## Geometric Embeddings for Faster and Better Multi-Way Netlist Partitioning (1993)

Venue: | Proc. ACM/IEEE Design Automation Conf |

Citations: | 30 - 15 self |

### BibTeX

@INPROCEEDINGS{Alpert93geometricembeddings,

author = {C. J. Alpert and A. B. Kahng},

title = {Geometric Embeddings for Faster and Better Multi-Way Netlist Partitioning},

booktitle = {Proc. ACM/IEEE Design Automation Conf},

year = {1993},

pages = {743--748}

}

### Years of Citing Articles

### OpenURL

### Abstract

We give new, effective algorithms for k-way circuit partitioning in the two regimes of k ø n and k = \Theta(n), where n is the number of modules in the circuit. We show that partitioning an appropriately designed geometric embedding of the netlist, rather than a traditional graph representation, yields improved results as well as large speedups. We derive d- dimensional geometric embeddings of the netlist via (i) a new "partitioning-specific" net model for constructing the Laplacian of the netlist, and (ii) computation of d eigenvectors of the netlist Laplacian; we then apply (iii) fast top-down and bottom-up geometric clustering methods. 1 Preliminaries In top-down layout synthesis of complex VLSI systems, the goal of partitioning/clustering is to reveal the natural circuit structure, via a decomposition into k subcircuits which minimizes connectivity between subcircuits. A generic problem statement is as follows: k-Way Partitioning: Given a circuit netlist G = (V; E) with jV j...

### Citations

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Citation Context ...nd [22], using their respective metrics. 1.2 The LKP Problem In the LKP regime, k = \Theta(n), e.g., k = n 5 . The LKP problem arises with two-phase enhancements of Fiduccia-Mattheyses (F-M) [2] [11] =-=[15]; a partit-=-ioning into small clusters induces a "compacted" netlist and reduces the solution space so that it can be searched more effectively. Thus, we wish to achieve LKP solutions with k just small ... |

305 | A linear time heuristic for improving network partitions - Fiduccia, Mattheyses - 1982 |

280 |
Clustering to minimize the maximum intercluster distance, Theoret
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- 1985
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Citation Context ...]. Fact 3: In general graphs whose edge weights do not satisfy the triangle inequality, neither Formulation 2 nor Formulation 3 may be approximated within any fixed constant factor of optimal for ks3 =-=[7]-=-. When distances satisfy the triangle inequality, as they do in geometry, several heuristics achieve performance ratio 2 for Formulation 2. We use the simple k-center (KC) technique of Gonzalez [7], w... |

247 |
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211 |
Fast spectral methods for ratio cut partitioning and clustering
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Citation Context ...nnected components obtained through the shortest-path deletion; its O(mn log n) time complexity also depends on two "accuracy" parameterssb and 1 \Delta . In [4], Chan et al. generalize the =-=result of [10] from 2-wa-=-y to k-way ratio cut partitioning. Chan et al. use the first k eigenvectors of the netlist Laplacian to construct an orthogonal "projector" which maps an ndimensional space into a k-dimensio... |

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Optimal algorithms for approximate clustering
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Citation Context ...ations 2 and 3, the following results are known. Fact 1: Formulations 2 and 3 are NP-Complete for ks3 and ds2 [16]. Fact 2: Solving Formulation 2 within a factor ! 2 of optimal is NP-complete for ds3 =-=[5]-=-. Fact 3: In general graphs whose edge weights do not satisfy the triangle inequality, neither Formulation 2 nor Formulation 3 may be approximated within any fixed constant factor of optimal for ks3 [... |

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- 1993
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Citation Context ...ive enumeration of all partitions of disconnected components obtained through the shortest-path deletion; its O(mn log n) time complexity also depends on two "accuracy" parameterssb and 1 \D=-=elta . In [4], Chan et -=-al. generalize the result of [10] from 2-way to k-way ratio cut partitioning. Chan et al. use the first k eigenvectors of the netlist Laplacian to construct an orthogonal "projector" which m... |

117 | On the complexity of some common geometric location problems
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- 1984
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Citation Context ...inated by the MST construction, e.g., O(n log n) in the plane [17]. With respect to Formulations 2 and 3, the following results are known. Fact 1: Formulations 2 and 3 are NP-Complete for ks3 and ds2 =-=[16]-=-. Fact 2: Solving Formulation 2 within a factor ! 2 of optimal is NP-complete for ds3 [5]. Fact 3: In general graphs whose edge weights do not satisfy the triangle inequality, neither Formulation 2 no... |

79 |
Ratio cut partitioning for hierarchical design
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- 1991
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Citation Context ...ble, we also include the "Recursive Ratio Cut" (RR) results that were reported in [22] (RSBipart is similar to RR; the former uses the spectral method of [10] while the latter uses the F-M v=-=ariant of [20]-=-). Here, as in [22], we assume uniform module areas. For each benchmark, and 2sks9, we report the best k-way partition obtained for 1sds10. Although our results are better than those of the recursive ... |

53 | A Proper Model for the Partitioning of Electrical Circuits - Schweikert, Kernighan - 1972 |

