K.M. Hall. ""An r-dimensional Quadratic Placement Algorithm"." Management Science, pp. 219--229, 1970.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....one can imagine the graph edges as springs that act to pull together(or apart) clips and features, so that the co occurring nodes are aligned, as shown in figure 3(b) There are many possible ways we can embed a graph in a lower dimensional space. Some of the techniques include graph drawing[1, 9, 10]. They di#er from each other in the criteria being minimized and computation e#ciency and accuracy. Here, we adopt a spectral graph method which has the an intuitive energy function: i,j)#E W (i, j) x(i) 2) where # x is the standard deviation of vector x. a) b) Figure 3: a) The ....

K.M. Hall, "An r-dimensional Quadratic Placement Algorithm." Management Science, 17:212-229, 1970.

....This is just the conventional definition of Laplacian, customary for undirected graphs. Indeed, since the Laplacian is independent of #, the definition does not distinguish between digraphs and undirected graphs. The Laplacian has a key role in some undirected graph drawing algorithms; see, e.g. [9], 14] 16] 17] and will be shown to play a fundamental role here, too. One of its most important properties is the following: Lemma 1: Let G(V, E; W,#) be a digraph. Its Laplacian is a positive semi definite matrix, and thus has non negative real eigenvalues. Moreover, when G is a connected ....

....a digraph. Its Laplacian is a positive semi definite matrix, and thus has non negative real eigenvalues. Moreover, when G is a connected digraph, L has exactly one zero eigenvalue, corresponding to the eigenvector c 1 n , where 1 n = 1, 1) R and c any constant. Proof: See Hall [9]. Naturally, a drawing algorithm has to deal only with connected digraphs. When a digraph is disconnected, one should draw each of its connected sub digraphs separately. Unless otherwise stated, we shall hereinafter always assume we have a connected digraph. Definition 2 (Balance) Let G(V, E; ....

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K. M. Hall, "An r-dimensional Quadratic Placement Algorithm", Management Science 17 (1970), 219--229.

....data. This is, mapping the data to points in a low dimensional space (mostly 2 D or 3 D) in a way that captures certain structured elements of the data. There are numerous techniques in this family, including principal component analysis [4, 11] multidimensional scaling [10] Eigen projection [5, 7, 8], and force directed placement [3, 6, 2, 9] We are particularly interested in the sub family of methods that use linear transformations to map the high dimensional data into a low dimensional space. This way, each low dimensional axis is some linear combination of the original axes. Linear ....

....LDA succeeds in showing the different intra structure of the clusters. while normalized LDA can produce a higher dimensional embedding. 5 Relation to Eigen projection Interestingly, the methods surveyed here have intimate relationships with the non linear Eigen projection visualization technique [5, 7, 8]. There, we use only pairwise relationships without utilizing the coordinates themselves. In fact, our methods reduce to the Eigen projection when we discard the coordinates matrix X , by setting it to the n n identity matrix. To make this apparent, substitute X = I in problem (6) or (5) and ....

K. M. Hall, "An r-dimensional Quadratic Placement Algorithm ", Management Science 17 (1970), 219--229.

....W,#) be a digraph. Its Laplacian is a positive semi definite matrix, and thus has non negative real eigenvalues. Moreover, when G is a connected digraph, L has exactly one zero eigenvalue, corresponding to the eigenvector c 1 n , where 1 n = 1, 1) and c any constant. Proof See Hall [8], or Koren et al. 12] Naturally, a drawing algorithm has to deal only with connected digraphs. When a digraph is disconnected, one should draw each of its connected sub digraphs separately. Unless otherwise stated, we shall hereinafter always assume we have a connected digraph. 5 Definition ....

....L and balance b, and let y = y 1 , y n ) be any vector of coordinates. The hierarchy energy is given by EH = E 0 y 2y b, 2) where E 0 = ij . Proof Expanding the hierarchy energy (1) we get EH = 1 y j ) 1 ij . The first term was shown in [8] to be just y Ly. The third term is, by definition, E 0 . The second term is y j ) w ij # ij y i = 2y b, 7 where the first equality stems from W being symmetric and # being antisymmetric. Exploiting the simple form of the hierarchy energy, we find an explicit formula for an ....

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K. M. Hall, "An r-dimensional Quadratic Placement Algorithm", Management Science 17 (1970), 219-229.

