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1

Hannah Honghua Yang and D. F. Wong. "Balanced partitioning." IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 15, no. 12 (1996): 1533–40. http://dx.doi.org/10.1109/43.552086.

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Andreev, Konstantin, and Harald Racke. "Balanced Graph Partitioning." Theory of Computing Systems 39, no. 6 (2006): 929–39. http://dx.doi.org/10.1007/s00224-006-1350-7.

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Avin, Chen, Marcin Bienkowski, Andreas Loukas, Maciej Pacut, and Stefan Schmid. "Dynamic Balanced Graph Partitioning." SIAM Journal on Discrete Mathematics 34, no. 3 (2020): 1791–812. http://dx.doi.org/10.1137/17m1158513.

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4

Robertson, Blair, Trent McDonald, Chris Price, and Jennifer Brown. "Halton iterative partitioning: spatially balanced sampling via partitioning." Environmental and Ecological Statistics 25, no. 3 (2018): 305–23. http://dx.doi.org/10.1007/s10651-018-0406-6.

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5

Dahlhaus, Elias, Jan Kratochvíl, Paul D. Manuel, and Mirka Miller. "Transversal partitioning in balanced hypergraphs." Discrete Applied Mathematics 79, no. 1-3 (1997): 75–89. http://dx.doi.org/10.1016/s0166-218x(97)00034-6.

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6

Soltan, Saleh, Mihalis Yannakakis, and Gil Zussman. "Doubly Balanced Connected Graph Partitioning." ACM Transactions on Algorithms 16, no. 2 (2020): 1–24. http://dx.doi.org/10.1145/3381419.

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7

El Moussawi, Adnan, Nacera Bennacer Seghouani, and Francesca Bugiotti. "BGRAP: Balanced GRAph Partitioning Algorithm for Large Graphs." Journal of Data Intelligence 2, no. 2 (2021): 116–35. http://dx.doi.org/10.26421/jdi2.2-2.

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The definition of effective strategies for graph partitioning is a major challenge in distributed environments since an effective graph partitioning allows to considerably improve the performance of large graph data analytics computations. In this paper, we propose a multi-objective and scalable Balanced GRAph Partitioning (\algo) algorithm, based on Label Propagation (LP) approach, to produce balanced graph partitions. \algo defines a new efficient initialization procedure and different objective functions to deal with either vertex or edge balance constraints while considering edge direction in graphs. \algo is implemented of top of the open source distributed graph processing system Giraph. The experiments are performed on various graphs with different structures and sizes (going up to 50.6M vertices and 1.9B edges) while varying the number of partitions. We evaluate \algo using several quality measures and the computation time. The results show that \algo (i) provides a good balance while reducing the cuts between the different computed partitions (ii) reduces the global computation time, compared to LP-based algorithms.
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Aydin, Kevin, MohammadHossein Bateni, and Vahab Mirrokni. "Distributed Balanced Partitioning via Linear Embedding †." Algorithms 12, no. 8 (2019): 162. http://dx.doi.org/10.3390/a12080162.

