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Spectral Partitioning Works Sp ectral Partitioni ng Works: Planar graphs and nite element meshes Preliminary Draft y Daniel A. Spielman Shang-Hua Teng February 13, 1996 Abstract Sp ectral partitioning metho ds use the Fiedler vector|the eigenvector of the second- smallest eigenvalue of the Laplacian matrix|to nd a small separator of a graph. These metho ds are imp ortant comp onents of many scienti c numerical algorithms and have b een demonstrated by exp erimenttowork extremely well. In this pap er, we show that sp ectral partitioning metho ds work well on b ounded-degree planar graphs and nite element meshes| the classes of graphs to which they are usually applied. While naive sp ectral bisection do es not necessarily work, we prove that sp ectral partitioning techniques can b e used to pro duce p n for b ounded-degree planar separators whose ratio of vertices removed to edges cut is O 1=d graphs and two-dimensional meshes and O n for well-shap ed d-dimensional meshes. The heart of our analysis is an upp er b ound on the second-smallest eigenvalues of the Laplacian matrices of these graphs. 1. Intro duction Sp ectral partitioning has b ecome one of the most successful heuristics for partitioning graphs and matrices. It is used in many scienti c numerical applications, such as mapping nite el- ement calculations on parallel machines [Sim91, Wil90], solving sparse linear systems [PSW92 ], and partitioning for domain decomp osition [CR87, CS93 ]. It is also used in VLSI circuit de- sign and simulation [CSZ93 , HK92, AK95]. Substantial exp erimental work has demonstrated that sp ectral metho ds nd go o d partitions of the graphs and matrices that arise in many ap- plications [BS92, HL92 , HL93 , PSL90 , Sim91, Wil90]. However, the quality of the partition that Computer Science Division, U.C. Berkeley, CA 94720, [email protected]. Supp orted in part byan NSF p ostdo c. After July 1996, Department of Mathematics, M.I.T. y Department of Computer Science, University of Minnesota, Minneap olis, MN 55455, [email protected]. Sup- p orted in part by an NSF CAREER award CCR-9502540 and an Intel research grant. Part of the work was done while visiting XeroxPARC. 1 these metho ds should pro duce has so far eluded precise analysis. In this pap er, we will prove that sp ectral partitioning metho ds give go o d separators for the graphs to which they are usually applied. The size of the separator pro duced by sp ectral metho ds can b e related to the Fiedler value|the second smallest eigenvalue of the Laplacian|of the adjacency structure to which they are applied. 2=d ,we By showing that well-shap ed meshes in d dimensions have Fiedler value at most O 1=n 11=d . show that sp ectral metho ds can b e used to nd bisectors of these graphs of size at most O n While a small Fiedler value do es not immediately imply that there is a cut along the Fiedler vector that is a balanced separator, it do es mean that there is a cut whose ratio of vertices separated to 1=d . By removing the vertices separated by this cut, computing a Fiedler vector of edges cut is O n 11=d edges. In particular, the new graph, and iterating as necessary, one can nd a bisector of O n we prove that b ounded-degree planar graphs have Fiedler value at most O 1=n, which implies that p sp ectral techniques can b e used to nd bisectors of size at most O n in these graphs. These b ounds are the b est p ossible for well-shap ed meshes and planar graphs. 1.1. History The sp ectral metho d of graph partitioning was b orn in the works of Donath and Ho man [DH72, DH73 ] who rst suggested using the eigenvectors of adjacency matrices of graphs to nd partitions. Fiedler [Fie73, Fie75a , Fie75b] asso ciated the second-smallest eigenvalue of the Laplacian of a graph with its connectivity and suggested partitioning by splitting vertices according to their value in the corresp onding eigenvector. Thus, we call this eigenvalue the Fied ler value and a corresp onding vector a Fied ler vector. A few years later, Barnes and Ho man [Bar82 , BH84] used linear programming in combination with an examination of the eigenvectors of the adjacency matrix of a graph. In a similar vein, Bop- pana [Bop87 ] analyzed eigenvector techniques in conjunction with convex programming. However, the use of linear and convex programming made these techniques impractical for most applications. By recognizing a relation b etween the Fiedler value and the Cheeger constant [Che70] of con- tinuous