J. Garbers, H. Promel, and A. Steger, "Finding Clusters in VLSI circuits", Proc. International Conference on Computer Aided Design, pp. 520-523, 1990  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The graph partitioning problem is NP complete. However, many algorithms have been developed that find a reasonably good partition. Recently, a number of researchers have investigated a class of algorithms that are based on multilevel graph partitioning that have moderate computational complexity [4, 5, 12, 13, 15, 7, 31, 20, 19]. In these schemes, the original graph is successively coarsened down until it has only a small number of vertices, a partition of this coarsened graph is computed, and then this initial partition is successively refined by using a Kernighan Lin type heuristic as it is being projected back to the ....

J. Garbers, H. J. Promel, and A. Steger. Finding clusters in VLSI circuits. In Proceedings of IEEE International Conference on Computer Aided Design, pages 520--523, 1990.

....of spatially related data items in large geographical information systems on disk to minimize disk I O requests, and mapping of task graphs to parallel processors. The graph partitioning problem is NP complete. However, many algorithms have been developed that find reasonably good partitionings [23, 22, 9, 24, 19, 18, 20, 2, 3, 7, 8, 12, 5, 21, 16, 13]. The k way partitioning problem is most frequently solved by recursive bisection. That is, we first obtain a 2 way partitioning of V, and then we recursively obtain a 2 way partitioning of each resulting partition. After log k phases, graph G is partitioned into k partitions. Thus, the problem ....

....refined. As the results in Section 3 show, despite the simplicity of our refinement algorithms, they produce high quality partitionings in small 1A partitioning is at a local minima, if movement of any vertex from one part to the other does not improve the edge cut. Partition 0 N(5) 0, 2 1D[5] = 2 ED[5]o = 2 ED[512 =3 Figure 3: Illustration of neighboring partitions, internal, and external vertex degrees. amount of time. In the rest of this section we describe some key concepts and definitions that are used in the description of our two k way partitioning refinement algorithms, ....

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J. Garbers, H. J. Promel, and A. Steger. Finding clusters in VLSI circuits. In Proceedings oflEEE International Conference on Computer Aided Design, pages 52(523, 1990.

.... and multiple minimum degree [16] The parallel formulation in this paper is described in the context of the serial multilevel graph partitioning algorithm presented in [16] However, nearly all of the discussion in this paper is applicable to other multilevel graph partitioning algorithms [4, 12, 7, 22]. The rest of the paper is organized as follows. Section 2 surveys the different types of graph partitioning algorithms that are widely used today. Section 2 briefly describes the serial multilevel algorithm that forms the basis for the parallel algorithm described in Sections 3 and 4 for graph ....

J. Garbers, H. J. Promel, and A. Steger. Finding clusters in VLSI circuits. In Proceedings oflEEE International Conference on Computer Aided Design, pages 52023.

....of large magnitude (relative to the sizes of y i and y j ) then these vectors likely belong in the same subset. Graph partitioning has no natural analogous similarity measure, as witnessed by the many ad hoc means in the literature (e.g. all pairs shortest paths, k Gamma l connectivity [9], degree=separation [10] etc. Because the min cut partitioning objective of Equation (1) requires us to minimize f and the max sum objective of Equation (3) attempts to maximize g, the objectives at first seem incompatible. However, following the work of [8] we may transform f into a ....

J. Garbers, H. J. Promel and A. Steger, "Finding Clusters in VLSI Circuits" Proc. IEEE Intl. Conf. on Computer-Aided Design, Santa Clara, Nov. 1990, pp. 520-523.

....concerned here with the 2 way min cut partitioning problem. Since 2 way min cut partitioning is NP complete [19] a number of approximate schemes have been proposed. These include iterative improvement methods [16, 17, 23, 24, 29, 30] simulated annealing [31, 32] and clustering based techniques [7, 5, 18, 20, 28, 29, 36, 35]. An excellent survey on partitioning techniques appears in [6] In iterative improvement, we start with a random 2 way partition of the circuit, and iteratively improve it by either swapping pairs of nodes between the subsets, or moving one node at a time between them so that the cutset size is ....

