C. Alpert and S. Yao, Spectral partitioning: the more eigenvectors the better. In Proceedings of 32nd ACM/IEEE Design Automation Conference, 1995, pp. 195-200.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....circuits show up to 70 improvement over FM in mincut, and average mincut improvements of about 35 over all circuits and 47 over large circuits. They also outperform state of the art non IIP techniques, the quadratic programming based method Paraboli [20] and the spectral partitioner MELO [3] by about 17 and 23 , respectively, with less CPU time. This demonstrates the potential of sophisticated IIP algorithms in dealing with the increasing complexity of emerging VLSI circuits. We also compare CLIP and CDIP to hMetis [16] one of the best of the recent state of the art partitioners ....

....of feasible and thus optimal solutions are different under the two assumptions, and hence the LSR MFFS results are not compatible with those of other partitioners. Delta 4 algorithms: 1) Classical IIP methods (FM [11] and LA [15] 2) State of the art non IIP techniques (Paraboli [20] and MELO [3]) and (3) Multilevel IIP algorithms (hMetis [16] Conclusions are in Section 5. 2. PREVIOUS ITERATIVE IMPROVEMENT ALGORITHMS A circuit netlist is usually modeled by a hypergraph G = V;E) where V is the set of cells (also called nodes) in the circuit, and E is the set of nets (also called ....

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C. J. Alpert and S-Z Yao, "Spectral Partitioning: The More Eigenvectors, the Better", Proc. ACM/IEEE Design Automation Conf., 1995.

....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 ....

....suites, which show that PROP performs an average of 30 better than FM and 27 better than LA, while SHRINK PROP obtains about 37 and 34 better results than FM and LA, respectively. We also compare our methods to some of the more recent techniques like EIG1 [20] WINDOW [7] PARABOLI [28] MELO [5] and GMetis [4] Results show that our new techniques also performs significantly better (by 4.5 to 67 ) than these techniques. The rest of this paper is organized as follows. In Sec. 2 we discuss two previous relevant iterative improvement methods FM and LA. Section 3 discusses the probability ....

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C.J. Alpert and S-Z Yao, "Spectral Partitioning: The more eigenvectors the better", Proc. Design Automation Conf., 1995, pp. 195-200.

....along with some new approaches and optimizations, can yield a bipartitioning algorithm significantly better than the current state of the art. Our algorithm achieved a 8 2 improvement over PROP [Dutt96] 16 improvement over FBB [Yang94] 22 improvement over Paraboli [Reiss94] and MELO [Alpert95b], 50 improvement over Fiduccia Mattheyses [Fiduccia82] and a 58 improvement over EIG1 [Hagen92] Alpert95b] some of the best current bipartitioning algorithms. In this paper, we seek to understand the critical issues in logic replication, the selective duplication of logic to reduce the ....

....better than the current state of the art. Our algorithm achieved a 8 2 improvement over PROP [Dutt96] 16 improvement over FBB [Yang94] 22 improvement over Paraboli [Reiss94] and MELO [Alpert95b] 50 improvement over Fiduccia Mattheyses [Fiduccia82] and a 58 improvement over EIG1 [Hagen92] [Alpert95b], some of the best current bipartitioning algorithms. In this paper, we seek to understand the critical issues in logic replication, the selective duplication of logic to reduce the resulting cutset. We examine most of the current logic replication literature, integrating it into our already ....

C. J. Alpert, S.-Z. Yao, "Spectral Partitioning: The More Eigenvectors, The Better", Design Automation Conference, pp. 195-200, 1995.

....so a hyperedge model is needed to approximate the hypergraph by a graph. The relation between the spectrum of a graph and other graph properties has been an area of active research [4, 8, 15, 25, 26, 43] but only recently spectrum based methods have been successfully applied to graph partitioning [1, 2, 21, 31, 46, 47, 53]. We present below a spectral graph K partitioning formulation that extends Rend and Wolkowicz s to graphs containing weighted vertices. Other Global Methods. Simulated annealing (SA) has been used for graph partitioning by Johnson et al. 34] and Bui et al. 10] with mixed results. They showed ....

C.J. Alpert and S.-Z. Yao, "Spectral Partitioning: The More Eigenvectors, The Better," to appear in Proceedings 32nd ACM/IEEE Design Automation Conference, June, 1995.