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Balanced partitioning is often a crucial first step in solving large-scale graph optimization problems, for example, in some cases, a big graph can be chopped into pieces that fit on one machine to be processed independently before stitching the results together, leading to certain suboptimality from the interaction among different pieces. In other cases, links between different parts may show up in the running time and/or network communications cost, hence the desire to have small cut size. We study a distributed balanced-partitioning problem where the goal is to partition the vertices of a given graph into k pieces so as to minimize the total cut size. Our algorithm is composed of a few steps that are easily implementable in distributed computation frameworks such as MapReduce. The algorithm first embeds nodes of the graph onto a line, and then processes nodes in a distributed manner guided by the linear embedding order. We examine various ways to find the first embedding, for example, via a hierarchical clustering or Hilbert curves. Then we apply four different techniques including local swaps, and minimum cuts on the boundaries of partitions, as well as contraction and dynamic programming. As our empirical study, we compare the above techniques with each other, and also to previous work in distributed graph algorithms, for example, a label-propagation method, FENNEL and Spinner. We report our results both on a private map graph and several public social networks, and show that our results beat previous distributed algorithms: For instance, compared to the label-propagation algorithm, we report an improvement of 15–25% in the cut value. We also observe that our algorithms admit scalable distributed implementation for any number of partitions. Finally, we explain three applications of this work at Google: (1) Balanced partitioning is used to route multi-term queries to different replicas in Google Search backend in a way that reduces the cache miss rates by ≈ 0.5 % , which leads to a double-digit gain in throughput of production clusters. (2) Applied to the Google Maps Driving Directions, balanced partitioning minimizes the number of cross-shard queries with the goal of saving in CPU usage. This system achieves load balancing by dividing the world graph into several “shards”. Live experiments demonstrate an ≈ 40 % drop in the number of cross-shard queries when compared to a standard geography-based method. (3) In a job scheduling problem for our data centers, we use balanced partitioning to evenly distribute the work while minimizing the amount of communication across geographically distant servers. In fact, the hierarchical nature of our solution goes well with the layering of data center servers, where certain machines are closer to each other and have faster links to one another.
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Borgwardt, Steffen, and Shmuel Onn. "Efficient solutions for weight-balanced partitioning problems." Discrete Optimization 21 (August 2016): 71–84. http://dx.doi.org/10.1016/j.disopt.2016.06.001.

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Conforti, Michele, Marco Di Summa, and Giacomo Zambelli. "Minimally Infeasible Set-Partitioning Problems with Balanced Constraints." Mathematics of Operations Research 32, no. 3 (2007): 497–507. http://dx.doi.org/10.1287/moor.1070.0250.

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11

Yang, Chuan-Kai. "An Optimal Balanced Partitioning of a Set of 1D Intervals." International Journal of Artificial Life Research 1, no. 2 (2010): 72–79. http://dx.doi.org/10.4018/jalr.2010040106.

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Given a set of 1D intervals and a desired partition number, in this paper, the author examines how to make an optimal partitioning of these intervals, such that the number of intervals between the largest partition and smallest partition is minimal among all possible partitioning schemes. This problem has its difficulty due to the fact that an interval “striding” multiple partitions should be counted multiple times. Previously the author proposed an approximated solution to this problem by employing a simulated annealing approach (Yang & Chiueh, 2006), which could give satisfactory results in most cases; however, there is no theoretical guarantee on its optimality. This paper proposes a method that could both optimally and deterministically partition a given set of 1D intervals into a given number of partitions. The author shows that some load balancing problems could also be formulated as a balanced interval partitioning problem.
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Gong, Zhi Xun, Wei Qin Tong, and Ying Li. "Automatic Load-Balanced Partitioning Strategy in Parallel Multilevel Fast Multipole Algorithm." Applied Mechanics and Materials 182-183 (June 2012): 893–97. http://dx.doi.org/10.4028/www.scientific.net/amm.182-183.893.

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In this paper, we propose an automatic load-balanced partitioning strategy for parallel multilevel fast multipole algorithm(MLFMA) based on distributed-memory architectures to solve the large scale electromagnetic scattering problems. We focus on the automatic load-balancing partitioning strategy because that our original scheme requires that users input the transition level, which is usually determined on the users’ experience and sometimes lead to a bad load-balancing partition. By introducing the automatic load-balancing algorithm to our pervious partitioning technique, our implementation can automatically achieve the best load-balancing and consequently attain better parallel efficiency. To present the effectiveness of the new strategy, we analyze results of the previous implementation according to different inputs of transition level and compare them with the result of implementation using the new algorithm.
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13

Geiser, Georg, and Wolfgang Schröder. "Structured multi-block grid partitioning using balanced cut trees." Journal of Parallel and Distributed Computing 138 (April 2020): 139–52. http://dx.doi.org/10.1016/j.jpdc.2019.12.010.