J. Garbers, H.J. Promel and A. Steger, "Finding clusters in VLSI circuits", Proc. Int'l. Conf. Computer-Aided Design, 1990, pp. 520-523.

....on a large system at the gate level. In the following, we discuss three system partitioning methodologies, local bottom up clustering, top down two way recursive partitioning, and the hybrid method of the former two. When circuit sizes are large, global clustering algorithms with high complexity [1, 6, 10, 12] become infeasible. Hence only local bottom up clustering with low computational overhead [3, 7, 8, 15, 17] can be performed. For circuit decomposition, local bottom up clustering is attractive because it is based 2 on local information and thus has very low time complexity. However, grouping ....

....low levels. For large circuits, it is very natural to combine the above two methods into a hybrid algorithm. That is, the algorithm applies local clustering to reduce the problem complexity until a good recursive partitioning is feasible in terms of time and space complexities. Hybrid algorithms [1, 3, 6, 7, 10, 12] have become popular and have reported encouraging quality using reasonable run time. As a rule of thumb, local clustering can provide good clustering results only when the cluster size is small, and recursive partitioning can work well only on top levels with a complexity of a few thousand ....

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J. Garbers, "Finding Clusters in VLSI Circuits," Proc. Int. Conf. Computer-Aided Design, 1990, pp. 520 - 523.

....vertices in tightly connected subgraphs, clusters of vertices can be treated as individual supernodes during the application of standard heuristics like KL or simulated annealing. The various incarnations of the clustering idea appear to show a marked superiority over the original KL algorithm [4, 6, 8, 11, 13, 16, 22, 26, 27], though the degree of superiority is unclear because the reported empirical results tend to sell the KL algorithm short, as we will argue below. The algorithms we present in this paper can be considered a synthesis of ideas from previous work. They are based on a simple constructive heuristic ....

....the number of edges added to the cut set separating X and 3 Y while maximizing the number of edges barred from future addition to the cut set. Thus the exploitation of structure is implicit in this heuristic, as compared to heuristics in which explicit clusters are computed and manipulated [4, 6, 8, 11, 13, 16, 22, 26, 27]. More formally, the algorithm can be given by the following pseudocode. To simplify notation, we assume jV j is even, although in practice we tolerate solutions with an imbalance of 1. The constant size specifies the cardinality of the seed sets. Input: An undirected graph G = V; E) Output: ....

J. Garbers, H. J. Promel, and A. Steger. Finding clusters in VLSI circuits. In Proceedings of the IEEE International Conference on Computer-Aided Design, pages 520--523, Santa Clara, California, Nov. 1990.

....The graph partitioning problem is NP complete. However, many algorithms have been developed that find a reasonably good partition. Recently, a number of researchers have investigated a class of algorithms that are based on multilevel graph partitioning that have moderate computational complexity [4, 5, 9, 10, 12, 7, 21, 15, 14]. In these schemes, the original graph is successively coarsened down until it has only a small number of vertices, a partition of this coarsened graph is computed, and then this initial partition is successively refined by using a Kernighan Lin type heuristic as it is being projected back to the ....

J. Garbers, H. J. Promel, and A. Steger. Finding clusters in VLSI circuits. In Proceedings of IEEE International Conference on Computer Aided Design, pages 520--523, 1990.

....The graph partitioning problem is NP complete. However, many algorithms have been developed that find a reasonably good partition. Recently, a number of researchers have investigated a class of algorithms that are based on multilevel graph partitioning that have moderate computational complexity [4, 5, 11, 12, 14, 7, 28, 17, 16]. In these schemes, the original graph is successively coarsened down until it has only a small number of vertices, a partition of this coarsened graph is computed, and then this initial partition is successively This work was supported by NSF CCR 9423082, by Army Research Office contract ....

J. Garbers, H. J. Promel, and A. Steger. Finding clusters in VLSI circuits. In Proceedings of IEEE International Conference on Computer Aided Design, pages 520--523, 1990.