....circuits show up to 70 improvement over FM in mincut, and average mincut improvements of about 35 over all circuits and 47 over large circuits. They also outperform state of the art non IIP techniques, the quadratic programming based method Paraboli [20] and the spectral partitioner MELO [3] by about 17 and 23 , respectively, with less CPU time. This demonstrates the potential of sophisticated IIP algorithms in dealing with the increasing complexity of emerging VLSI circuits. We also compare CLIP and CDIP to hMetis [16] one of the best of the recent state of the art partitioners ....

....higher level lookahead gains and improved the results for small circuits. All these algorithms improve an initial partition through a sequence of node moves (based on node gains ) and thus fall under the class of iterative improvement partitioners (IIPs) Recently, a number of non IIP algorithms [3, 4, 12, 20, 21] have been proposed and excellent results have been obtained. FM and LA are the most commonly used two way partitioning algorithms largely due to their excellent run times, simple implementations and flexibility. However, this class of IIP algorithms have a common weakness, viz. they only find ....

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C. J. Alpert and S-Z Yao, "Spectral Partitioning: The More Eigenvectors, the Better", Proc. ACM/IEEE Design Automation Conf., 1995.

....ACM SIGDA benchmark circuits show up to 70 improvement over FM in cutsize, with an average of per circuit percent improvements of about 25 , and a total cut improvement of about 35 . They also outperform the recent placement based partitioning tool Paraboli [13] and the spectral partitioner MELO [14] by about 17 and 23 , respectively, with less CPU time. This demonstrates the potential of iterative improvement algorithms in dealing with the increasing complexity of modern VLSI circuitry. 1. Introduction The essence of VLSI circuit partitioning is to divide a circuit into a number of ....

....FM by adding higher level lookahead gains and improved the results for small circuits, while Hagen et al. 15] investigated implementation and tie breaking techniques for improving the performance of FM type algorithms. A number of clustering based, i.e. bottom up, partitioning algorithms [9, 10, 12, 13, 14, 18] have also been proposed and good results have been obtained. FM and LA are the most commonly used two way partitioning algorithms largely due to their excellent run times, simple implementations and flexibility. However, this class of iterative improvement algorithms have a common weakness, ....

[Article contains additional citation context not shown here]

C. J. Alpert and S-Z Yao, "Spectral Partitioning: The More Eigenvectors, the Better", Proc. ACM/IEEE Design Automation Conf., 1995.

.... 2 [3, 7, 13, 14] We will thus be concerned here with the 2 way min cut partitioning problem. Since the problem is NP complete, a number of approximate schemes have been proposed. These include iterative improvement methods [3, 6, 9, 10] simulated annealing [12] and clustering based techniques [1, 2, 7, 11, 13, 14]. In iterative improvement methods, 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 reduced. Clustering based methods try to find natural ....

....either FM or LA.We also run tests on circuit netlists from the ACM SIGDA benchmark, which show that PROP performs an average of 30 better than FM and 27 better than LA. Comparison of PROP to some of the more recent clustering based techniques like EIG1 [7] WINDOW [1] PARABOLI [11] and MELO [2] show that PROP also performs significantly better (by 15 to 57 ) than them. The rest of this paper is organized as follows. In Sec. 2. we discuss two previous iterative improvement methods FM and LA, and thereby set the stage for discussing the PROP technique in Sec. 3. where we also derive its ....

[Article contains additional citation context not shown here]

C.J. Alpert and S-Z Yao, "Spectral Partitioning: The more eigenvectors the better", Proc. Design Automation Conf., 1995, pp. 195-200.

....ACM SIGDA benchmark circuits show up to 70 improvement over FM in cutsize, with an average of per circuit percent improvements of about 25 , and a total cut improvement of about 35 . They also outperform the recent placement based partitioning tool Paraboli [11] and the spectral partitioner MELO [12] by about 17 and 23 , respectively, with less CPU time. This demonstrates the potential of iterative improvement algorithms in dealing with the increasing complexity of modern VLSI circuitry. 1. INTRODUCTION The essence of VLSI circuit partitioning is to divide a circuit into a number of ....

....with respect to the number of pins in the circuit. This is done by moving one cell at a time and using an efficient bucket data structure. Krishnamurthy [4] enhanced FM by adding higher level lookahead gains and improved the results for small circuits. Recently, a number of clustering algorithms [9, 10, 11, 12, 15] have been proposed and excellent results have been obtained. FM and LA are the most commonly used two way partitioning algorithms largely due to their excellent run times, simple implementations and flexibility. However, this class of iterative improvement algorithms have a common weakness, viz. ....

