Partition Shape Research Articles

Guest Editorial: Special Issue on Algorithms and Computation

This special issue is dedicated to the themes of Algorithms and Computation. The eight papers in this issue cover a wide range of areas within the themes such as graph partitioning, graph matching, graph coloring, tree partitioning, graph minor theory, polynomial multiplication, algorithmic self-assembly, and the complexity of counting problems. The paper “Beyond good partition shapes: an analysis of diffusive graph partitioning” by Meyerhenke and Sauerwald (doi:10.1007/s00453-012-9666-y) studies the problem of graph partitioning and gives a theoretical analysis of the diffusionbased partitioning heuristic Bubble-FOS/C which yields good experimental results in practice. In the paper “Improved algorithms for even factors and square-free simple bmatchings” (doi:10.1007/s00453-012-9642-6), Babenko considers the problem of finding an even factor of maximum cardinality, which is a generalization of path matchings. He gives an improved algorithm to find a maximum cardinality even factor in an odd-cycle symmetric digraph. The paper “3-coloring AT-free graphs in polynomial time” by Stacho (doi:10.1007/ s00453-012-9624-8) deals with graph coloring and gives a polynomial time algorithm for the 3-coloring problem of asteroidal-triple-free graphs. In “Minimum cost partitions of trees with supply and demand” (doi:10.1007/ s00453-011-9573-7), Ito, Hara, Zhou and Nishizeki consider the problem of parti-

Beyond Good Partition Shapes: An Analysis of Diffusive Graph Partitioning

In this paper we study the prevalent problem of graph partitioning by analyzing the diffusion-based partitioning heuristic Bubble-FOS/C, a key component of a practical successful graph partitioner (Meyerhenke et al. in J. Parallel Distrib. Comput. 69(9):750---761, 2009). We begin by studying the disturbed diffusion scheme FOS/C, which computes the similarity measure used in Bubble-FOS/C and is therefore the most crucial component. By relating FOS/C to random walks, we obtain precise characterizations of the behavior of FOS/C on tori and hypercubes. Besides leading to new knowledge on FOS/C (and therefore also on Bubble-FOS/C), these characterizations have been recently used for the analysis of load balancing algorithms (Berenbrink et al. in Proceedings of the 22nd Annual Symposium on Discrete Algorithms, pp. 429---439, 2011). We then regard Bubble-FOS/C, which has been shown in previous experiments to produce solutions with good partition shapes and other favorable properties. In this paper we prove that it computes a relaxed solution to an edge cut minimizing binary quadratic program (BQP). This result provides the first substantial theoretical insight why Bubble-FOS/C yields good experimental results in terms of graph partitioning metrics. Moreover, we show that in bisections computed by Bubble-FOS/C, at least one of the two parts is connected. Using the aforementioned relation between FOS/C and random walks, we prove that in vertex-transitive graphs both parts must be connected components.

Tableaux and plane partitions of truncated shapes

We consider a new kind of straight and shifted plane partitions/Young tableaux – ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function.

The Measurement of Local Variation in Shape.

Geometric morphometrics comprises tools for measuring and analyzing shape as captured by an entire set of landmark configurations. Many interesting questions in evolutionary, genetic, and developmental research, however, are only meaningful at a local level, where a focus on "parts" or "traits" takes priority over properties of wholes. To study variational properties of such traits, current approaches partition configurations into subsets of landmarks which are then studied separately. This approach is unable to fully capture both variational and spatial characteristics of these subsets because interpretability of shape differences is context-dependent. Landmarks omitted from a partition usually contain information about that partition's shape. We present an interpolation-based approach that can be used to model shape differences at a local, infinitesimal level as a function of information available globally. This approach belongs in a large family of methods that see shape differences as continuous "fields" spanning an entire structure, for which landmarks serve as reference parameters rather than as data. We show, via analyses of simulated and real data, how interpolation models provide a more accurate representation of regional shapes than partitioned data. A key difference of this interpolation approach from current morphometric practice is that one must assume an explicit interpolation model, which in turn implies a particular kind of behavior of the regions between landmarks. This choice presents novel methodological challenges, but also an opportunity to incorporate and test biomechanical models that have sought to explain tissue-level processes underlying the generation of morphological shape.