.... and multiple minimum degree [16] The parallel formulation in this paper is described in the context of the serial multilevel graph partitioning algorithm presented in [16] However, nearly all of the discussion in this paper is applicable to other multilevel graph partitioning algorithms [4, 12, 7, 22]. The rest of the paper is organized as follows. Section 2 surveys the different types of graph partitioning algorithms that are widely used today. Section 2 briefly describes the serial multilevel algorithm that forms the basis for the parallel algorithm described in Sections 3 and 4 for graph ....

J. Garbers, H. J. Promel, and A. Steger. Finding clusters in VLSI circuits. In Proceedings of IEEE International Conference on Computer Aided Design, pages 520--523, 1990.

.... static nature: it is not clear how to adapt them to construct a different clustering if some other objective is desired. A number of works have proposed explicit objectives for netlist clustering. Of particular note 2 : 2 In addition to the three objectives that we discuss, Garbers et al. [10] have proposed a k l connectivity criterion. Two vertices are k l connected if there are k edge disjoint paths of length l between them. While there is some flexibility in that k and l are specified by the user, evaluating the objective is infeasible for l 2 (and is known to be NP hard for l ....

J. Garbers, H. J. Promel and A. Steger, "Finding Clusters in VLSI Circuits" Proc. IEEE Intl. Conf. on Computer-Aided Design, Santa Clara, Nov. 1990, pp. 520-523.

....capable of producing a large number of clusters of nearly equal size has been proposed [8] Although efficient and effective, storage requirements are prohibitive for large circuits. Heuristics for producing natural clusters (the number of clusters is unknown until termination) have been proposed [11, 12]. Natural clustering produces few clusters which vary greatly in size. A heuristic based on random walks has been proposed [13] which produces good clusters. Its complexity is O(c 3 ) which is undesireable for a preprocessing step. 3.2. Proposed Heuristic We consider a simple heuristic based ....

J. Garbers, H. J. Promel, and A. Steger. Finding clusters in vlsi circuits. In Proc. IEEE Int. Conf. on Computer-Aided Design, pages 520--523, 1990.

....October 1993. ffl Second, input constructions for which optimum solution costs are known (i.e. for which the error of a heuristic can be quantified) are often considered artificial . For example, mesh or chainlike topologies for placement, and the instances of Bui et al. 6] Garbers et al. [8] and Ackley [1] for partitioning, may be helpful in measuring algorithmic suboptimality but do not always give meaningful performance estimates for real examples. In this paper, we propose a general measure of heuristic performance, based on the notion of the scaling suboptimality of a given ....

....suboptimality have centered on the construction of instances for which the optimal solution cost is known. As noted above, such methods have included the use of mesh and chain topologies for placement, and difficult (highly clustered or even disconnected) classes of partitioning instances [6, 8, 1]. Historically, the major objection to the constructive approach has been that the instances are artifactual and not realistic . Two recent methods which start with real instances are thus of interest: ffl Nakatake and Kajitani [11] generate a sequence of global routing instances, each of ....

J. Garbers, H. J. Promel, and A. Steger, "Finding Clusters in VLSI Circuits", Proc. IEEE Intl. Conf. on ComputerAided Design, 1990, pp. 520--523. Extended version.

....CA 90024 1596 USA jenhsin cs.ucla.edu, abk cs.ucla.edu Abstract Recent research on multi way partitioning has focused on the minimum cut [20, 26, 27] or generalized ratio cut [28, 29, 5] cost metrics. At the same time, clustering research has focused on such objectives as k l connectivity [12], DS metric [6] or clique finding [8] In this paper, we make the basic observation that cut objectives in partitioning, and density objectives in clustering, are fundamentally incompatible. Moreover, for multi way decomposition applications (e.g. decomposing a system onto multiple FPGA chips) ....