[Article contains additional citation context not shown here]

C. J. Alpert and S-Z Yao, "Spectral Partitioning: The More Eigenvectors, the Better", Proc. ACM/IEEE Design Automation Conf., 1995.

.... techniques, along with some new approaches and optimizations, can yield a bipartitioning algorithm significantly better than the current state of the art [12] Our algorithm achieved a 8 improvement over PROP [5] 16 improvement over FBB [19] 22 improvement over Paraboli [18] and MELO [2], 50 improvement over Fiduccia Mattheyses [7] and a 58 improvement over EIG1 [10] 2] some of the best current bipartitioning algorithms. In this paper, we seek to understand the critical issues in logic replication, the selective duplication of logic to reduce the resulting cutset. We examine ....

.... algorithm significantly better than the current state of the art [12] Our algorithm achieved a 8 improvement over PROP [5] 16 improvement over FBB [19] 22 improvement over Paraboli [18] and MELO [2] 50 improvement over Fiduccia Mattheyses [7] and a 58 improvement over EIG1 [10] [2], some of the best current bipartitioning algorithms. In this paper, we seek to understand the critical issues in logic replication, the selective duplication of logic to reduce the resulting cutset. We examine most of the current logic replication literature, integrating it into our already ....

C. J. Alpert, S.-Z. Yao, "Spectral Partitioning: The More Eigenvectors, The Better", Design Automation Conference, pp. 195-200, 1995.

....so a hyperedge model is needed to approximate the hypergraph by a graph. The relation between the spectrum of a graph and other graph properties has been an area of active research [4, 8, 15, 25, 26, 43] but only recently spectrum based methods have been successfully applied to graph partitioning [1, 2, 21, 30, 31, 46, 47, 53]. We present below a spectral graph K partitioning formulation that extends Rend and Wolkowicz s to graphs containing weighted vertices. Other Global Methods. Simulated annealing (SA) has been used for graph partitioning by Johnson et al. 34] and Bui et al. 10] with mixed results. They showed ....

C.J. Alpert and S.-Z. Yao, "Spectral Partitioning: The More Eigenvectors, The Better," to appear in Proceedings 32nd ACM/IEEE Design Automation Conference, June, 1995.

.... scheme [Kr83] and the simulated annealing approach [KiGV83, GrSu84] The analytical methods use a linear placement formulation with either the quadratic wirelength objective function, which is solved by computing the second smallest eigenvector of the Laplacian matrix of the given circuit [Ba82, Bo87, DoHo73, HaKa92, AlYa95], or a linear wirelength objective function, which is solved by an iterative method in [RiDJ94, LiLC95] The min cut based method uses the maximum flow algorithm to compute a series of minimum cuts in the given circuit in order to obtain an area balanced cut with small cut size [YaWo94] The ....

....the cell movement, and the updated gain, generated from the cell movement afterwards. By focusing on the updated gain when choosing cells to move, they were very successful in removing big clusters from the cutset without any prior clustering. Their method outperforms Paraboli [RiDJ94] and MELO [AlYa95] substantially. Our research revealed that cluster removal can be accomplished more efficiently by focusing on removing nets in the cutset through our specially designed FM LSRb algorithm. Our FM LSRb algorithm combines hierarchical clustering based on Maximum Fanout Free Subgraph (MFFS) ....

C.J. Alpert, and S.-Z. Yao, "Spectral Partitioning: The More Eigenvectors, the Better", Proc. ACM/IEEE Design Automation Conference, pp. 195-200, 1995

....the techniques discussed in this paper, an algorithm based upon KLFM can produce results better than the current state of the art. In Table 10 we present the results of our algorithm (Strawman) along with results of four of the best current methods (Paraboli [Riess94] EIG1 [Hagen92] MELO [Alpert95b] and FBB [Yang94] on a set of standard benchmarks. Those benchmarks beginning with s are the XNF versions of the MCNC partitioning benchmark suite [MCNC93] while the rest were obtained from Charles Alpert s benchmark set [Alpert96] in NET format and translated into XNF. All partitioners were ....

