The modern integrated circuit is one of the most complex products that has been engineered to-date. It continues to grow in complexity as the years progress. As a result, very large-scale integrated (VLSI) circuit design now involves massive design teams employing state-of-the art computer-aided design (CAD) tools. One of the oldest, yet most important CAD problems for VLSI circuits is physical design automation, where one needs to compute the best physical layout of millions to billions of circuit components on a tiny silicon surface Lim08 . The process of mapping an electronic design to a chip involves a number of physical design stages, one of which is clustering. In this paper, we focus on problems in clustering which are critical for more sustainable chips. The clustering problem in combinatorial circuits alone is a source of multiple models. In particular, we consider the problem of clustering combinatorial circuits for delay minimization, when logic replication is not allowed ( CN ). The problem of delay minimization when logic replication is allowed ( CA ) has been well studied, and is known to be solvable in polynomial-time Wong1 . However, unbounded logic replication can be quite expensive. Thus, CN is an important problem. We show that selected variants of CN are NP-hard . We also obtain approximability and inapproximability results for these problems. A preliminary version of this paper appeared in Don15 .
In @cite_8 a polynomial-time optimal solution to the clustering problem is given assuming the unit delay model. In this model, there is no delay that is associated with any gate or with any edge linking two gates within a cluster. A delay of one unit of time is assumed along any edge crossing a cluster boundary. Let us note that in that paper replication of the logic is allowed, that is, a node may be present in many clusters. The more general delay model considered in this paper is presented in @cite_5 . In that paper an algorithm is given that constructs a clustering that aims to minimize the delay under the general delay model. Let us note that the algorithm achieves optimum delay under some specific conditions. In @cite_2 another clustering algorithm is presented, and in the same paper it is proved that the algorithm always finds the optimum clustering.
@cite_5 @cite_2 @cite_8
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This paper addresses the problem of circuit clustering for delay minimization, subject to capacity constraints. We use the general delay model, for which only heuristic solutions were known. We present an optimal polynomial time algorithm for combinational circuits under this model. Our algorithm can be generalized to solve the problem under any monotone clustering constraint. In a clustering problem, the aim is to partition a given set of n points in d -dimensional space into k groups, called clusters, so that points within each cluster are near each other. Two objective functions frequently used to measure the performance of a clustering algorithm are, for any L 4 metric, (a) the maximum distance between pairs of points in the same cluster, and (b) the maximum distance between points in each cluster and a chosen cluster center; we refer to either measure as the cluster size. We show that one cannot approximate the optimal cluster size for a fixed number of clusters within a factor close to 2 in polynomial time, for two or more dimensions, unless P=NP. We also present an algorithm that achieves this factor of 2 in time O ( n log k ), and show that this running time is optimal in the algebraic decision tree model. For a fixed cluster size, on the other hand, we give a polynomial time approximation scheme that estimates the optimal number of clusters under the second measure of cluster size within factors arbitrarily close to 1. Our approach is extended to provide approximation algorithms for the restricted centers, suppliers, and weighted suppliers problems that run in optimal O ( n log k ) time and achieve optimal or nearly optimal approximation bounds. The authors address the problem of clustering a circuit for minimizing its delay, subject to capacity constraints on the clusters. They present an algorithm for combinational circuits and give sufficient conditions under which it is optimum. In addition, they address the problem of minimizing the number of clusters and nodes without increasing the maximum delay found by the algorithm. Finally, they extend the clustering algorithm to minimize the clock cycle of a sequential synchronous circuit. > We consider two closely related problems: planted clustering and submatrix localization. The planted clustering problem assumes that a random graph is generated based on some underlying clusters of the nodes; the task is to recover these clusters given the graph. The submatrix localization problem concerns locating hidden submatrices with elevated means inside a large real-valued random matrix. Of particular interest is the setting where the number of clusters submatrices is allowed to grow unbounded with the problem size. These formulations cover several classical models such as planted clique, planted densest subgraph, planted partition, planted coloring, and stochastic block model, which are widely used for studying community detection and clustering bi-clustering. For both problems, we show that the space of the model parameters (cluster submatrix size, cluster density, and submatrix mean) can be partitioned into four disjoint regions corresponding to decreasing statistical and computational complexities: (1) the regime, where all algorithms fail; (2) the regime, where the computationally expensive Maximum Likelihood Estimator (MLE) succeeds; (3) the regime, where the polynomial-time convexified MLE succeeds; (4) the regime, where a simple counting thresholding procedure succeeds. Moreover, we show that each of these algorithms provably fails in the previous harder regimes. Our theorems establish the minimax recovery limit, which are tight up to constants and hold with a growing number of clusters submatrices, and provide a stronger performance guarantee than previously known for polynomial-time algorithms. Our study demonstrates the tradeoffs between statistical and computational considerations, and suggests that the minimax recovery limit may not be achievable by polynomial-time algorithms.