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Avdiukhin, Dmitrii, Sergey Pupyrev, and Grigory Yaroslavtsev. "Multi-dimensional balanced graph partitioning via projected gradient descent." Proceedings of the VLDB Endowment 12, no. 8 (2019): 906–19. http://dx.doi.org/10.14778/3324301.3324307.

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15

Caraballo, Luis Evaristo, José-Miguel Díaz-Báñez, and Nadine Kroher. "A polynomial algorithm for balanced clustering via graph partitioning." European Journal of Operational Research 289, no. 2 (2021): 456–69. http://dx.doi.org/10.1016/j.ejor.2020.07.031.

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16

Caldwell, Andrew E., Andrew B. Kahng, and Igor L. Markov. "Iterative Partitioning with Varying Node Weights." VLSI Design 11, no. 3 (2000): 249–58. http://dx.doi.org/10.1155/2000/15862.

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The balanced partitioning problem divides the nodes of a [hyper]graph into groups of approximately equal weight (i.e., satisfying balance constraints) while minimizing the number of [hyper]edges that are cut (i.e., adjacent to nodes in different groups). Classic iterative algorithms use the pass paradigm [24] in performing single-node moves [16, 13] to improve the initial solution. To satisfy particular balance constraints, it is usual to require that intermediate solutions satisfy the constraints. Hence, many possible moves are rejected.Hypergraph partitioning heuristics have been traditionally proposed for and evaluated on hypergraphs with unit node weights only. Nevertheless, many real-world applications entail varying node weights, e.g., VLSI circuit partitioning where node weight typically represents cell area. Even when multilevel partitioning [3] is performed on unit-node-weight hypergraphs, intermediate clustered hypergraphs have varying node weights. Nothing prevents the use of conventional move-based heuristics when node weights vary, but their performance deteriorates, as shown by our analysis of partitioning results in [1].We describe two effects that cause this deterioration and propose simple modifications of well-known algorithms to address them. Our baseline implementations achieve dramatic improvements over previously reported results (by factors of up to 25); explicitly addressing the described harmful effects provides further improvement. Overall results are superior to those of the PROP-REXest algorithm reported in [14], which addresses similar problems.
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17

Velupillai, Viveka. "Partitioning the timeline." Studies in Language 40, no. 1 (2016): 93–136. http://dx.doi.org/10.1075/sl.40.1.04vel.

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This article presents the findings of a cross-linguistic survey of tense. In an areally and genetically balanced sample, 318 languages were investigated for whether they have tense and, if so, how they partition the timeline with respect to the deictic centre. Three quarters of the languages have tense: the majority partition the timeline into three sections: before, during and after the deictic centre (effectively past, present and future tense). Those languages with only two tenses most commonly have future/nonfuture tense. Interestingly, a group of languages have only one tense, the majority of them the future. This might indicate that there is a stronger motivation for the future tense to grammaticalize than for other tenses, mirroring a real/unreal world divide: real world events are easier to characterize through aspect than events that are yet to happen, which might create a need for a device that locates an event in future time.
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18

Tözün, Pınar, Ippokratis Pandis, Ryan Johnson, and Anastasia Ailamaki. "Scalable and dynamically balanced shared-everything OLTP with physiological partitioning." VLDB Journal 22, no. 2 (2012): 151–75. http://dx.doi.org/10.1007/s00778-012-0278-6.

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19

Zhang, Jilian, Kaimin Wei, and Xuelian Deng. "Heuristic algorithms for diversity-aware balanced multi-way number partitioning." Pattern Recognition Letters 136 (August 2020): 56–62. http://dx.doi.org/10.1016/j.patrec.2020.05.022.

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20

Feldmann, Andreas Emil. "Fast balanced partitioning is hard even on grids and trees." Theoretical Computer Science 485 (May 2013): 61–68. http://dx.doi.org/10.1016/j.tcs.2013.03.014.

