Density-friendly Graph Decomposition

Decomposing a graph into a hierarchical structure via k-core analysis is a standard operation in any modern graph-mining toolkit. k-core decomposition is a simple and efficient method that allows to analyze a graph beyond its mere de-gree distribution. More specifically, it is used to identify areas in the graph of increasing centrality and connected-ness, and it allows to reveal the structural organization of the graph. Despite the fact that k-core analysis relies on vertex de-grees, k-cores do not satisfy a certain, rather natural, density property. Simply put, the most central k-core is not nec-essarily the densest subgraph. This inconsistency between k-cores and graph density provides the basis of our study. We start by defining what it means for a subgraph to be locally-dense, and we show that our definition entails a nested chain decomposition of the graph, similar to the one given by k-cores, but in this case the components are ar-ranged in order of increasing density. We show that such a locally-dense decomposition for a graph G = (V,E) can be computed in polynomial time. The running time of the exact decomposition algorithm is O(|V |2|E|) but is signifi-cantly faster in practice. In addition, we develop a linear-time algorithm that provides a factor-2 approximation to the optimal locally-dense decomposition. Furthermore, we show that the k-core decomposition is also a factor-2 ap-proximation, however, as demonstrated by our experimental evaluation, in practice k-cores have different structure than locally-dense subgraphs, and as predicted by the theory, k-cores are not always well-aligned with graph density

Layout Decomposition for Quadruple Patterning Lithography and Beyond

For next-generation technology nodes, multiple patterning lithography (MPL) has emerged as a key solution, e.g., triple patterning lithography (TPL) for 14/11nm, and quadruple patterning lithography (QPL) for sub-10nm. In this paper, we propose a generic and robust layout decomposition framework for QPL, which can be further extended to handle any general K-patterning lithography (K$>$4). Our framework is based on the semidefinite programming (SDP) formulation with novel coloring encoding. Meanwhile, we propose fast yet effective coloring assignment and achieve significant speedup. To our best knowledge, this is the first work on the general multiple patterning lithography layout decomposition.Comment: DAC'201

Methodology for standard cell compliance and detailed placement for triple patterning lithography

As the feature size of semiconductor process further scales to sub-16nm technology node, triple patterning lithography (TPL) has been regarded one of the most promising lithography candidates. M1 and contact layers, which are usually deployed within standard cells, are most critical and complex parts for modern digital designs. Traditional design flow that ignores TPL in early stages may limit the potential to resolve all the TPL conflicts. In this paper, we propose a coherent framework, including standard cell compliance and detailed placement to enable TPL friendly design. Considering TPL constraints during early design stages, such as standard cell compliance, improves the layout decomposability. With the pre-coloring solutions of standard cells, we present a TPL aware detailed placement, where the layout decomposition and placement can be resolved simultaneously. Our experimental results show that, with negligible impact on critical path delay, our framework can resolve the conflicts much more easily, compared with the traditional physical design flow and followed layout decomposition

A High-Performance Triple Patterning Layout Decomposer with Balanced Density

Triple patterning lithography (TPL) has received more and more attentions from industry as one of the leading candidate for 14nm/11nm nodes. In this paper, we propose a high performance layout decomposer for TPL. Density balancing is seamlessly integrated into all key steps in our TPL layout decomposition, including density-balanced semi-definite programming (SDP), density-based mapping, and density-balanced graph simplification. Our new TPL decomposer can obtain high performance even compared to previous state-of-the-art layout decomposers which are not balanced-density aware, e.g., by Yu et al. (ICCAD'11), Fang et al. (DAC'12), and Kuang et al. (DAC'13). Furthermore, the balanced-density version of our decomposer can provide more balanced density which leads to less edge placement error (EPE), while the conflict and stitch numbers are still very comparable to our non-balanced-density baseline

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient $(1+\varepsilon)$-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density

Distance-generalized Core Decomposition

The $k$-core of a graph is defined as the maximal subgraph in which every vertex is connected to at least $k$ other vertices within that subgraph. In this work we introduce a distance-based generalization of the notion of $k$-core, which we refer to as the $(k,h)$-core, i.e., the maximal subgraph in which every vertex has at least $k$ other vertices at distance $\leq h$ within that subgraph. We study the properties of the $(k,h)$-core showing that it preserves many of the nice features of the classic core decomposition (e.g., its connection with the notion of distance-generalized chromatic number) and it preserves its usefulness to speed-up or approximate distance-generalized notions of dense structures, such as $h$-club. Computing the distance-generalized core decomposition over large networks is intrinsically complex. However, by exploiting clever upper and lower bounds we can partition the computation in a set of totally independent subcomputations, opening the door to top-down exploration and to multithreading, and thus achieving an efficient algorithm