....as we continue research in partitioning we properly place new concepts and optimizations in the context of what has already been discovered. Table 10. Quality comparison of partitioning methods. Values for basic FM and Strawman 2 are the best of ten trials. The EIG1 and MELO results are from [Alpert95b] (though EIG1 was proposed in [Hagen92] the Paraboli results are from [Riess94] and the FBB results are from [Yang94] All tests require partition sizes to be between 45 and 55 of the total circuit sizes, and assume that all non I O nodes have unit area. Example Nodes Nets Pins FM EIG1 ....

C. J. Alpert, S.-Z. Yao, "Spectral Partitioning: The More Eigenvectors, The Better", Dsign Automation Conference, pp. 195-200, 1995.

....layout strategy using the divide andconquer technique is indispensable in order to make the VLSI layout problem tractable. The existing circuit partitioning algorithms in the literature can be roughly classified into two classes; constructive methods, such as the spectral based methods [11, 3] and the network flow based method [19] and iterative improvement methods (also referred as group migration or move based methods) In practice, iterative improvement based partitioning (IIP) algorithms have been used extensively due to the following advantages over the other approaches; i) 1 ....

C. J. Alpert and S. Z. Yao. "Spectral partitioning: The more eigenvectors, the better". In Proc. ACM/IEEE Design Automation Conf., pages 195--200, 1995.

....by a graph. The relation between the spectrum of a graph and other graph properties has been an area of active research (Barnes, 1982; Boppana, 1987; Donath and Hoffman, 1973) but only recently spectrum based methods has been successfully applied to graph partitioning (Alpert and Kahng, 1993; Alpert and Yao, 1995; Hagen and Kahng, 1992; Hendrickson and Leland, 1992; Pothen et al. 1990; Simon, 1991) Michelena and Papalambros (1995) have extended formulations given by Rendl and Wolkowicz (1990) and Falkner et al. 1994) to account for weighted vertices. In the powertrain application below, the software ....

Alpert, C.J. and Yao, S.-Z., 1995, "Spectral Partitioning: The More Eigenvectors, The Better," Proceedings, 32nd ACM/IEEE Design Automation Conference.

....on FPGA can be used to implement different types of nodes in a netlist. Such multiple choices of implementation of a netlist on an FPGA can not be accurately captured by a simple area or gate count capacity metric. Though many algorithms have been proposed for circuit partitioning problems [1,2,3,4,5,7,8,9,10,11,12,13], the multiple resource types in an FPGA are not taken into consideration. A partitioning algorithm with simple resource capacity metric may produce partitioning results that actually violate resource constraints and thus render the results unusable. For a partitioning algorithm to be useful for ....

....are balanced by the total area of each subset, where the area of each cell is the number of resources (e:g LUTs) used. Many heuristic approaches have been proposed to solve this problem, such as the iterative improvement method (K L, FM) 1,2,8,9] simulated annealing[3] spectral based method[4,7,11] and network flow based method (FBB) 10] Even this special case is well known to be a NP complete problem, so is the general problem stated above. This leads to Lemma 1. Lemma 1: The problem of two way circuit partitioning with complex resource constraints is NP complete. Another special case ....

C. J. Alpert and S. Z. Yao, "Spectral Partitioning: The More Eigenvectors, the Better", Proc. ACM/IEEE Design Automation Conference, pp.195-200, 1995.

....system must be partitioned into subsystems such that elements in the same subsystem are strongly interconnected, whereas elements in different subsystems are weakly interconnected. Such applications include computer logic and page partitioning (Donath 1988) VLSI layout and packaging of circuits (Alpert and Yao, 1995; Dunlop and Kernighan, 1985) machine layout in manufacturing systems, assignment of computations to multiple processors (Hendrickson and Leland, 1992) and domain decomposition of finite element or OPTIMAL MODEL BASED DECOMPOSITION OF POWERTRAIN SYSTEM DESIGN 168 finite volume meshes for ....

....applied to hypergraphs, so a hyperedge model is needed to approximate the hypergraph by a graph. The relation between the spectrum of a graph and other graph properties has been an area of active research, but only recently spectrum based methods has been successfully applied to graph partitioning (Alpert and Yao, 1995; Hendrickson and Leland, 1992; Pothen et al. 1990) Michelena and Papalambros (1995a) have extended formulations given by Rendl and Wolkowicz (1990) and Falkner et al. 1994) to account for weighted vertices. In the powertrain application below, the software package Chaco (Hendrickson and ....

Alpert, C.J. and Yao, S.-Z., 1995, "Spectral Partitioning: The More Eigenvectors, The Better," Proceedings, 32nd ACM/IEEE Design Automation Conference.

