a license. A Spectrum Access System (SAS) is responsible for managing spectrum among the users. The Federal Communications Commission (FCC) rules specify that GAA users be allocated resources in a fair manner, but the method to do so is left open. It is envisioned that there will be multiple SAS operators in a given region. Thus, fair partitioning of resources among SASs and fair allocation of spectrum to GAA users are important factors. In this paper, we present methods to partition resources among SASs and mechanisms that can be used by a SAS to allocate resources to a GAA user to ensure fairness.

Over the last decade, hyperspectral images (HSIs) have attracted considerable attention in the remote sensing community, making HSI classification and segmentation a major concern. Convolutional neural networks (CNNs) have been widely used in HSI classification and segmentation due to their effectiveness and efficiency in feature extraction and expression. However, traditional patch-wise classification methods may easily lead to information leakage, rendering overoptimistic results and making it imperative to address data leakage issues to ensure classification performance. In this paper, we propose a novel data partitioning strategy along with a lightweight, attention-aided CNN-based model for HSI classification without any information leakage. Not only does the proposed data partition method avoid information leakage, but it also generates balanced data splits even if the training samples are limited and datasets are imbalanced. Though our model only exploits spectral features instead of extracting spectral-spatial information, it is very effective and efficient since there is abundant spectral information; moreover, the spatial information is not easy to utilize effectively due to the limited availability of training samples and low spatial resolution in datasets. Experimental results, collected by applying the proposed novel data partitioning scheme and attention-aided CNN networks to three standard hyperspectral datasets, outperforms the state-of-the-art CNN-based models without any information leakage.