Separators in two and three dimensions

We show that every graph that is the 1-skeleton of a simplicial complex K in 3-dimensions has a separator of size O(c 2/3 + ~), where c is the number of 3-simplexes in K and 0 is the number of 0simplexes on the boundary of K, if every 3-simplex has bounded aspect-ratio. This is natural generalization of the separator results for planar graphs, such as the Lipton and Tarjan planar separator theorem. We also show that a family of separators of size O(c 2/3) exists and is constructible. Using this family of separators we get an O(n 2) time algorithm for solving linear systems that arise from the finite element method. In particular, we solve linear systems in O(n 2) time where the underlying graph is the 1-skeleton of a simplicial complex having bounded aspect-ratio and small boundary. All the constructions work in RNC with a reasonably small number of processors. 1 I n t r o d u c t i o n The Divide-and-Conquer paradigm is fundamental for a large number of both sequential as well as *This work was s u p p o r t e d in p a r t by Na t iona l Science F o u n d a t i o n g r a n t DCR-8713489. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. parallel algorithm design. Divide-and-Conquer can give both fast and efficient algorithms. For graph algorithms and numerical analysis the efficiency of the algorithm is determined by the size and quality of the separator used in the algorithm. D e f i n i t i o n 1.1 A subset of vertices B of a graph with n vertices is said to & s e p a r a t e if the remaining vertices can be partitioned into 2 sets A and C such that there are no edges from A to C, IA[, ICl < ~f. n. The subset B is an f ( n ) s e p a r a t o r if there exist a constant ~f < 1 such that B $-separates and IBI < f (n) . Two of the most important classes of graphs with small separators have been trees and planar graphs. It is well known that a tree has a single vertex separator that 2/3-separates. Another natural class of graphs with smM1 separators are the planar graphs. Lipton and Tarjan showed that any planar graph has x/-8" n-separator that 2/3-separates, [LT79]. They gave a linear time algorithm to find this separator. There have been many extensions of this work, [Mi186, GM87, GM, Dji82, Gaz86]. All these results only consider planar graphs. There have also been several results on finding separators for graphs of a given genus, [GHT82, HM86]. Many applications of these separators exist, [Lei83, FJ86] One of the main applications of these separators is the finite element method, [LRT79, PR85a, PR85b, GT87]. We show that our separators do not generate too much fill-in and, therefore, can also be used for solving these linear systems in O(n 2) sequential time or O(log2nloglogn) time using O(n 2) processors on a PRAM. Thus, we show that n 2 direct © 1990 ACM 089791-361-2/90/0005/0300 $1.50 300 methods exist for the finite element method in the 3-dimensions (possibly the most important dimension). The algorithms we present for finding these family are randomized. But linear system solvers are otherwise deterministic. To motivate our result we view the planar separator theorem as a statement about 2-dimensional simplicial complexes. We first give a few definitions. Def in i t ion 1.2 A k-dimensional simplex (k-s implex) is the convex hull of k + 1 affinely independent points in ~d space. A simplicial complex is a collection of simplexes closed under subsimplex and intersection. A k-complex K is a simplicial complex such that for every kl-simplex in K, k I <_ k. Thus, a 3-complex is a collection of tetrahedra or cells (3-simplexes), triangular patches or faces (2-simplexes), edges (1-simplexes), and vertices (0-simplexes). The k -ske le ton of a simplicial complex K is the k-complex consisting of all k Isimplexes in K for k I _< k. Thus, the 1-skeleton of a 2-complex in the plane can be viewed as a graph that is planar. On the other hand, by F£ry's Theorem we know that every planar graph can be embedded in the plane such that each edge maps to a straight line, [Tho80a]. Thus, if G is a triangulated planar graph then it can be embedded in a 1-skeleton of a 2-complex in the plane. Thus we can view the planar separator theorem as statements about the 1-skeletons in 2-dimensions. The main goal of this paper is to show that under reasonable assumptions small separators exist and can be found for 1-skeletons of 3-complexes embedded 3-dimensions. We next discuss the restriction we place on the 3:omplex. It is not hard to see that any graph can be embedded in 3-dimensions. In particular, one can show that the complete graph can be embedded in ~;he 1-skeleton of a 3-complex in ~3. The 3-complex will have O(n 2) 3-simplexes. We can accomodate r, his example by allowing the size of the separator 1;o be a function of the number of 3-simplexes. This restriction alone is not sufficient to insure ~;he existence of small separators. In Section 6 we exhibit a 3-complex such that its 1-skeleton has only separators for size >_ t . c~ log c, where c is the number of 3-simplexes and t is some fixed constant. The 3-simplexes in the example are long and thin. For many applications we can restrict our attention to those complexes where the simplexes have bounded aspect-ratio. There are many equivalent definitions of the aspect-ratio of a simplex such as: all angles have minimum size, the number of simplexes that can share a point is bounded below, and ratio between the diameter of the circumscribing sphere and diameter of the inscribing sphere is bounded. We have picked the following definition: Def in i t ion 1.3 The d i a m e t e r Dia(S) of a ksimplex ,,q is the maximum distance between any pair for points in S. While the a spec t r a t i o equals