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Minmax graph partitioning and small set expansion
, 2011
"... We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal s ..."
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Cited by 14 (2 self)
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We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal size, and (ii) the parts must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O ( √ log n log k)approximation algorithm. This improves over an O(log 2 n) approximation for the second version due to Svitkina and Tardos [22], and roughly O(k log n) approximation for the first version that follows from other previous work. We also give an improved O(1)approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the SmallSet Expansion problem. In this problem, we are given a graph G and the goal is to find a nonempty set S ⊆ V of size S  ≤ ρn with minimum edgeexpansion. We give an O ( √ log n log (1/ρ)) bicriteria approximation algorithm for the general case of SmallSet Expansion, and O(1) approximation algorithm for graphs that exclude any fixed minor.
A linear work, O(n^1/6) time, parallel algorithm for solving planar Laplacians
"... We present a linear work parallel iterative algorithm for solving linear systems involving Laplacians of planar graphs. In particular, if Ax = b, where A is the Laplacian of any planar graph with n nodes, the algorithm produces a vector ¯x such that x − ¯xA ≤ ɛ, in O(n 1/6+c log(1/ɛ)) parallel t ..."
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Cited by 10 (3 self)
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We present a linear work parallel iterative algorithm for solving linear systems involving Laplacians of planar graphs. In particular, if Ax = b, where A is the Laplacian of any planar graph with n nodes, the algorithm produces a vector ¯x such that x − ¯xA ≤ ɛ, in O(n 1/6+c log(1/ɛ)) parallel time, doing O(n log(1/ɛ)) work, where c is any positive constant. One of the key ingredients of the solver, is an O(nk log 2 k) work, O(k log n) time, parallel algorithm for decomposing any embedded planar graph into components of size O(k) that are delimited by O(n / √ k) boundary edges. The result also applies to symmetric diagonally dominant matrices of planar structure.
Multiway Partitioning Using Bipartition Heuristics
 IN PROCEEDINGS OF ASPDAC
, 2000
"... The multiway partition problem is very important in various applications. In this paper, we use analytical and experimental results to study the kway partition problem. We introduce the concept of embedding graph for the the kway partition problem. Based on this concept, we explain different scen ..."
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Cited by 8 (1 self)
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The multiway partition problem is very important in various applications. In this paper, we use analytical and experimental results to study the kway partition problem. We introduce the concept of embedding graph for the the kway partition problem. Based on this concept, we explain different scenarios of using a bipartition heuristic to solve the kway partition problem. If C denote the optimal cut cost for the kway partition problem and the bipartition heuristics we use are ffiapprozimation heuristics (defined in Section 2), we prove that the cut cost from the hierarchical approach has an approximate upper bound of ffiC \Deltalog k while the cut cost from the allway bipartition, or flat approach, has an upper bound of ffiCk. This is contrary to some claims made in recent literature ( and CAD tools designed based on it). Experimental results strongly support our theoretical analysis. Our results show that for large target graph, the hierarchical approach is about 77% better tha...
Combinatorial and algebraic tools for optimal multilevel algorithms
, 2007
"... This dissertation presents combinatorial and algebraic tools that enable the design of the first linear work parallel iterative algorithm for solving linear systems involving Laplacian matrices of planar graphs. The major departure of this work from prior suboptimal and inherently sequential approac ..."
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Cited by 4 (1 self)
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This dissertation presents combinatorial and algebraic tools that enable the design of the first linear work parallel iterative algorithm for solving linear systems involving Laplacian matrices of planar graphs. The major departure of this work from prior suboptimal and inherently sequential approaches is centered around: (i) the partitioning of planar graphs into fixed size pieces that share small boundaries, by means of a local ”bottomup ” approach that improves the customary ”topdown ” approach of recursive bisection, (ii) the replacement of monolithic global preconditioners by graph approximations that are built as aggregates of miniature preconditioners. In addition, we present extensions to the theory and analysis of Steiner tree preconditioners. We construct more general Steiner graphs that lead to natural linear time solvers for classes of graphs that are known a priori to have certain structural properties. We also present a graphtheoretic approach to classical algebraic multigrid algorithms. We show that their design can be
MINMAX GRAPH PARTITIONING AND SMALL SET EXPANSION∗
"... Abstract. We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be ..."