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21

Leng, Yonglin, Chen Zhikui, Fangming Zhong, Xiongjiu Li, Yueming Hu, and Chao Yang. "BRGP: a balanced RDF graph partitioning algorithm for cloud storage." Concurrency and Computation: Practice and Experience 29, no. 14 (2016): e3896. http://dx.doi.org/10.1002/cpe.3896.

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22

Benlic, Una, and Jin-Kao Hao. "An effective multilevel tabu search approach for balanced graph partitioning." Computers & Operations Research 38, no. 7 (2011): 1066–75. http://dx.doi.org/10.1016/j.cor.2010.10.007.

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23

Balasubramaniam, Karthikeyan, Abdlmnam Abdlrahem, Ramtin Hadidi, and Elham B. Makram. "Balanced, non-contiguous partitioning of power systems considering operational constraints." Electric Power Systems Research 140 (November 2016): 456–63. http://dx.doi.org/10.1016/j.epsr.2016.06.001.

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24

Yang, Wenyin, Guojun Wang, Kim-Kwang Raymond Choo, and Shuhong Chen. "HEPart: A balanced hypergraph partitioning algorithm for big data applications." Future Generation Computer Systems 83 (June 2018): 250–68. http://dx.doi.org/10.1016/j.future.2018.01.009.

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25

Zou, Jian, De Min Li, and Min Zhang. "An Energy-Balanced Routing Algorithm for ZigBee Audio Guide System in Ad Hoc Social Network." Advanced Engineering Forum 6-7 (September 2012): 1177–82. http://dx.doi.org/10.4028/www.scientific.net/aef.6-7.1177.

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In order to avoid the network partitioning and nodes died too early in ZigBee audio guide system, we designed an energy-balanced routing algorithm in this paper, which can manage the rest energy of all network nodes effectively. The simulation results indicate that this algorithm can balance the entire network energy, extend the survival time of the whole network and increase the stability of the system relative to the original routing protocol.
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Ji, Shengwei, Chenyang Bu, Lei Li, and Xindong Wu. "Local Graph Edge Partitioning." ACM Transactions on Intelligent Systems and Technology 12, no. 5 (2021): 1–25. http://dx.doi.org/10.1145/3466685.

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Graph edge partitioning, which is essential for the efficiency of distributed graph computation systems, divides a graph into several balanced partitions within a given size to minimize the number of vertices to be cut. Existing graph partitioning models can be classified into two categories: offline and streaming graph partitioning models. The former requires global graph information during the partitioning, which is expensive in terms of time and memory for large-scale graphs. The latter creates partitions based solely on the received graph information. However, the streaming model may result in a lower partitioning quality compared with the offline model. Therefore, this study introduces a Local Graph Edge Partitioning model, which considers only the local information (i.e., a portion of a graph instead of the entire graph) during the partitioning. Considering only the local graph information is meaningful because acquiring complete information for large-scale graphs is expensive. Based on the Local Graph Edge Partitioning model, two local graph edge partitioning algorithms—Two-stage Local Partitioning and Adaptive Local Partitioning—are given. Experimental results obtained on 14 real-world graphs demonstrate that the proposed algorithms outperform rival algorithms in most tested cases. Furthermore, the proposed algorithms are proven to significantly improve the efficiency of the real graph computation system GraphX.
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Ma, Yung-Cheng, Chung-Ping Chung, and Tien-Fu Chen. "Load and storage balanced posting file partitioning for parallel information retrieval." Journal of Systems and Software 84, no. 5 (2011): 864–84. http://dx.doi.org/10.1016/j.jss.2011.01.028.

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Mehrdoost, Zahra, and Seyed Saied Bahrainian. "A multilevel tabu search algorithm for balanced partitioning of unstructured grids." International Journal for Numerical Methods in Engineering 105, no. 9 (2015): 678–92. http://dx.doi.org/10.1002/nme.5003.

