Balanced Partitioning Research Articles

Characterizing and Mining the Citation Graph of the Computer Science Literature

Citation graphs representing a body of scientific literature convey measures of scholarly activity and productivity. In this work we present a study of the structure of the citation graph of the computer science literature. Using a web robot we built several topic-specific citation graphs and their union graph from the digital library ResearchIndex. After verifying that the degree distributions follow a power law, we applied a series of graph theoretical algorithms to elicit an aggregate picture of the citation graph in terms of its connectivity. We discovered the existence of a single large weakly-connected and a single large biconnected component, and confirmed the expected lack of a large strongly-connected component. The large components remained even after removing the strongest authority nodes or the strongest hub nodes, indicating that such tight connectivity is widespread and does not depend on a small subset of important nodes. Finally, minimum cuts between authority papers of different areas did not result in a balanced partitioning of the graph into areas, pointing to the need for more sophisticated algorithms for clustering the graph.

An improved circuit-partitioning algorithm based on min-cut equivalence relation

Network flow is an excellent approach to finding min-cuts in network because of the celebrated max-flow/min-cut theorem. For a long time, however, it has not been a practical circuit partitioning method for its relatively high computational complexity. Algorithm FBB is the first one to apply network flow to solve two-way balanced circuit partitioning successfully. Later, its improved version DMC efficiently solved the problem of multi-way circuit partitioning. In this paper, an improved circuit-partitioning algorithm MCER is presented. MCER is based on a min-cut equivalence relation. The algorithm not only inherits to take advantage of flow theory, but also incorporates a subsets movement-based method to utilize internal equivalent topological structure of circuit. Based on the equivalence relation and subsets movement between initial min-cuts, MCER will find more min-cuts. A series of experiments demonstrate that searching space of MCER is bigger than that of former algorithms and MCER outperforms FBB or DMC in terms of the size of min-cuts. Furthermore, MCER will degrade to DMC or FBB under special conditions.

Balanced multi-level multi-way partitioning of analog integrated circuits for hierarchical symbolic analysis

This paper considers the problem of partitioning analog integrated circuits for hierarchical symbolic analysis based on determinant decision diagrams (DDDs). The objective is to use DDDs with the minimum number of vertices to represent all the symbolic expressions. We show that the problem can be formulated as that of multi-level multi-way hyper graph partitioning with balance constraints, and be solved in two phases by connectivity-oriented initial clustering and iterative improvement. Our new contribution consists of a fast and effective heuristic for constructing a balanced initial partition, a potential gain formulae that can be computed efficiently, and a multiple-vertex moving strategy for relaxing and enforcing balance constraints. The proposed algorithm has been implemented and applied to symbolic analysis of several practical analog integrated circuits. Experimental results are described and compared to the contour tableau method of Sangiovanni-Vincentelli, Chen and Chua, and the SCAPP algorithm of Hassoun and Lin. The resulting hierarchical symbolic analyzer outperforms SPICE in numerical evaluations for a number of large analog circuits.

Balanced Convex Partitions of Measures in ℝ 2

Let n≥2 be an integer and let μ1 and μ2 be measures in ℝ2 such that each μ i is absolutely continuous with respect to the Lebesgue measure and μ1(ℝ2)=μ2(ℝ2)=n. Let u≠0 be a vector on the plane. We show that if μ1(B)=μ2(B)=n for some bounded domain B, then there exist positive integers n1,n2 with n1+n2=n and disjoint open half-planes D1,D2 such that \(\), μ1(D1)=μ2(D1)=n1 and μ1(D2)=μ2(D2)=n2; or there exist positive integers n1,n2,n3 with n1+n2+n3=n and disjoint open convex domains D1,D2,D3 such that \(\), μ1(D1)=μ2(D1)=n1, μ1(D2)= μ2(D2)=n2, μ1(D3)=μ2(D3)=n3 and such that the ray \(\) is parallel to u. We also show a similar result for partitioning of point sets on the plane.

Vector Balancing Games with Aging

In this article we study an extension of the vector balancing game investigated by Spencer and Olson (which corresponds to the on-line version of the discrepancy problem for matrices). We assume that decisions in earlier rounds become less and less important as the game continues. For an aging parameter q⩾1 we define the current move to be q times more important than the previous one. We consider two variants of this problem: First, the objective is a balanced partition at the end of the game, and second, it is to ensure a balanced partition throughout the game. We concentrate on the case q⩾2. We give an optimal solution for the first problem and a nearly optimal one for the second.