....end for end for end for Retrieve optimal ordering: for i = n to 1 table[S] right vtx ordering[i] v end Fig. 1. An algorithm for finding optimal MinLA using dynamic programming 5 Let v be the normalized eigenvector of L corresponding to the second smallest eigenvalue. Hall [9] has shown that v is the best non trivial minimum of the following quadratic energy: E = v i v j ) 1) subject to: #v# 2 =1 As a matter of fact, v is a vector of fundamental importance to many fields as it reflects connectivity properties of the graph, see, e.g. 16] So ....

K. M. Hall, "An r-dimensional Quadratic Placement Algorithm", Management Science 17 (1970), 219-229.

....a different set of properties. In this paper we concentrate on one particular form of the energy function, characterized by being simple and smooth, thus enabling rigorous analytical analysis and a straightforward implementation. This particular function was first applied to graph drawing by Hall [13], and we therefore term it Hall s energy. It was further studied in [26] where its results are shown to satisfy several desired aesthetic properties. Most graph drawing methods suffer from lengthy computation times when applied to really large graphs. Several publications in the graph drawing ....

....follows in Section 5. Appendix A summarizes some results of linear algebra that we will be needing, and Appendix B addresses related issues of graph connectivity. 2 The Eigenprojection Method In this section we briefly introduce the original method of energy minimization derived by Hall [13]; we use somewhat different notation, in order to facilitate later derivations. We will call his method as the eigenprojection method, for reasons that will become clear shortly. Let G(V, be a graph, with a set of n nodes, and a set of weighted edges, and with w ij 0 being the weight of ....

K. M. Hall, "An r-dimensional Quadratic Placement Algorithm", Management Science 17 (1970), 219-229.

....) 4 ( 2.5. # # # # # # Fig. 2. a) A small example of an unweighted cyclic digraph; b) its optimal arrangement. The idea of using energy minimization to determine a vector of coordinates on one axis was already exploited in undirected graph drawing by Tutte [12] and Hall [5], both utilizing the same quadratic energy function, ETH = y j ) Ly. Comparing this energy with the hierarchy energy (1) it is clear that they become identical for a digraph with # ij =0, E, which is really an undirected graph. It is instructive to adopt a different viewpoint in ....

....reasoning, minimizing edge lengths is known to reduce edge crossings. We present two alternative variants of the method: Minimizing edge squared lengths: This means minimizing the already familiar Tutte Hall energy function, ETH = i,j=1 w ij (x i x j ) x Lx. As suggested by Hall [5, 7], the non trivial minimizer of this energy function is the Fiedler vector, which is the eigenvector of the Laplacian associated with the smallest positive eigenvalue. We find the minimizer of ETH using ACE [8] an extremely fast multiscale algorithm for undirected graph drawing. Minimizing ....

K. M. Hall, "An r-dimensional Quadratic Placement Algorithm", Management Science 17 (1970), 219-229.

....Given a graph G(V,E) with V = 1, n , the corresponding Laplacian is the symmetric n n matrix L defined as follows: L ij = # # #i,k##E w ik i = j ij j##E i, j =1, n. Let v be the normalized eigenvector of L corresponding to the second smallest eigenvalue. Hall [11] has shown that v is the best non trivial minimum of the following quadratic energy: E = v i v j ) 1) subject to: #v# 2 =1 In fact, the vector v called the Fiedler vector is of fundamental importance in many fields, as it reflects connectivity properties of the graph; see, ....

K. M. Hall, "An r-dimensional Quadratic Placement Algorithm", Management Science 17 (1970), 219-229.

....a different set of properties. In this paper we concentrate on one particular form of an energy function, characterized by being simple and smooth, thus enabling rigorous analytical analysis and a straightforward implementation. This particular function was first applied to graph drawing by Hall [10], and we therefore term it Hall s energy. It was further studied in [22] where its results are shown to satisfy several desired aesthetic properties. Most graph drawing methods suffer from lengthy computation times when applied to really large graphs. Several publications in the graph drawing ....

....follows in section 5. Appendix A summarizes some results of linear algebra that we will be needing, and appendix B addresses related issues of graph connectivity. 2 The Eigenprojection Method In this section we briefly introduce the original method of energy minimization derived by Hall [10], but put his results in somewhat different notations, to facilitate later derivations. We will call his method as the eigenprojection method, for reasons that will become clear shortly. Let G(V, E) be a graph, with V a set of n nodes, and E a set of weighted edges, and with 0 being the weight ....

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K. M. Hall, "An r-dimensional Quadratic Placement Algorithm", Management Science 17 (1970), 219-229.