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Abstract. We study graph partitioning problems from a minmax perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the k parts need to be of equal size, and where they must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O( logn log k) approximation algorithm. This improves over an O(log2 n) approximation for the second version due to Svitkina and Tardos [Minmax multiway cut, in APPROXRANDOM, 2004, Springer, Berlin, 2004], and roughly O(k logn) approximation for the first version that follows from other previous work. We also give an O(1) approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the smallset expansion problem. In this problem, we are given a graph G and the goal is to find a nonempty set S ⊆ V of size S  ≤ ρn with minimum edge expansion. We give an O( logn log (1/ρ)) bicriteria approximation algorithm for smallset expansion in general graphs, and an improved factor of O(1) for graphs that exclude any fixed minor.
Optimal CacheOblivious Mesh Layouts
, 2011
"... A mesh is a graph that divides physical space into regularlyshaped regions. Meshes computations form the basis of many applications, including finiteelement methods, image rendering, collision detection, and Nbody simulations. In one important mesh primitive, called a mesh update, each mesh verte ..."
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A mesh is a graph that divides physical space into regularlyshaped regions. Meshes computations form the basis of many applications, including finiteelement methods, image rendering, collision detection, and Nbody simulations. In one important mesh primitive, called a mesh update, each mesh vertex stores a value and repeatedly updates this value based on the values stored in all neighboring vertices. The performance of a mesh update depends on the layout of the mesh in memory. Informally, if the mesh layout has good data locality (most edges connect a pair of nodes that are stored near each other in memory), then a mesh update runs quickly. This paper shows how to find a memory layout that guarantees that the mesh update has asymptotically optimal memory performance for any set of memory parameters. Specifically, the cost of the mesh update is roughly the cost of a sequential memory scan. Such a memory layout is called cacheoblivious. Formally, for a ddimensional mesh G, block size B, and cache size M (where M = Ω(B d)), the mesh update of G uses O(1+G/B) memory transfers. The paper also shows how the meshupdate performance degrades for smaller caches, where M = o(B d). The paper then gives two algorithms for finding cacheoblivious mesh layouts. The first layout
Tight Bounds on the MinMax Boundary Decomposition Cost of Weighted Graphs
, 2008
"... Many load balancing problems that arise in scientific computing applications boil down to the problem of partitioning a graph with weights on the vertices and costs on the edges into a given number of equallyweighted parts such that the maximum boundary cost over all parts is small. Here, this part ..."
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Many load balancing problems that arise in scientific computing applications boil down to the problem of partitioning a graph with weights on the vertices and costs on the edges into a given number of equallyweighted parts such that the maximum boundary cost over all parts is small. Here, this partitioning problem is considered for graphs G = (V, E) with edge costs c: E → R+, that have bounded maximum degree and a pseparator theorem for some p> 1, i.e., any (arbitrarily weighted) subgraph of G can be separated into two parts of roughly the same weight by removing a separator S ⊆ V such that the edges incident to S in the subgraph have total cost at most proportional to ( ∑ e cpe) 1/p, where the sum is over all edges in the subgraph. For arbitrary weights w: V → R+, we show that the vertices of such graphs can be partitioned into k parts such that the weight of each part differs from the average weight ∑ v∈V wv/k by at most (1 − 1 k)maxv∈V wv, and the boundary edges of each part have total cost at most proportional to ( ∑ e∈E cpe/k) 1/p + maxe∈E ce. The partition can be computed in time nearly proportional to the time for computing separators S for G as above. Our upper bound is shown to be tight up to a constant factor for infinitely many instances with a broad range of parameters. Previous results achieved this bound only if one has c ≡ 1, w ≡ 1, and one allows parts of weight as large as a constant multiple of the average weight. We also give a separator theorem for ddimensional grid graphs with arbitrary edge costs, which is the first result of its kind for nonplanar graphs. 1
for solving planar Laplacians∗
"... A linear work, O(n1/6) time, parallel algorithm for ..."
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