Three-dimensional Imaging of Dam Structure and its Application

With many high-hazard potential dams and long river embankments in China, the task of reinforcement of hydraulic projects needs tremendous efforts. Thus, it is of great significance to carry out effective structure quality control and hidden hazard detection. Based on GSR-3D imaging system and dam engineering practice in China, three-dimensional imaging system for dam structure with China's national conditions was created through secondary development. The system had already been applied in several reservoirs. And the application results show that the system can well realize real-time presenting in all material partition shape of dam and has a high imaging ability. Therefore, the three-dimensional imaging of dam structure provides complete real-time analysis and imaging tools for the internal structure and the orientation of hidden trouble of dams and embankments, which can be used for detecting construction quality and safety hazard, for experts to determine the security state of engineering and for leadership to make decisions by providing the scientific basis.

Tableaux and plane partitions of truncated shapes (extended abstract)

We consider a new kind of straight and shifted plane partitions/Young tableaux — ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function. Nous considérons un nouveau type de partitions planes, ou de tableaux de Young, droits ou décalés, obtenus en privant leurs diagrammes de certaines cellules en haut à droite, et dans certains cas nous trouvons des formules d'énumération pour les tableaux standard. Les preuves impliquent le calcul de la fonction génératrice pour les partitions planes correspondantes, en utilisant des interprétations et des formules pour les sommes de fonctions de Schur restreintes et leurs spécialisations. Le nombre de tableaux standard est alors obtenu comme une certaine limite de cette fonction.

Improved Video Compression Efficiency Through Flexible Unit Representation and Corresponding Extension of Coding Tools

This paper proposes a novel video compression scheme based on a highly flexible hierarchy of unit representation which includes three block concepts: coding unit (CU), prediction unit (PU), and transform unit (TU). This separation of the block structure into three different concepts allows each to be optimized according to its role; the CU is a macroblock-like unit which supports region splitting in a manner similar to a conventional quadtree, the PU supports nonsquare motion partition shapes for motion compensation, while the TU allows the transform size to be defined independently from the PU. Several other coding tools are extended to arbitrary unit size to maintain consistency with the proposed design, e.g., transform size is extended up to 64 × 64 and intraprediction is designed to support an arbitrary number of angles for variable block sizes. Other novel techniques such as a new noncascading interpolation Alter design allowing arbitrary motion accuracy and a leaky prediction technique using both open-loop and closed-loop predictors are also introduced. The video codec described in this paper was a candidate in the competitive phase of the high-efficiency video coding (HEVC) standardization work. Compared to H.264/AVC, it demonstrated bit rate reductions of around 40% based on objective measures and around 60% based on subjective testing with 1080 p sequences. It has been partially adopted into the first standardization model of the collaborative phase of the HEVC effort.

Graph partitioning and disturbed diffusion

The NP -hard graph partitioning problem is an important subtask in load balancing and many other applications. It requires the division of a graph’s vertex set into P equally sized subsets such that some objective function is optimized. State-of-the-art libraries addressing this problem show several deficiencies: they are hard to parallelize, focus on small edge-cuts instead of few boundary vertices, and often produce disconnected partitions. This work introduces our novel graph partitioning and repartitioning heuristic Bubble-FOS/C. In contrast to other libraries, Bubble-FOS/C does not try to minimize the edge-cut explicitly, but focuses instead implicitly on good partition shapes. The shapes are optimized by diffusion processes that are embedded into an iterative framework. This approach incorporates a high degree of parallelism. Besides describing the evolution process that led to the new diffusion scheme FOS/C used by Bubble-FOS/C, we reveal some of FOS/C’s properties and propose a number of enhancements for a fast and reliable implementation. Our experiments, in which we compare sequential and parallel Bubble-FOS/C implementations to the state-of-the-art libraries Metis and Jostle, reveal that our new heuristic is slower, but generates high-quality solutions that are often superior.