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Lum, Oliver, Carmine Cerrone, Bruce Golden, and Edward Wasil. "Partitioning a street network into compact, balanced, and visually appealing routes." Networks 69, no. 3 (2017): 290–303. http://dx.doi.org/10.1002/net.21730.

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Miyazawa, Flávio K., Phablo F. S. Moura, Matheus J. Ota, and Yoshiko Wakabayashi. "Partitioning a graph into balanced connected classes: Formulations, separation and experiments." European Journal of Operational Research 293, no. 3 (2021): 826–36. http://dx.doi.org/10.1016/j.ejor.2020.12.059.

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Dzamic, Dusan, Bojana Cendic, Miroslav Maric, and Aleksandar Djenic. "Solving balanced multi-weighted attribute set partitioning problem with variable neighborhood search." Filomat 33, no. 9 (2019): 2875–91. http://dx.doi.org/10.2298/fil1909875d.

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This paper considers the Balanced Multi-Weighted Attribute Set Partitioning (BMWASP) problem which requires finding a partition of a given set of objects with multiple weighted attributes into a certain number of groups so that each attribute is evenly distributed amongst the groups. Our approach is to define an appropriate criterion allowing to compare the degree of deviation from the ?perfect balance? for different partitions and then produce the partition that minimizes this criterion. We have proposed a mathematical model for the BMWASP and its mixed-integer linear reformulation. We evaluated its efficiency through a set of computational experiments. To solve instances of larger problem dimensions, we have developed a heuristic method based on a Variable Neighborhood Search (VNS). A local search procedure with efficient fast swap-based local search is implemented in the proposed VNS-based approach. Presented computational results show that the proposed VNS is computationally efficient and quickly reaches all optimal solutions for smaller dimension instances obtained by exact solver and provide high-quality solutions on large-scale problem instances in short CPU times.
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ANDERSSON, MATTIAS, JOACHIM GUDMUNDSSON, CHRISTOS LEVCOPOULOS, and GIRI NARASIMHAN. "BALANCED PARTITION OF MINIMUM SPANNING TREES." International Journal of Computational Geometry & Applications 13, no. 04 (2003): 303–16. http://dx.doi.org/10.1142/s0218195903001190.

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To better handle situations where additional resources are available to carry out a task, many problems from the manufacturing industry involve dividing a task into a number of smaller tasks, while optimizing a specific objective function. In this paper we consider the problem of partitioning a given set [Formula: see text] of n points in the plane into k subsets, [Formula: see text], such that [Formula: see text] is minimized. Variants of this problem arise in applications from the shipbuilding industry. We show that this problem is NP-hard, and we also present an approximation algorithm for the problem, in the case when k is a fixed constant. The approximation algorithm runs in time O(n log n) and produces a partition that is within a factor (4/3+ε) of the optimal if k=2, and a factor (2+ε) otherwise.
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Kurian, Annie P., and V. Jeyabalaraja. "A Survey on Analyzing and Processing Data Faster Based on Balanced Partitioning." International Journal of Data Mining Techniques and Applications 4, no. 2 (2015): 78–81. http://dx.doi.org/10.20894/ijdmta.102.004.002.007.

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34

Pande, Santosh, and Tareq Bali. "A Computation+Communication Load Balanced Loop Partitioning Method for Distributed Memory Systems." Journal of Parallel and Distributed Computing 58, no. 3 (1999): 515–45. http://dx.doi.org/10.1006/jpdc.1999.1567.

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Yang, Zhipeng, Rongrong Zheng, and Yinglong Ma. "Parallel Heuristics for Balanced Graph Partitioning Based on Richness of Implicit Knowledge." IEEE Access 7 (2019): 96444–54. http://dx.doi.org/10.1109/access.2019.2926753.

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Matty, K. R., and L. P. Kondi. "Balanced multiple description video coding using optimal partitioning of the DCT coefficients." IEEE Transactions on Circuits and Systems for Video Technology 15, no. 7 (2005): 928–34. http://dx.doi.org/10.1109/tcsvt.2005.848343.