....with very dense connections among them. The goal of a clustering algorithm is to identify the good clusters in a circuit, often via some bottom up approach. The key issue is how to assess whether a cluster is good. Several clustering metrics have been proposed in the literature. Garbers et al. [12] used k l connectivity to measure the cluster quality. In a graph, two nodes are k l connected if there exist k edge disjoint paths connecting them such that each path has length l. The transitive closure of this relation induces a heuristic clustering. Cong et al. 6] proposed measuring cluster ....

J. Garbers, H. J. Promel, and A. Steger, "Finding Clusters in VLSI Circuits", Proc. IEEE Intl. Conf. on Computer-Aided Design, 1990, pp. 520-523.

....of another 5 clustering, but an unclustered node can choose to join a grouping with an already clustered node. Note that with random clustering a separate clustering is always generated for each run of the KLFM algorithm. Numerous more intelligent clustering algorithms exist. K L clustering [Garbers90] (not to be confused with KL, the Kernighan Lin algorithm) is a method that looks for multiple independent short paths between nodes, expecting that these nodes should be placed into the same partition. Otherwise, each of these paths will have a net in the cutset, degrading the partition quality. ....

....C are K L connected with node B, while A and C are not directly K L connected) In our study of K L clustering we ignored all nets with fanout greater than 10, and used k = 2, l 1 = 1, l 2 = 3. The values of maximum considered fanout and l 1 were chosen to give reasonable computation times. While [Garbers90] recommends k = 3, l 1 = 1, l 2 = 3, l 3 = 3, we have found that this yielded few clustering opportunities (this will be discussed later) and the parameters we chose gave the greatest clustering opportunities with reasonable run time. Using l 2 = 4 would increase the clustering opportunities, but ....

J. Garbers, H. J. Prömel, A. Steger, "Finding Clusters in VLSI Circuits", International Conference on Computer-Aided Design , pp. 520-523, 1990.

....is not used as a source of a clustering, but an unclustered node can choose to join a grouping with a node already clustered. Note that with random clustering a new clustering is generated for each run of the KLFM algorithm. Numerous more intelligent clustering algorithms exist. K L clustering [15] (not to be confused with KL, the Kernighan Lin algorithm) is a method that looks for multiple independent short paths between nodes, expecting that these nodes should be placed into the same partition. Otherwise, each of these paths will have a net in the cutset, degrading the partition quality. ....

....both A C are K L connected with node B, while A C are not K L connected) In our study of K L clustering we ignored all nets with fanout greater than 10, and used k = 2, l 1 = 1, l 2 = 3. The values of maximum considered fanout and l 1 were chosen to give reasonable computation times. While [15] recommends k = 3, l 1 = 1, l 2 = 3, l 3 = 3, we have found that this yielded few clustering opportunities (this will be discussed later) and the parameters we chose gave the greatest clustering opportunities with reasonable run time. Using l 2 = 4 would increase the clustering opportunities, but ....

J. Garbers, H. J. Prömel, A. Steger, "Finding Clusters in VLSI Circuits", ICCAD, pp. 520-523, 1990.

....in each cluster are not separated during the partitioning process. 1.2 Basic Concepts and Terminology A netlist is best represented by a hypergraph with each component being represented by a node and each net represented by a hyperedge. However, many clustering and partitioning algorithms [3, 10, 14, 11, 6], including the one presented in this paper, use a graph representation of the netlist rather than a hypergraph representation. An r terminal net is represented by an r clique in the graph representation. An r clique is a complete graph with r nodes, and the number of edges in an r clique is i ....

....and most intuitive metric, this metric is biased toward small clusters since the value of M c increases rapidly as c increases. k l connectedness. In a graph, two nodes are k l connected if and only if there exist k edge disjoint paths connecting them such that each path has length at most l [10]. The idea is that if two nodes are connected by many short paths, they are strongly connected. However, it is not obvious what values should be assigned to k and l for a given circuit. Degree Separation. A more recent metric for determining the quality of clusters is the degree separation (DS) ....

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J. Garbers, H.J. Promel, and A. Steger. Finding Clusters in VLSI Circuits. ICCAD'90, pages 520--523, 1990.