44 |
Improving the performance of the Kernighan-Lin and simulated annealing graph bisection algorithms
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- 1989
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Citation Context ... in [4] and [22], using their respective metrics. 1.2 The LKP Problem In the LKP regime, k = \Theta(n), e.g., k = n 5 . The LKP problem arises with two-phase enhancements of Fiduccia-Mattheyses (F-M) =-=[2] [11] [15]-=-; a partitioning into small clusters induces a "compacted" netlist and reduces the solution space so that it can be searched more effectively. Thus, we wish to achieve LKP solutions with k j... |

43 | On the Complexity of Clustering Problems - Brucker - 1978 |

25 |
An efficient eigenvector approach for finding netlist partitions
- Hadley, Mark, et al.
- 1992
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Citation Context ...ensional Euclidean space via the well-established relationship between eigenvectors of the netlist Laplacian 2 and minimum ratio cut partitionings (or minimum squared wirelength placements) (see [11] =-=[9]-=- for surveys). Every eigenvector of the Laplacian gives a distinct, one-dimensional spatial embedding of the circuit graph wherein strongly connected modules will tend to be placed close to each other... |

23 |
A General Purpose Multiple Way Partitioning Algorithm
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- 1991
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Citation Context ...d epitaxial growth, extensions of the FiducciaMattheyses iterative interchange bipartitioning algorithm [18], and a primal-dual iteration motivated by a generalization of the minimum ratio cut metric =-=[21]-=-. These methods use simple objective functions based Partial support for this work was provided by a Department of Defense Graduate Fellowship, by NSF MIP-9110696, NSF Young Investigator Award MIP-925... |

15 |
A Probabilistic Multicommodity-Flow Solution to Circuit Clustering
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- 1992
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Citation Context ...ation under the State of California MICRO program. only on the number of nets crossing partition boundaries; they moreover require the partition sizes to be specified in advance. Recently, Yeh et al. =-=[22] proposed a "sh-=-ortest-path clustering" (SPC) method, where "shortest paths" between random pairs of modules are iteratively deleted from the netlist graph until it becomes disconnected into multiple c... |

11 |
Construction d’une classification ascendante hiérarchique par la recherche en chaîne des voisins réciproques, Les Cahiers de l’Analyse des Données, VII
- Benzecri
- 1982
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Citation Context ...Linkage algorithm of Johnson [14]), which begins with each point in its own cluster and then iteratively merges a pair of clusters so as to minimize the increase in maximum cluster diameter. Benzecri =-=[1]-=- has given an O(n 2 ) implementation of this algorithm based on constructions of chains of nearest neighbors. AGG Algorithm 1. Initialize Pn = fC 1 ; C 2 ; : : : ; Cng s.t. each C i contains exactly o... |

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Citation Context ... 2 is NP-complete only for ks3. Indeed, for k = 2, there are wellknown efficient algorithms, e.g., bicoloring a maximum spanning tree divides the point set into an optimal min-diameter bipartitioning =-=[8]-=-. We may obtain a heuristic k-way partitioning by iteratively applying optimal bipartitioning to the largest current cluster; we call this the Divisive Min-Diameter approach. This technique illustrate... |

9 |
Kahng, "A New Approach to Effective Circuit Clustering
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Citation Context ...[4] and [22], using their respective metrics. 1.2 The LKP Problem In the LKP regime, k = \Theta(n), e.g., k = n 5 . The LKP problem arises with two-phase enhancements of Fiduccia-Mattheyses (F-M) [2] =-=[11] [15]; a p-=-artitioning into small clusters induces a "compacted" netlist and reduces the solution space so that it can be searched more effectively. Thus, we wish to achieve LKP solutions with k just s... |

8 |
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Citation Context ... 24.8 26.0 17.4 17.9 13.5 KC 2 [8] 34.6 33.6 34.4 30.7 27.5 16.4 17.4 13.5 Single-Linkage 1 [14] 49.5 39.2 39.9 23.7 13.5 Divisive Min-Diameter 2 [9] 59.5 46.8 42.9 32.8 13.5 Divisive Sum-Diameters 3 =-=[13]-=- 96.0 74.9 59.8 72.3 13.5 Agglom Sum-Diameters 3 -- 154.5 127.2 88.3 72.3 13.5 Table 1: Comparison of the various clustering objectives and algorithms for the Primary1 benchmark netlist, using the par... |

2 |
Multiple-Way Network Partitioning," in
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Citation Context ...blem arises in high-level system partitioning and floorplanning. Early approaches involved seeded epitaxial growth, extensions of the FiducciaMattheyses iterative interchange bipartitioning algorithm =-=[18]-=-, and a primal-dual iteration motivated by a generalization of the minimum ratio cut metric [21]. These methods use simple objective functions based Partial support for this work was provided by a Dep... |

1 |
Spectral K-Way Ratio Cut Graph Partitioning
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Citation Context ...new clique net model for constructing the adjacency matrix A. Recall that the clique net model 1 Recently, the authors of [4] have improved their algorithm by combining geometric and graph clustering =-=[23]-=-; see Section 5 for further discussion. 2 We represent the circuit netlist by a simple undirected graph G = (V; E) with jV j = n vertices v 1 ; : : : ; vn representing the n modules, and edges in E ca... |