A new diffusion-based multilevel algorithm for computing graph partitions

Graph partitioning requires the division of a graph’s vertex set into k equally sized subsets s. t. some objective function is optimized. High-quality partitions are important for many applications, whose objective functions are often N P -hard to optimize. Most state-of-the-art graph partitioning libraries use a variant of the Kernighan–Lin (KL) heuristic within a multilevel framework. While these libraries are very fast, their solutions do not always meet all user requirements. Moreover, due to its sequential nature, KL is not easy to parallelize. Its use as a load balancer in parallel numerical applications therefore requires complicated adaptations. That is why we developed previously an inherently parallel algorithm, called Bubble-FOS/C [H. Meyerhenke, B. Monien, S. Schamberger, Accelerating shape optimizing load balancing for parallel FEM simulations by algebraic multigrid, in: Proceedings of the 20th IEEE International Parallel and Distributed Processing Symposium, IPDPS’06, IEEE Computer Society, 2006, p. 57 (CD)], which optimizes partition shapes by a diffusive mechanism. However, it is too slow for practical use, despite its high solution quality. In this paper, besides proving that Bubble-FOS/C converges towards a local optimum of a potential function, we develop a much faster method for the improvement of partitionings. This faster method called TruncCons is based on a different diffusive process, which is restricted to local areas of the graph and also contains a high degree of parallelism. By coupling TruncCons with Bubble-FOS/C in a multilevel framework based on two different hierarchy construction methods, we obtain our new graph partitioning heuristic DibaP. Compared to Bubble-FOS/C, DibaP shows a considerable acceleration, while retaining the positive properties of the slower algorithm. Experiments with popular benchmark graphs show that DibaP computes consistently better results than the state-of-the-art libraries METIS and JOSTLE. Moreover, with our new algorithm, we have improved the best known edge-cut values for a significant number of partitionings of six widely used benchmark graphs.

Adaptive Global Elimination Algorithm for Low Power Motion Estimation

Motion estimation typically consumes 50% to 70% of total power in video encode application. Optimizing the power consumption of motion estimation process is of great importance to low power video applications. Power dissipation increases with computational complexity. Reduction in motion estimation complexity is usually associated with increase in bit rate and a loss of quality. We explore a set of algorithms that reduce the complexity of motion estimation by adaptively changing the matching complexity based on macro-block features yet have only a modest cost in terms of bit rate increase and quality loss. The adaptive techniques are applied to the global elimination algorithm, which is a well known motion estimation algorithm. The global elimination algorithm uses fixed partition sizes and shapes irrespective of the nature of the macro-block. We show that by adapting the partition sizes and shapes according to the macro-block features such as variance and Hadamard coefficients, the computational complexity of global elimination algorithm can be significantly reduced with only a small increase in bit rate. We also propose a novel center-biased search order that uses early termination method designed to work with the global elimination algorithm. The adaptive match and center-biased search together result in around 57% reduction in computational complexity and 50% reduction in power dissipation compared to the original global elimination algorithm.