Instance space of the number partitioning problem

Within the replica framework we study analytically the instance space of the number partitioning problem. This classic integer programming problem consists of partitioning a sequence of N positive real numbers $\{a_1, a_2,..., a_N}$ (the instance) into two sets such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized. We show that there is an upper bound $\alpha_c N$ to the number of perfect partitions (i.e. partitions for which that difference is zero) and characterize the statistical properties of the instances for which those partitions exist. In particular, in the case that the two sets have the same cardinality (balanced partitions) we find $\alpha_c=1/2$. Moreover, we show that the disordered model resulting from hte instance space approach can be viewed as a model of replicators where the random interactions are given by the Hebb rule.

Divide-and-conquer approximation algorithms via spreading metrics

We present a novel divide-and-conquer paradigm for approximating NP-hard graph optimization problems. The paradigm models graph optimization problems that satisfy two properties: First, a divide-and-conquer approach is applicable. Second, a fractional spreading metric is computable in polynomial time. The spreading metric assigns lengths to either edges or vertices of the input graph, such that all subgraphs for which the optimization problem is nontrivial have large diameters. In addition, the spreading metric provides a lower bound, τ, on the cost of solving the optimization problem. We present a polynomial time approximation algorithm for problems modeled by our paradigm whose approximation factor is O (min{log τ, log log τ, log k log log k }) where k denotes the number of “interesting” vertices in the problem instance, and is at most the number of vertices. We present seven problems that can be formulated to fit the paradigm. For all these problems our algorithm improves previous results. The problems are: (1) linear arrangement; (2) embedding a graph in a d -dimensional mesh; (3) interval graph completion; (4) minimizing storage-time product; (5) subset feedback sets in directed graphs and multicuts in circular networks; (6) symmetric multicuts in directed networks; (7) balanced partitions and p -separators (for small values of p ) in directed graphs.

Balanced partitions for Markov chains

We call a finite partition of the state space of a (discrete-time) Markov chain balanced if the flows in both directions between any two of its classes are equal in equilibrium. If a Markov chain is reversible then any finite partition is balanced. We use this notion in order to gain insight into the structure of the stationary distributions of not necessarily reversible transition kernels. We illustrate our theory with an asymptotic analysis of a non-reversible Markovian star network with loss.1

Structure of Supercritical Fluid Krypton at Small Scattering Angle Using Parallel Molecular Dynamics Simulation

We present a parallel algorithm for molecular dynamics involving short-range two- and three-body potentials and the pair-correlation function, g(r). The method is based on a spatial decomposition of the simulation box that takes advantage of a linked-cell list, and allows a load balanced partition of the computations of both the forces and g(r) over the processors. The tests of the program is conducted by evaluating the efficiency for both the thermalization phase and the production phase of the simulation. This method is successfully applied to the calculation of the direct correlation function of fluid krypton at small scattering angle along the T = 297 K supercritical isotherm.

Iterative Partitioning with Varying Node Weights

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.

Balanced partitions of two sets of points in the plane

We prove the following theorem. Let m≥2 and q≥1 be integers and let S and T be two disjoint sets of points in the plane such that no three points of S∪ T are on the same line, | S|=2 q and | T|= mq. Then S∪ T can be partitioned into q disjoint subsets P 1, P 2,…, P q satisfying the following two conditions: (i) conv( P i )∩conv( P j )= φ for all 1≤ i< j≤ q, where conv( P i ) denotes the convex hull of P i ; and (ii) | P i ∩ S|=2 and | P i ∩ T|= m for all 1≤ i≤ q.

A Computation+Communication Load Balanced Loop Partitioning Method for Distributed Memory Systems