....bigger the circuit is, the smaller the probability is to find a global minimum. To cope with this problem, multiple approaches of clustering grouping of the circuit elements before the partitioning starts were proposed to reduce the complexity. However, bottom up clustering methods [13] 3] [7] suffer from the absence of the global circuit view, and top down clustering approaches [4] still are faced with the same complexity problem. In [5] an algorithm is proposed which combines the top down and bottom up clustering approaches and allows rapidly the partitioning of FPGA netlists of 100 ....

J. Garbers, H.J. Promel, A. Steger, "Finding Clusters in VLSI circuits", Proc. Int. Conf. on Computed- Aided Design (1990): 520-523.

....Another class of graph partitioning algorithms reduces the size of the graph (i.e. coarsen the graph) by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph. These are called multilevel graph partitioning schemes [4, 7, 19, 20, 26, 10, 43]. Some researchers investigated multilevel schemes primarily to decrease the partitioning time, at the cost of somewhat worse partition quality [43] Recently, a number of multilevel algorithms have been proposed [4, 26, 7, 20, 10] that further refine the partition during the uncoarsening phase. ....

.... These are called multilevel graph partitioning schemes [4, 7, 19, 20, 26, 10, 43] Some researchers investigated multilevel schemes primarily to decrease the partitioning time, at the cost of somewhat worse partition quality [43] Recently, a number of multilevel algorithms have been proposed [4, 26, 7, 20, 10] that further refine the partition during the uncoarsening phase. These schemes tend to give good partitions at a reasonable cost. Bui and Jones [4] use random maximal matching to successively coarsen the graph down to a few hundred vertices; they partition the smallest graph and then uncoarsen ....

[Article contains additional citation context not shown here]

J. Garbers, H. J. Promel, and A. Steger. Finding clusters in VLSI circuits. In Proceedings of IEEE International Conference on Computer Aided Design, pages 520--523, 1990.

....is not used as a source of a clustering, but an unclustered node can choose to join a grouping with a node already clustered. Note that with random clustering, a new clustering is generated for each run of the KLFM algorithm. Numerous more intelligent clustering algorithms exist. K L clustering [Garbers90] (not to be confused with KL, the Kernighan Lin algorithm) is a method that looks for multiple independent short paths between nodes, expecting that these nodes should be placed into the same partition. Otherwise, each of these paths will have a net in the cutset, degrading the partition quality. ....

....both A C are K L connected with node B, while A C are not K L connected) In our study of K L clustering we ignored all nets with fanout greater than 10, and used k = 2, l 1 = 1, l 2 = 3. The values of maximum considered fanout and l 1 were chosen to give reasonable computation times. While [Garbers90] recommends k = 3, l 1 = 1, l 2 = 3, l 3 = 3, we have found that this yielded few clustering opportunities (this will be discussed later) and the parameters we chose gave the greatest clustering opportunities with reasonable run time. Using l 2 = 4 would increase the clustering opportunities, but ....

J. Garbers, H. J. Prömel, A. Steger, "Finding Clusters in VLSI Circuits", International Conference on Computer-Aided Design, pp. 520-523, 1990.

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J. Garbers, H. Promel, and A. Steger, "Finding Clusters in VLSI circuits", Proc. International Conference on Computer Aided Design, pp. 520-523, 1990

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Garbers, J., Promel, H. J., and Steger, A.: Finding clusters in VLSI circuits. In Proc. IEEE Intl. Conf. on Computer Aided Design, pages 520--523, (1990)

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J. Garbers, H. Promel, and A. Steger, "Finding Clusters in VLSI circuits", Proc. International Conference on Computer Aided Design, pp. 520-523, 1990

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J orn Garbers, Hans J urgen Pr omel, and Angelika Steger. Finding clusters in vlsi circuits. In Proceedings of the IEEE International Conference on Computer-Aided Design (ICCAD 1990), pages 520--523, 1990.

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J. Garbers, H. J. Promel, and A. Steger. Finding clusters in VLSI circuits. In Proceedings of IEEE International Conference on Computer Aided Design, pages 520--523, 1990.

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