manifolds, Alon [Alo86] and Sinclair and Jerrum [SJ89 ] demonstrated that if the Fiedler value of a graph is small, then directly partitioning the graph according to the values of vertices in the eigenvector will pro duce a cut with a go o d ratio of cut edges to separated vertices see also [AM85, Fil91, DS91 , Mih89, Moh89 ]. Around the same time, improvements in algorithms for approximately computing eigenvectors, such as the Lanczos algorithm, made the computation of eigenvectors practical [PSS82 , Sim91]. In the next few years, a wealth of exp erimental work demonstrated that sp ectral partitioning metho ds work well on graphs that usually arise in prac- tice [BS92, HL92 , PSL90 , Sim91, Wil90]. Sp ectral partitioning b ecame a standard to ol for mesh partitioning in many areas [HL93 ]. Still, researchers were unable to prove that sp ectral partition- ing techniques would work well on the graphs encountered in practice. This failure is partially explained by results of Guattery and Miller [GM95] demonstrating that naive applications of sp ec- tral partitioning, such as sp ectral bisection, will fail miserably on some graphs that could con- ceivably arise in practice. By b ounding the Fiedler values of the graphs of interest in scienti c applications|b ounded-degree planar graphs and well-shap ed meshes|we are able to show that 2 sp ectral partitioning metho ds will successfully nd go o d partitions of these graphs. In a related line of research, algorithms were develop ed along with pro ofs that they will always nd small separators in various families of graphs. The seminal work in this area was that of Lipton and Tarjan [LT79 ], who constructed a linear-time algorithm that pro duces a 1=3-separator p of 8n no des in any n-no de planar graph. Their result improved a theorem of Ungar [Ung51 ] which p demonstrated that every planar graph has a separator of size O n log n . Gilb ert, Hutchinson, and Tarjan [GHT84] extended these results to show that every graph of genus at most g has a separator p of size O gn . Another generalization was obtained by Alon, Seymour, and Thomas [AST90], p 3=2 who showed that graphs that do not haveanh-clique minor have separators of O h n no des. 3=2 Plotkin, Rao, and Smith [PRS94] reduced the dep endency on h from h to h. Using geometric techniques, Miller, Teng, Thurston, and Vavasis [MT90, MTTV96a , MTTV96b , MTV91 , MV91, Ten91 ] extended the planar separator theorem to graphs emb edded in higher dimensions and showed d 11=d has a 1=d + 2-separator of size O n . Using multicommo dity that every well-shap ed mesh in R ow, Leighton and Rao [LR88 ] designed a partitioning metho d guaranteed to return a cut whose ratio of cut size to vertices separated is within logarithmic factors of optimal. While sp ectral metho ds have b een favored in practice, they lacked a pro of of e ectiveness. 1.2. Outline of pap er In Section 2, weintro duce the concept of a graph partition, review some facts from linear algebra that we require, and describ e the class of sp ectral partitioning metho ds. In Section 3, we prove the embedding lemma, which relates the quality of geometric emb eddings of a graph with its Fiedler value. We then show using the main result of Section 4 that every planar graph has a \nice" emb edding as a collection of spherical caps on the surface of a unit sphere in three dimensions. By applying the emb edding lemma to this emb edding, we prove that the Fiedler value of every b ounded-degree planar graph is O 1=n. In Section 4, we show that, for almost every arrangement of spherical caps on the unit sphere in d R , there is a sphere-preserving map that transforms the caps so that the center of the sphere is the centroid of their centers. It is this fact that enables us to nd nice emb eddings of planar graphs. In Sections 5 and 6, we extend our sp ectral planar separator theorem to the class of overlap graphs of k -ply neighb orho o d systems emb edded in any xed dimension. This extension enables us 1=d to show that the sp ectral metho d nds cuts of ratio O 1=n for k -nearest neighb or graphs and well-shap ed nite element meshes. In Section 7, we present an elementary pro of that from anyvector p erp endicular to the all-ones vector with small Rayleigh quotient, one can obtain a cut of small ratio. In Section 8, we extend the results of Guattery and Miller to show that our results are essentially the b est p ossible given currentcharacterizations of well-shap ed meshes.
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