....of a set of objects according to their pair wise similarities (see, e.g. Fig. 1) We have focused on spectral graph drawing methods, which construct the layout using eigenvectors of certain matrices associated with the graph. This approach is quite old, originating with the work of Hall [5] in 1970. However, since then it has not been used much. In fact, spectral graph drawing algorithms are almost absent in the graph drawing literature (e.g. they are not mentioned in the two books [1, 10] that deal with graph drawing) It seems that in most visualization research the spectral ....

....normalized eigenvectors.It can be shown (see Appendix A) that all the generalized eigenvalues are real non negative, and that all the degree normalized eigenvectors are D orthogonal, i.e. u i Du j =0, j. 3 Spectral Graph Drawing The earliest spectral graph drawing algorithm was that of Hall [5]; it uses the low eigenvectors of the Laplacian. Henceforth, we will refer to this method as the eigen projection method. A few other researchers utilize the top eigenvectors of the adjacency matrix instead of those of the Laplacian. For example, consider the work of [8] which uses the adjacency ....

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K. M. Hall, "An r-dimensional Quadratic Placement Algorithm", Management Science 17 (1970), 219--229.

....this to avoid 1 2 terms throughout this work. Min cut graph partitioning is known to be NP complete, so heuristic methods must be invoked. Previous approaches have included seeded epitaxial growth, iterative improvement [16] genetic algorithms [6] etc. Spectral methods [1] 2] 4] 7] 8] 11] [13] [15] have been successful in recent years and are of particular interest for our present work. These works share a common trait of using eigenvectors to construct some type of geometric representation of the graph. We note four such representations: ffl Linear ordering or 1 dimensional placement: ....

....have been successful in recent years and are of particular interest for our present work. These works share a common trait of using eigenvectors to construct some type of geometric representation of the graph. We note four such representations: ffl Linear ordering or 1 dimensional placement: Hall [13] showed that the second eigenvector of the Laplacian yields an optimum 1 dimensional placement in terms of squared wirelength. Barnes [4] proposed a method using multiple eigenvectors that reduces to sorting the coordinates of the adjacency matrix s largest eigenvector when k = 2. Hagen and Kahng ....

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K. M. Hall, "An r-dimensional Quadratic Placement Algorithm", Manag. Sci., 17(1970), pp. 219-229.

....between two movable cells i and j is #x i , x j # 2 #y i ,y j # 2 .Ifcelliis connected to a fixed cell with coordinates #x f ; y f #, the distance is #x i , x f # 2 #y i ,y f # 2 . Consequently, the objective function sums up the cost of all edges and can be written in matrix notation [11] 1 2 p T C p d T p const (1) by using the 2n # 2n symmetric matrix C and the 2n dimensional vector d. For example, the x part of the connection between two movable cells i and j is #x i ,x j # 2 = x 2 i ,2 # x i # x x 2 j .Thefirstterm contributes to the diagonal of C ....

K. M. Hall, "An r-dimensional quadratic placement algorithm, " Management Science, vol. 17, pp. 219--229, Nov. 1970.

....and partitioning approaches [15] Similar orderings have been applied to the one dimensional gate assignment problem (e.g. 9] Finally, there are several vertex orderings that are inferred from heuristic global minimization of wirelength functions in one dimensional module placements. Hall [13] showed that the second eigenvector of the netlist discrete Laplacian yields the natural module ordering for minimum squared wirelength; we call this the EIG1 ordering, following [12] who used this eigenvector for ratio cut partitioning. Recently, Riess et al. 19] have used an analytical ....

K.M. Hall, "An r-dimensional Quadratic Placement Algorithm", Manag. Sci., 17(1970), pp.219-229.

....unit vector basis (the n nodes of a graph) into k distinct points (the partitions) to minimize the weighted quadratic displacement. This is essentially the formulation given by Barnes [2, 3] However, unlike Barnes formulation we do not assume any pre determined partition sizes. By Hall s result [4], using the k eigenvectors of the graph s Laplacian, corresponding to the smallest k eigenvalues provides a projection which minimizes the weighted quadratic displacement under the orthonormality constraint. In the case of a partition, this amounts to the number of edges cut. We show that a k ....