Due to a significant communication overhead of sending and receiving data, the loop partitioning approaches on distributed memory systems must guarantee not just the computation load balance but computation+communication load balance. The previous approaches in loop partitioning have achieved a communication-free, computation load balanced iteration space partitioning solution for a limited subset of DOALL loops. But a large category of DOALL loops inevitably result in communication and the trade-offs between computation and communication must be carefully analyzed for these loops in order to balance out the combined computation time and communication overheads. In this work, we describe a partitioning approach based on the above motivation for the general cases of DOALL loops. Our goal is to achieve a computation+communication load balanced partitioning through static data and iteration space distribution. Our approach first performs partitioning of iteration and data spaces of a loop nest by analyzing communication and parallelism; it then performs architecture-dependent analysis to adjust the granularity of partitions, load balance each partition with respect to total computation+communication, and then performs mapping of partitions onto the available number of processors. This multiphase partitioning method works as follows. First, the code partitioning phase analyzes the references in the body of the DOALL loop nest and determines a set of directions for reducing a larger degree of communication by trading a lesser degree of parallelism. The partitioning is carried out in the iteration space of the loop by cyclically following a set of direction vectors such that the data references are maximally localized and reused, eliminating a larger communication volume than parallelism. We then perform data space partitioning based on a new larger partition owns rule to minimize the communication overhead for a compute intensive partition by localizing its references relatively more than a smaller noncompute intensive partition. A partition interaction graph is then constructed which is used by the architecture-dependent analysis phase to merge the partitions to achieve granularity adjustment, computation+communication load balance, and mapping on the actual number of available processors. Relevant theory and algorithms are developed along with a performance evaluation on the Cray T3D.

An Associative Block Design ABD(8,5)

Previous article Next article An Associative Block Design ABD(8,5)A. E. BrouwerA. E. Brouwerhttps://doi.org/10.1137/S0097539797316622PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractAn associative block design is a certain balanced partition of a hypercube into smaller hypercubes. We construct such a design, thus settling the smallest open case.[1] A. Brouwer, On associative block designs, Colloq. Math. Soc. János Bolyai, Vol. 18, North‐Holland, Amsterdam, 1978, 173–184 80d:05009 Google Scholar[2] J. La Poutré, A theorem on associative block designs, Discrete Math., 58 (1986), 205–208 10.1016/0012-365X(86)90164-0 87d:05034 CrossrefISIGoogle Scholar[3] J. La Poutré and , J. van Lint, An associative block design ABD(10,5) does not exist, Utilitas Math., 31 (1987), 219–225 88i:05047 ISIGoogle Scholar[4] Ronald Rivest, On hash‐coding algorithms for partial‐match retrieval (extended abstract), IEEE Comput. Soc., Long Beach, Calif., 1974, 95–103 55:1881 Google Scholar[5] Google Scholar[6] Ronald Rivest, Partial‐match retrieval algorithms, SIAM J. Comput., 5 (1976), 19–50 52:16195 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Volume 28, Issue 6| 1999SIAM Journal on Computing History Published online:28 July 2006 InformationCopyright © 1999 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/S0097539797316622Article page range:pp. 1970-1971ISSN (print):0097-5397ISSN (online):1095-7111Publisher:Society for Industrial and Applied Mathematics

Network-flow-based multiway partitioning with area and pin constraints

Network flow is an excellent approach to finding min-cuts because of the celebrated max-flow min-cut theorem. For a long time, however, it was perceived as computationally expensive and deemed impractical for circuit partitioning. Recently, the algorithm FBB successfully applied network flow to two-way balanced partitioning. It for the first time demonstrated that network flow was a viable approach to circuit partitioning. In this paper, we present FBB-MW, which is an extension of FBB, to solve the problem of multiway partitioning with area and pin constraints. Experimental results show that FBB-MW outperforms previous approaches for multiple field programmable gate array partitioning. In particular, although FBB-MW does not employ logic replication and logic resynthesis, it still outperforms some other algorithms, which allow replication and resynthesis for optimization.

How Good is Recursive Bisection?

The most commonly used p-way partitioning method is recursive bisection (RB). It first divides a graph or a mesh into two equal-sized pieces, by a "good" bisection algorithm, and then recursively divides the two pieces. Ideally, we would like to use an optimal bisection algorithm. Because the optimal bisection problem that partitions a graph into two equal-sized subgraphs to minimize the number of edges cut is NP-complete, practical RB algorithms use more efficient heuristics in place of an optimal bisection algorithm. Most such heuristics are designed to find the best possible bisection within allowed time. We show that the RB method, even when an optimal bisection algorithm is assumed, may produce a p-way partition that is very far way from the optimal one. Our negative result is complemented by two positive ones: first we show that for some important classes of graphs that occur in practical applications, such as well-shaped finite-element and finite-difference meshes, RB is within a constant factor of the optimal one "almost always." Second, we show that if the balance condition is relaxed so that each block in the p-way partition is bounded by 2n/p, where n is the number of vertices of the graph, then a modified RB finds an approximately balanced p-way partition whose cost is within an O(log p) factor of the cost of the optimal p-way partition.