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Sangamuang, Sumalee, Pruet Boonma, Juggapong Natwichai, and Wanpracha Art Chaovalitwongse. "Impact of minimum-cut density-balanced partitioning solutions in distributed webpage ranking." Optimization Letters 14, no. 3 (2019): 521–33. http://dx.doi.org/10.1007/s11590-019-01399-9.

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38

FLORIDO, J. P., H. POMARES, and I. ROJAS. "GENERATING BALANCED LEARNING AND TEST SETS FOR FUNCTION APPROXIMATION PROBLEMS." International Journal of Neural Systems 21, no. 03 (2011): 247–63. http://dx.doi.org/10.1142/s0129065711002791.

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In function approximation problems, one of the most common ways to evaluate a learning algorithm consists in partitioning the original data set (input/output data) into two sets: learning, used for building models, and test, applied for genuine out-of-sample evaluation. When the partition into learning and test sets does not take into account the variability and geometry of the original data, it might lead to non-balanced and unrepresentative learning and test sets and, thus, to wrong conclusions in the accuracy of the learning algorithm. How the partitioning is made is therefore a key issue and becomes more important when the data set is small due to the need of reducing the pessimistic effects caused by the removal of instances from the original data set. Thus, in this work, we propose a deterministic data mining approach for a distribution of a data set (input/output data) into two representative and balanced sets of roughly equal size taking the variability of the data set into consideration with the purpose of allowing both a fair evaluation of learning's accuracy and to make reproducible machine learning experiments usually based on random distributions. The sets are generated using a combination of a clustering procedure, especially suited for function approximation problems, and a distribution algorithm which distributes the data set into two sets within each cluster based on a nearest-neighbor approach. In the experiments section, the performance of the proposed methodology is reported in a variety of situations through an ANOVA-based statistical study of the results.
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Nakajima, Y., Y. Kawano, H. Sekigawa, M. Nakanishi, S. Yamashita, and Y. Nakashima. "Synthesis of quantum circuits for $d$-level systems by using cosine-sine." Quantum Information and Computation 9, no. 5&6 (2009): 423–43. http://dx.doi.org/10.26421/qic9.5-6-6.

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We study the problem of designing minimal quantum circuits for any operations on $n$ qudits by means of the cosine-sine decomposition. Our method is based on a divide-and-conquer strategy. In that strategy, the size of the produced quantum circuit depends on whether the partitioning is balanced. We provide a new cosine-sine decomposition based on a balanced partitioning for $d$-level systems. The produced circuit is not asymptotically optimal except when $d$ is a power of two, but, when the number of qudits $n$ is small, our method can produce the smallest quantum circuit compared to the circuits produced by other synthesis methods. For example, when $d=3$ (three-level systems) and $n=2$ (two qudits), then the number of two-qudit operations called CINC, which is a generalized versions of CNOT, is 36 whereas the previous method needs 156 CINC gates. Moreover, we show that our method is useful for designing a polynomial-size quantum circuit for the radix-$d$ quantum Fourier transform.
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Kim, Yumi, Seungmin Han, Miji Yeom, et al. "Balanced Nucleocytosolic Partitioning Defines a Spatial Network to Coordinate Circadian Physiology in Plants." Developmental Cell 26, no. 1 (2013): 73–85. http://dx.doi.org/10.1016/j.devcel.2013.06.006.

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Onuekwusi, Nnaemeka Chiemezie, Michael Chukwudi Ndinechi, Gordon Chiagozie Ononiwu, and Onyebuchi Chikezie Nosiri. "An Energy Balanced Routing Hole and Network Partitioning Mitigation Model for Homogeneous Hierarchical Wireless Sensor Networks." International Journal of Interdisciplinary Telecommunications and Networking 12, no. 1 (2020): 28–42. http://dx.doi.org/10.4018/ijitn.2020010103.