A variation on the tableau switching and a Pak-Vallejo's conjecture

Pak and Vallejo have defined fundamental symmetry map as any Young tableau bijection for the commutativity of the Littlewood-Richardson coefficients $c_{\mu,\nu}^{\lambda}=c_{\nu, \mu}^{\lambda}$. They have considered four fundamental symmetry maps and conjectured that they are all equivalent (2004). The three first ones are based on standard operations in Young tableau theory and, in this case, the conjecture was proved by Danilov and Koshevoy (2005). The fourth fundamental symmetry, given by the author in (1999;2000) and reformulated by Pak and Vallejo, is defined by nonstandard operations in Young tableau theory and will be shown to be equivalent to the first one defined by the involution property of the Benkart-Sottile-Stroomer tableau switching. The proof of this equivalence provides, in the case the first tableau is Yamanouchi, a variation of the tableau switching algorithm which shows $\textit{switching}$ as an operation that takes two tableaux sharing a common border and moves them trough each other by decomposing the first tableau into a sequence of tableaux whose sequence of partition shapes defines a Gelfand-Tsetlin pattern. This property leads to a $\textit{jeu de taquin-chain sliding}$ on Littlewood-Richardson tableaux. Pak et Vallejo ont défini la transformation de la symétrie fondamentale comme une bijection de tableaux de Young pour la comutativité des coefficients de Littlewood-Richardson $c_{\mu,\nu}^{\lambda}=c_{\nu, \mu}^{\lambda}$. Ils ont considéré quatre bijections fondamentaux et ont conjecturé qu’elles sont équivalentes (2004). Les trois premières sont basées sur des opérations standard de la théorie des tableaux de Young et, dans ce cas, la conjecture a été confirmée par Danilov et Koshevoy (2005). La quatrième symétrie fondamentale, donnée par l’auteur (1999;2000) et reformulée par Pak et Vallejo, est définie par des opérations $\textit{nonstandard}$ dans la théorie des tableaux de Young. Cette bijection sera montrée équivalente à la première définie pour la propriété involutoire du $\textit{tableau switching}$ de Benkart-Sottile-Stroomer. La preuve de cette équivalence, dans le cas le premier tableau est de Yamanouchi, donne une variation du algorithme de $\textit{tableau switching}$ qui montre $\textit{switching}$ comme une opération qui prendre deux tableaux avec une même borde et meut un à travers de l’autre en décomposant le premier dans une séquence de tableaux dont la séquence des partitions des formats définit une diagramme de Gelfand-Tsetlin. Cette propriété conduit à un algorithme du type $\textit{jeu de taquin-glissements sur chaînes}$ pour les tableaux de Littlewood-Richardson.

Signed differential posets and sign-imbalance

We study signed differential posets, a signed version of differential posets. These posets satisfy enumerative identities which are signed analogues of those satisfied by differential posets. Our main motivations are the sign-imbalance identities for partition shapes originally conjectured by Stanley, now proven in [T. Lam, Growth diagrams, domino insertion and sign-imbalance, J. Combin. Theory Ser. A 107 (2004) 87–115; A. Reifergerste, Permutation sign under the Robinson–Schensted–Knuth correspondence, Ann. Comb. 8 (2004) 103–112; J. Sjöstrand, On the sign-imbalance of partition shapes, J. Combin. Theory Ser. A 111 (2005) 190–203]. We show that these identities result from a signed differential poset structure on Young's lattice, and explain similar identities for Fibonacci shapes.

Motion‐compensated video coding using sliced blocks

AbstractIn international standards for video coding such as MPEG‐4 and H.264, motion compensation is performed by partitioning macroblocks (square pixel sets) into multiple areas. In this paper, we propose motion compensation using sliced blocks as a new method for macroblock‐based motion compensation. The partition shape is not restricted to squares or other patterns; instead, macroblocks are sliced by arbitrary line segments. This produces better agreement in shape between partitions and moving objects, as well as better motion compensation performance. In addition, when partition data are described in bitstream, adaptive prediction of segments based on partition shape of neighbor macroblocks is applied, aiming at reduction of the overhead caused by partition data. In the proposed method, coding is performed at the macroblock level, and therefore, coding mode selection and other related algorithms of conventional rate distortion optimization can be used unaltered. The proposed method was applied to forward‐prediction picture coding (basic algorithm of H.264), and an improvement of coding efficiency by about 5% was confirmed. © 2007 Wiley Periodicals, Inc. Syst Comp Jpn, 38(7): 12– 22, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.20714

On the sign-imbalance of skew partition shapes

Let the sign of a skew standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. We examine how the sign property is transferred by the skew Robinson–Schensted correspondence invented by Sagan and Stanley. The result is a remarkably simple generalization of the ordinary non-skew formula. The sum of the signs of all standard tableaux on a given skew shape is the sign-imbalance of that shape. We generalize previous results on the sign-imbalance of ordinary partition shapes to skew ones.