Design of minimal-level PLA self-testing checkers for m-out-of-n codes

This paper presents the design of minimal-level PLA self-testing checkers (STCs) for incomplete m-out-of-n (m/n) codes and 1-out-of-n (1/n) codes. All checkers are selftesting for three classes of typical PLA faults and hence they are all crosspoint irredundant. A number of various incomplete m/n codes which exhibit the two-closure property with balanced partitioning are constructed, which allow one to build two-level area optimal PLA STCs for incomplete m/n codes with virtually any capacity. These new PLA STC's for m/n codes are then used to build a family of efficient three-level PLA STCs for most 1/n codes. In most cases, the new checkers offer area and/or active device number reduction, compared to existing designs which rely upon m/2m codes only. Obviously, all new minimal-level checkers can be implemented using logic gates as well.

The Differencing Algorithm LDM for Partitioning: A Proof of a Conjecture of Karmarkar and Karp

The algorithm LDM (largest differencing method) divides a list of n random items into two blocks. The parameter of interest is the expected difference between the two block sums. It is shown that if the items are i.i.d. and uniform then the rate of convergence of this parameter to zero is n−Θ(log n). An algorithm for balanced partitioning is constructed, with the same rate of convergence to zero.

Balanced partitioning

We consider the problem of bipartitioning a circuit into two balanced components that minimizes the number of crossing nets. Previously, Kernighan and Lin type (K&L) heuristics, simulated annealing approach, and analytical methods were given to solve the problem. However, network flow (max-flow min-cut) techniques were overlooked as viable heuristics to min-cut balanced bipartition due to their high complexity. In this paper we propose a balanced bipartition heuristic based on repeated max-flow min-cut techniques, and give an efficient implementation that has the same asymptotic time complexity as that of one max-flow computation. We implemented our heuristic algorithm in a package called FBB. The experimental results demonstrate that FBB outperforms K&L heuristics and analytical methods in terms of the number of crossing nets, and our efficient implementation makes it possible to partition large circuit netlists with reasonable runtime. For example, the average elapsed time for bipartitioning a circuit S35932 of almost 20 K gates is less than 20 min on a SPARC10 with 32 MB memory.

General design of area-efficient PLA self-testing checkers for 1-out-of-n codes

A general study of designing area-efficient self-testing checkers (STCs) for 1-out-of-n (1/n) codes implemented using PLAs is presented. The checkers are shown to be self-testing under a fault model which covers four classes of PLA faults: stuck-at faults, missing or extra crosspoint devices, shorts between two adjacent or crossing lines and break faults. It is shown that the principal prerequisite for designing a PLA-based STC for a 1/n code is that the code is translated into an unordered code whose codewords are formed by concatenating codewords of some k≥.2 2-closed unordered codes with balanced partitioning. The latter class of codes includes a newly proposed incomplete 3/7 code with only 26 codewords. The resulting PLA-based STCs are less complex than existing designs for many 1/n codes.

On parallelizing the EM algorithm for PET image reconstruction

The expectation maximization (EM) algorithm is one of the most suitable iterative methods for positron emission tomography (PET) image reconstruction; however, it requires a long computation time and an enormous amount of memory space. To overcome these problems, we present two classes of highly efficient parallelization schemes: homogeneous and inhomogeneous partitionings. The essential difference between these two classes is that the inhomogeneous partitioning schemes may partially overlap the communication with computation by deliberate exploitation of the inherent data access pattern with a multiple-ring communication pattern. In theory, the inhomogeneous partitioning schemes may outperform the homogeneous partitioning schemes. However, the latter require a simpler communication pattern. In an attempt to estimate the achievable performance and to analyze the performance degradation factors without actual implementation, we have derived efficiency prediction formulas for closely estimating the performance for the proposed parallelization schemes. We propose new integration and broadcasting algorithms for hypercube, ring, and n-D mesh topologies, which are more efficient than the conventional algorithms when the link setup time is relatively negligible. The concept of the proposed task and data partitioning schemes, the integration and broadcasting algorithms, and the efficiency estimation methods can be applied to many other problems that are rich in data parallelism, but without balanced exclusive partitioning. >