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This article addresses the challenges of routing hole and network partitioning often experienced in hierarchical wireless sensor networks (WSNs). This developed model classifies network nodes into sets for effective energy management and formulates two cluster networks namely: switching and non-switching networks. Both networks are considered homogeneous and static WSNs and adopted approaches of residual energy, multi-hop and minimal distance as routing decision parameters. The switching network in addition introduces an energy switching factor as a major decision parameter for the switching of cluster head roles amongst cluster nodes. Network simulation was done using Truetime 2.0 and energy dissipation of the respective nodes and cluster heads was observed against a threshold. Results showed the introduction of the energy switching factor gave a significant energy balancing effect as nodes exhibited uniform energy dissipation. Furthermore, the residual energies for most nodes were above the threshold eliminating the possibility of the presence of routing hole and network partitioning.
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Belussi, Alberto, Sara Migliorini, and Ahmed Eldawy. "Skewness-Based Partitioning in SpatialHadoop." ISPRS International Journal of Geo-Information 9, no. 4 (2020): 201. http://dx.doi.org/10.3390/ijgi9040201.

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In recent years, several extensions of the Hadoop system have been proposed for dealing with spatial data. SpatialHadoop belongs to this group of projects and includes some MapReduce implementations of spatial operators, like range queries and spatial join. the MapReduce paradigm is based on the fundamental principle that a task can be parallelized by partitioning data into chunks and performing the same operation on them, (map phase), eventually combining the partial results at the end (reduce phase). Thus, the applied partitioning technique can tremendously affect the performance of a parallel execution, since it is the key point for obtaining balanced map tasks and exploiting the parallelism as much as possible. When uniformly distributed datasets are considered, this goal can be easily obtained by using a regular grid covering the whole reference space for partitioning the geometries of the input dataset; conversely, with skewed distributed datasets, this might not be the right choice and other techniques have to be applied. for instance, SpatialHadoop can produce a global index also by means of a Quadtree-based grid or an Rtree-based grid, which in turn are more expensive index structures to build. This paper proposes a technique based on both a box counting function and a heuristic, rooted on theoretical properties and experimental observations, for detecting the degree of skewness of an input spatial dataset and then deciding which partitioning technique to apply in order to improve as much as possible the performance of subsequent operations. Experiments on both synthetic and real datasets are presented to confirm the effectiveness of the proposed approach.
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Ababei, Cristinel. "Speeding Up FPGA Placement via Partitioning and Multithreading." International Journal of Reconfigurable Computing 2009 (2009): 1–9. http://dx.doi.org/10.1155/2009/514754.

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One of the current main challenges of the FPGA design flow is the long processing time of the placement and routing algorithms. In this paper, we propose a hybrid parallelization technique of the simulated annealing-based placement algorithm of VPR developed in the work of Betz and Rose (1997). The proposed technique uses balanced region-based partitioning and multithreading. In the first step of this approach placement subproblems are created by partitioning and then processed concurrently by multiple worker threads that are run on multiple cores of the same processor. Our main goal is to investigate the speedup that can be achieved with this simple approach compared to previous approaches that were based on distributed computing. The new hybrid parallel placement algorithm achieves an average speedup of2.5×using four worker threads, while the total wire length and circuit delay after routing are minimally degraded.
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44

Tan, Sheldon X. D., and C. J. Richard Shi. "Balanced multi-level multi-way partitioning of analog integrated circuits for hierarchical symbolic analysis." Integration 34, no. 1-2 (2003): 65–86. http://dx.doi.org/10.1016/s0167-9260(03)00002-6.

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Hada, Romas James, Hongyi Wu, and Miao Jin. "Scalable Minimum-Cost Balanced Partitioning of Large-Scale Social Networks: Online and Offline Solutions." IEEE Transactions on Parallel and Distributed Systems 29, no. 7 (2018): 1636–49. http://dx.doi.org/10.1109/tpds.2017.2694835.