K. M. Hall, "An r-dimensional quadratic placement algorithm," Management Science, vol. 17, pp. 219--229, Nov. 1970.

....[3] In general, these previous works formulate the partitioning problem as the assignment or placement of nodes into bounded size clusters, i.e. chip locations. The problem is then transformed into a quadratic optimization, and Lagrangian relaxation is used to derive an eigenvector formulation [13]. In [11] Hagen and Kahng established a close relationship between the optimal ratio cut cost and the second smallest eigenvalue of the matrix Q = D Gamma A, where D and A are as defined above: Theorem 1 (Hagen Kahng) Given a netlist graph G = V; E) with adjacency matrix A, diagonal degree ....

K. M. Hall, "An r-dimensional Quadratic Placement Algorithm ", Management Science 17(1970), pp. 219-229.

....lengths of the edges of the given adjacency graph: EQ 2) where c ij represents the total number of connections between vertex v i and v j . x i ,y i ) and (x j ,y j ) represent the locations of v i and v j . The cost function can then be rewritten using matrix notation as follows [13]: EQ 3) where x is a vector of the x coordinates of the vertex locations and y is a vector of the y coordinates. B is a symmetric matrix with B = D C where C = c ij ] is the connectivity matrix and D is a X Y Z X Y Z a b c c a b Figure 3: Optimal placement for different objectives (a) ....

....function L 1 2 c ij x i x j ( 2 y i y j ( 2 v v j i j ( L x y , x T Bx y T By = 7 diagonal matrix with . It has been shown that if the nodes (vertices) cannot be partitioned into disconnected subsets, then B is positive semi definite [13]. That means the objective function is a convex function. This fact allows the calculation of a unique global optimum solution and plays an important role in our approach. In addition, B is almost always sparse for practical cases. This enables efficient numerical techniques to be applied to the ....

K. M. Hall, "An r-Dimensional Quadratic Placement Algorithm," Management Science, vol. 17, pp.219-229, 1970.

....methodology consists of two phases the first phase generates several candidate partitioning solutions and the second phase evaluates them and selects the best one. The generation of candidate partitions is based on a spectral partitioning technique used in a number of partitioning algorithms [7, 8, 9]. The technique was introduced by Hall [7] who proved that the second smallest eigenvector of the Laplacian of a graph gives a one dimensional placement of graph nodes such that the sum of squares of edge lengths is minimized. Large gaps in this ordering are used to delimit the clusters. For ....

....phase generates several candidate partitioning solutions and the second phase evaluates them and selects the best one. The generation of candidate partitions is based on a spectral partitioning technique used in a number of partitioning algorithms [7, 8, 9] The technique was introduced by Hall [7] who proved that the second smallest eigenvector of the Laplacian of a graph gives a one dimensional placement of graph nodes such that the sum of squares of edge lengths is minimized. Large gaps in this ordering are used to delimit the clusters. For example, Fig. 2 shows an eighth order cascade ....

K. M. Hall, "An r-Dimensional Quadratic Placement Algorithm," Management Science, Vol. 17, No. 3, Nov. 1970, pp. 219-229.

....cluster, iteratively adds the vertex that minimizes the perimeter of the resulting cluster. Alternatively, adding the vertex with the most connections to the current cluster yields a max adjacency approach; cf. Nagamochi and Ibaraki [12] Other constructions been given for VLSI layout. Hall [11] showed that the second eigenvector of the netlist discrete Laplacian yields a minimum squaredwirelength ordering; 10] used this ordering in ratio cut partitioning. Riess et al. 14] used an analytical conjugate gradient method to construct orderings according to a linear wirelength objective. In ....

K.M. Hall, "An r-dimensional Quadratic Placement Algorithm ", Manag. Sci., 17(1970), pp.219-229.

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K.M. Hall. ""An r-dimensional Quadratic Placement Algorithm"." Management Science, pp. 219--229, 1970.

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K. M. Hall, "An r-dimensional quadratic placement algorithm," Manag. Sci., Vol. 17, pp. 219-229, 1970.

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K. M. Hall, "An r-Dimensional Quadratic Placement Algorithm," Management Science, vol. 17, pp.219-229, 1970.

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K. M. Hall, "An r-dimensional Quadratic Placement Algorithm", Management Science 17 (1970), 219--229.

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K. M. Hall, "An r-dimensional Quadratic Placement Algorithm", Management Science 17 (1970), 219--229.

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K. M. Hall, "An r-dimensional Quadratic Placement Algorithm", Management Science 17 (1970), 219--229.

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K.M. Hall, "An r-dimensional quadratic placement algorithm", Management Science, 17, 219-229, 1970.

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