.... the simulated annealing based approach [KGV83, GS84] The analytical method uses a linear placement formulation with either (i) the quadratic wire length objective function solved by computing the second smallest eigenvector of the Laplacian matrix of the given circuit [Bar82, Bop87, DH73, HK92, AY95] or (ii) the linear wire length objective function solved by an iterative method [RDJ94, LLC95] The mincut based method uses the maximum flow algorithm to compute a series of minimum cuts in the given circuit in order to obtain an area balanced cut with the smallest cutsize [YW94] The net ....

....the cell move, and the updated gain generated from the cell move afterwards. Then, by focusing on the updated gain when choosing cells to move, they were very successful in removing big clusters from the cutset without any prior clustering. Their method outperforms Paraboli [RDJ94] and MELO [AY95] substantially. However, our research reveals that cluster removal can be accomplished more efficiently by focusing intensively on removing nets in the cutset instead of cells. Every net in the cutset follows the free loose locked cycle, where a free net contains only free cells, a loose net ....

C. J. Alpert and S. Z. Yao. "Spectral partitioning: The more eigenvectors, the better". In Proc. ACM/IEEE Design Automation Conf., pages 195--200, 1995.

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C. J. Alpert and S.-Z. Yao, "Spectral Partitioning: The More Eigenvectors, the Better," Proc. ACM/IEEE Design Automation Conf., 1995, pp. 195-200.

....of the i th entries in a set of d eigenvectors. Then, modules with the same code are assigned to the same cluster. Despite its simplicity, this new clustering algorithm is strongly motivated by theoretical results for both spectral bipartitioning [6] and multi dimensional vector partitioning [4]. The algorithm also has linear time complexity (not including the eigenvector computation) and is at least as effective as previous clustering algorithms in terms of two phase Fiduccia Mattheyses bipartitioning. 1. INTRODUCTION Clustering of netlist hypergraphs can effectively reduce the ....

....assigned to the same cluster. In contrast to complicated geometric clustering techniques, our algorithm is completely obvious and, in addition, has strong theoretical motivation. We show that this algorithm is a natural extension of spectral bipartitioning [6] and also follows the recent result of [4] which establishes the equivalence of min cut graph partitioning and an eigenvector based vector partitioning formulation. We have tested the quality of these clusterings via two phase FM; our experiments show that simple eigenvector based clustering produces bipartitionings with cuts that are at ....

[Article contains additional citation context not shown here]

C. J. Alpert and S.-Z. Yao, "Spectral Partitioning: The More Eigenvectors, the Better," Proc. ACM/IEEE Design Automation Conf., 1995, pp. 195-200.

.... y d i remaining in Y , i.e. the one that maximizes the magnitude of the sum of the vectors in S plus y d i . Throughout MELO s execution, S should consist of a reasonably good subset for vector partitioning. MELO s complexity is no worse that O(dn 2 ) although speedups are possible (see [5]) This idea of iteratively constructing a cluster (i.e. subset of vectors) seems to be the most direct method for constructing a linear ordering. MELO may not seem much different from a greedy graph traversals (cf. 3] however, with our approach each added vector incorporates global ....

....to C 1 that optimizes this approximation. Such an approach is exactly equivalent to MELO. We have not yet discussed how to choose H. When d = n, H is inconsequential; however, when d 6= n, the choice of H can affect the ordering. We tried four different schemes for scaling the eigenvectors (see [5] for more details) H = 1, H = d 2 , H chosen to minimize P d j=1 ff 2 j1 j , and H = E1 Gamma P d j=1 ff 2 j1 j jCh j Gamma P d j=1 ff 2 j1 for a given P 2 [8] Our multi way partitioning results indicate the second scheme is slightly better, averaging 1.70 , 0.01 and ....

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C. J. Alpert and S.-Z. Yao, "Spectral Partitioning: The More Eigenvectors, the Better," UCLA CS Dept. Technical Report, #940036, October 1994.

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C. Alpert and S. Yao, Spectral partitioning: the more eigenvectors the better. In Proceedings of 32nd ACM/IEEE Design Automation Conference, 1995, pp. 195-200.

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C. Alpert and S. Yao, "Spectral partitioning: The more eigenvectors, the better," in 32nd ACM/IEEE Design Automation Conference, (San Francisco), pp. 195--200, 1995.

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