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46

Suharjono, Amin, Wirawan, and Gamantyo Hendrantoro. "A NEW UNEQUAL CLUSTERING ALGORITHM USING ENERGY-BALANCED AREA PARTITIONING FOR WIRELESS SENSOR NETWORKS." International Journal on Smart Sensing and Intelligent Systems 6, no. 5 (2013): 1808–29. http://dx.doi.org/10.21307/ijssis-2017-616.

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47

Tavakkoli-Moghaddam, R., M. B. Aryanezhad, H. Kazemipoor, and A. Salehipour. "Partitioning machines in tandem AGV systems based on “balanced flow strategy” by simulated annealing." International Journal of Advanced Manufacturing Technology 38, no. 3-4 (2007): 355–66. http://dx.doi.org/10.1007/s00170-007-1094-9.

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48

Xu, Qin, and Jie Cao. "Semibalance Model in Terrain-Following Coordinates." Journal of the Atmospheric Sciences 69, no. 7 (2012): 2201–6. http://dx.doi.org/10.1175/jas-d-12-012.1.

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Abstract By partitioning the hydrostatically balanced flow into a nonlinearly balanced primary-flow part and a remaining secondary-flow part and then truncating the secondary-flow vorticity advection and stretching–tilting terms in the vector vorticity equation, the previous semibalance model (SBM) in pseudoheight coordinates is rederived in terrain-following pressure coordinates, called η coordinates. The involved truncation is topologically the same as that in pseudoheight coordinates but the truncated terms in η coordinates are not equivalent to those in pseudoheight coordinates. Because its potential vorticity (PV) is conserved and invertible, the rederived SBM is suitable for studying balanced dynamics via “PV thinking” in real weather events, such as slowly varying vortices and curved fronts in which the primary-flow velocity and secondary-flow vorticity are nearly parallel in η coordinates.
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Wang, Zecong, Hamid Parvin, Sultan Noman Qasem, Bui Anh Tuan, and Kim-Hung Pho. "Cluster ensemble selection using balanced normalized mutual information." Journal of Intelligent & Fuzzy Systems 39, no. 3 (2020): 3033–55. http://dx.doi.org/10.3233/jifs-191531.

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A bad partition in an ensemble will be removed by a cluster ensemble selection framework from the final ensemble. It is the main idea in cluster ensemble selection to remove these partitions (bad partitions) from the selected ensemble. But still, it is likely that one of them contains some reliable clusters. Therefore, it may be reasonable to apply the selection phase on cluster level. To do this, a cluster evaluation metric is needed. Some of these metrics have been recently introduced; each of them has its limitations. The weak points of each method have been addressed in the paper. Subsequently, a new metric for cluster assessment has been introduced. The new measure is named Balanced Normalized Mutual Information (BNMI) criterion. It balances the deficiency of the traditional NMI-based criteria. Additionally, an innovative cluster ensemble approach has been proposed. To create the consensus partition considering the elected clusters, a set of different aggregation-functions (called also consensus-functions) have been utilized: the ones which are based upon the co-association matrix (CAM), the ones which are based on hyper graph partitioning algorithms, and the ones which are based upon intermediate space. The experimental study indicates that the state-of-the-art cluster ensemble methods are outperformed by the proposed cluster ensemble approach.
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Kel’manov, A. V., A. V. Pyatkin, and V. I. Khandeev. "On the complexity of some partition problems of a finite set of points in Euclidean space into balanced clusters." Доклады Академии наук 488, no. 1 (2019): 16–20. http://dx.doi.org/10.31857/s0869-5652488116-20.

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We consider some problems of partitioning a finite set of N points in d-dimension Euclidean space into two clusters balancing the value of (1) the quadratic variance normalized by a cluster size, (2) the quadratic variance, and (3) the size-weighted quadratic variance. We have proved the NP-completeness of all these problems.
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