Recent research on FPGA partitioning has focussed on finding minimum cuts between partitions without regard to the routability of the partitioned subcircuits. In this paper we develop a spectral approach to multi-way partitioning in which the primary goal is to produce routable subcircuits while maximizing FPGA device utilization. To assist the partitioner in assessing the routability of the partitioned subcircuits, we have developed a theory to predict the routability of the partitioned subcircuits prior to partitioning. Advancement over the current work is evidenced by results of experiments on the standard MCNC benchmarks. I FPGA partitioning The design flow using commercial FPGA tools involves technology mapping, placement and routing. Since FPGA devices have relatively low density, the use of multiple FPGAs is often required to implement a large circuit. A large circuit has to be decomposed or partitioned into subcircuits for a multiple-FPGA realization, as shown in Fig. 1. Mod...

According to the Darwinian theory of evolution, adaptation results from spontaneously generated genetic variation and natural selection. Mathematical models of this process can be seen as describing a dynamics on an algebraic structure which in turn is defined by the processes which generate genetic variation (mutation and/or recombination). The theory of complex adaptive system has shown that the properties of the algebraic structure induced by mutation and recombination is more important for understanding the dynamics than the differential equations themselves. This has motivated new directions in the mathematical analysis of evolutionary models in which the algebraic properties induced by mutation and recombination are at the center of interest. In this paper we summarize some new results on the algebraic properties of recombination spaces. It is shown that the algebraic structure induced by recombination can be represented by a map from the pairs of types to the power set of the ty...

In this paper we present and analyze new sequential and parallel heuristics to approximate the Minimum Linear Arrangement problem (MinLA). The heuristics consist in obtaining a first global solution using Spectral Sequencing and improving it locally through Simulated Annealing. In order to accelerate the annealing process, we present a special neighborhood distribution that tends to favor moves with high probability to be accepted. We show how to make use of this neighborhood to parallelize the Metropolis stage on distributed memory machines by mapping partitions of the input graph to processors and performing moves concurrently. The paper reports the results obtained with this new heuristic when applied to a set of large graphs, including graphs arising from finite elements methods and graphs arising from VLSI applications. Compared to other heuristics, the measurements obtained show that the new heuristic improves the solution quality, decreases the running time and o#ers an excellent speedup when ran on a commodity network made of nine personal computers.

. A new spectral algorithm for reordering a sparse symmetric matrix to reduce its envelope size was described in [2]. The ordering is computed by associating a Laplacian matrix with the given matrix and then sorting the components of a specified eigenvector of the Laplacian. In this paper we provide an analysis of the spectral envelope reduction algorithm. We describe related 1- and 2-sum problems; the former is related to the envelope size, while the latter is related to an upper bound on the work in an envelope Cholesky factorization. We formulate the latter two problems as quadratic assignment problems, and then study the 2-sum problem in more detail. We obtain lower bounds on the 2-sum by considering a relaxation of the problem, and then show that the spectral ordering finds a permutation matrix closest to an orthogonal matrix attaining the lower bound. This provides stronger justification of the spectral envelope reduction algorithm than previously known. The lower bound on the 2-...

Survey article for the proceedings of Discrete Optimization '99 where some of these results were presented as a plenary address. y

Consider the following load balancing problem. We are given a graph with arbitrary topology and an arbitrary load distribution on the nodes. In each time step, a node can send any amount of load to each neighbor in parallel. This problem is a simple formalization of dynamic load balancing problems arising in various applications on parallel and distributed machines. Earlier work on this problem [Cyb89, SS94, HT93] provides relaxation-type diffusive algorithms which use local and/or global information about the structure of the graph. In this paper we present Algorithm SLB which uses an over-relaxation technique to perform load balancing. For any particular graph, Algorithm SLB needs only a limited amount of global information to begin with but subsequently uses only local information to schedule the load movement. Extensive simulations on different graphs and load distributions confirm that Algorithm SLB is substantially faster than earlier relaxation-based algorithms, especially on...

Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early works in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis of mesh geometry based on a combinatorial Laplacian aids our understanding of the low-pass filtering approach to mesh smoothing. Over the past ten years or so, the list of applications in the area of geometry processing which utilize the eigenstructures of a variety of mesh operators in different manners have been growing steadily. Many works presented so far draw parallels from developments in fields such as graph theory, computer vision, machine learning, graph drawing, numerical linear algebra, and high-performance computing. This state-of-the-art report aims to provide a comprehensive survey on the spectral approach, focusing on its power and versatility in solving geometry processing problems and attempting to bridge the gap between relevant research in computer graphics and other fields. Necessary theoretical background will be provided and existing works will be classified according to different criteria — the operators or eigenstructures employed, application domains, or the dimensionality of the spectral embeddings used — and described in adequate length. Finally, despite much empirical success, there still remain many open questions pertaining to the spectral approach, which we will discuss in the report as well.

A landscape is rugged if it has many local optima, if it gives rise to short adaptive walks, and if it exhibits a rapidly decreasing pair-correlation function (and hence if it has a short correlation length). The "correlation length conjecture" allows to estimate the number of meta-stable states from the correlation length, provided the landscape is "typical". Isotropy, originally introduced as a geometrical condition on the covariance matrix of a random field, can be re-interpreted as maximum entropy condition that lends a precise meaning to the notion of a "typical" landscape. The XY-Hamiltonian, which violates isotropy only to a relatively small extent, is an ideal model for investigating the influence of anisotropies. Numerical estimates for the number of local optima and predictions obtained from the correlation length conjecture indeed show deviations that increase with the extent of anisotropies in the model.

Abstract. Image segmentation based on graph representations has been a very active field of research recently. One major reason is that pairwise similarities (encoded by a graph) are also applicable in general situations where prototypical image descriptors as partitioning cues are no longer adequate. In this context, we recently proposed a novel convex programming approach for segmentation in terms of optimal graph cuts which compares favorably with alternative methods in several aspects. In this paper we present a fully elaborated version of this approach along several directions: first, an image preprocessing method is proposed to reduce the problem size by several orders of magnitude. Furthermore, we argue that the hierarchical partition tree is a natural data structure as opposed to enforcing multiway cuts directly. In this context, we address various aspects regarding the fully automatic computation of the final segmentation. Experimental results illustrate the encouraging performance of our approach for unsupervised image segmentation. 1

We show a Faber-Krahn-type inequality for regular trees with boundary. Leydold: A Faber-Krahn-type inequality for regular trees 1. Introduction The eigenvectors of the Laplacian \Delta on graphs have received little attention compared to the spectrum of this operator (see e.g. [6, 9, 10]) or the eigenfunctions of the "classical" Laplacian differential operator on Riemannian manifolds (e.g. [1, 2]). Recently these eigenvectors seem to become more important. Grover [7] has discovered that the cost function of a number of well-studied combinatorial optimisation problems, e.g. the travelling salesman problem, are eigenvectors of the Laplacian of certain graphs. Thus global properties of such eigenvectors are of interest. In the last years some results for the Laplacian on manifolds have been shown to hold also for the graph Laplacian, e.g. Courant's nodal domain theorem ([3, 5]) or Cheeger's inequality ([4]). In [5] Friedman described the idea of a "graph with boundary" (see below). With...