On the sign-imbalance of partition shapes

Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2 ⌊ n / 2 ⌋ . We present a stronger theorem with a purely combinatorial proof using the Robinson–Schensted correspondence and a new concept called chess tableaux. We also prove a sharpening of another conjecture by Stanley concerning weighted sums of squares of sign-imbalances. The proof is built on a remarkably simple relation between the sign of a permutation and the signs of its RS-corresponding tableaux.

Optimizing overall loop schedules using prefetching and partitioning

In this paper, a method combining the loop pipelining technique with data prefetching, called Partition Scheduling with Prefetching (PSP), is proposed. In PSP, the iteration space is first divided into regular partitions. Then a two-part schedule, consisting of the ALU and memory parts, is produced and balanced to produce high throughput. These two parts are executed simultaneously, and hence, the remote memory latencies are overlapped. We study the optimal partition shape and size so that a well-balanced overall schedule can be obtained. Experiments on DSP benchmarks show that the proposed methodology consistently produces optimal or near optimal solutions.

Development of an Idiotypic Network in Shape Space

Based upon the shape-space formalism, a model of an idiotypic network including both bound and free immunoglobulins is simulated. Our point of interest is the network development in the context of self antigens. The investigations are organized around simulations initiated by various spatial configurations of antigens; the behavior of the system with respect to antigens is analyzed in terms of morphogenetic processes occurring in the shape space. For certain values of the parameters, the network expands by traveling waves. The resulting spatial pattern is a partition of the shape space into zones where introduction of an antigen entails an infinite growth of the clones binding to it, and into zones where, on the contrary, the anti-antigen idiotypes decrease. Among the parameter combinations tested, some produce a partition that remains static whereas others produce a partition that changes in time. For other values of the parameters, the patterns generated do not partition shape space into zones; in these cases, it is observed that the system systematically explodes when an antigen is present.

A partitioning strategy for explicit difference methods

Two partition shapes, slices and rectangles, are analyzed for the case of explicit difference methods on regular grids. The question is how to partition the grid in order to minimize the execution time on MIMD computers with distributed memory. The amount of computational work at the boundary points is assumed to be different from that the interior points. It is shown that a good strategy is to nevertheless do the partitioning as if the work was equal at all point. This strategy is nearly optimal, in the sense that it will in most cases give only a small reduction of the efficiency. At the same time it has the advantage of making it straightforward to compute the partitioning. The only situation where this strategy is not recommendable is when the work is much larger at the boundary points than in the interior part of the computational domain.

Compiler techniques for data partitioning of sequentially iterated parallel loops

This paper uses bottom-up, static program partitioning to minimize the execution time of parallel programs by reducing interprocessor communication. Program partitioning is applied to a parallel programming construct known as a sequentially iterated parallel loop. This paper develops and evaluates compiler techniques to automatically generate data partitions for sequentially iterated parallel loops that minimize interprocessor communication. These techniques could be included as a communication-reducing back end for existing parallelizing compilers. A multiprocessor model is defined to analyze program execution factors that lead to interprocessor communication. A model for the sequentially iterated parallel loop is defined to quantify communication patterns within a program. A vector notation is chosen to quantify communication across a global data set. Communication parameters used in generating data partitions are found and used within equations which are shown to generate rectangular and hexagonal partition shapes that minimize interprocessor communication. Experiments measuring the impact of communication on program performance and the improvements gained by using the automatically generated data partitions were conducted on a sixteen-processor Encore Multimax. The new method outperformed established data partitioning methods.

Stencils and Problem Partitionings: Their Influence on the Performance of Multiple Processor Systems

Given a discretization stencil, partitioning the problem domain is an important first step for the efficient solution of partial differential equations on multiple processor systems. We derive partitions that minimize interprocessor communication when the number of processors is known a priori and each domain partition is assigned to a different processor. Our partitioning technique uses the stencil structure to select appropriate partition shapes. For square problem domains, we show that nonstandard partitions (e.g., hexagons) are frequently preferable to the standard square partitions for a variety of commonly used stencils. We conclude with a formalization of the relationship between partition shape, stencil structure, and architecture, allowing selection of optimal partitions for a variety of parallel systems.