Utilizing Macromodels in Floating Random Walk Based Capacitance Extraction

The non-Manhattan conductor geometry existing in some capacitance extraction problems brings difficulty to the floating random walk (FRW) method using cubic transition domains. In this chapter, techniques are presented to enhance the FRW method for handling the structures with non-Manhattan conductors. Based on the aligned-box distances and corresponding calculating approaches, the techniques for generating the Gaussian surface and constructing axis-aligned transition cubes are presented. A practical strategy is then presented to judge the domination relationship of non-Manhattan conductor blocks for building the space management structure with candidate list. Finally, the strategy using rotated transition cube and related space management technique are presented to make further acceleration. Experiments on three-dimensional (3-D) interconnect structures including from eight to one thousand non-Manhattan blocks show that the developed method is from 2.9X to 96X faster than a simple extension of the original FRW method. The developed FRW solver is also up to 39X faster than a boundary element method based solver. Additional experiments are carried out to further validate the accuracy and efficiency of the presented techniques, and to demonstrate their suitability for large and multi-dielectric structures.

Capacitance extraction for interconnect parasitics modeling is the basis of accurate circuit simulation and physical verification for quality integrated circuit (IC) design. Due to its scalability and reliability, the floating random walk (FRW) based solver has been widely used for the capacitance extraction. In this paper, the recently developed techniques on distributed parallel FRW solver and for handling complex floating metals are presented. With them, the efficiency and accuracy of capacitance extraction/simulation for very large-scale integration (VLSI) circuit design are remarkably improved.

Due to the advantages on scalability and reliability, the floating random walk (FRW) algorithm has been widely adopted for calculating the capacitances among three-dimensional (3-D) conductors. This is evidenced by the industrial practice of interconnect capacitance extraction during the design of high-performance very large-scale integrated (VLSI) circuits. In this work, the FRW algorithm is enhanced through the distributed parallel computing. With an efficient and adaptive task allocation scheme, the communication among different computer nodes is largely reduced. A distributed algorithm for accelerating the space management is also proposed. They have been implemented with Message Passing Interface (MPI) and applied to the high-precision capacitance simulation for touchscreen design and the interconnect capacitance extraction of VLSI circuits. Experiments on a computer cluster show that the proposed techniques achieve up to 114X speedup while using 120 cores, and build up the space management structure for a VLSI case including two million conductor blocks in just 22 seconds (37X parallel speedup on 60 cores).

With the advancement of fabrication technology, the electrostatic coupling has increasing impact on the performance of very large-scale integrated (VLSI) circuits and micro-electromechanical systems (MEMS). For the structures in VLSI circuits which are mostly rectilinear geometries, the floating random walk (FRW) method using cubic transition domains has been successfully applied to calculate the electric capacitances among interconnect wires. It is also applicable to the calculation of electric field intensity, which becomes important for nowadays nanometer-technology circuits. In this chapter, an efficient approach is presented to parallelize the FRW algorithm with the graphic processing units (GPUs). GPU-friendly algorithmic flow and data structure are designed to reduce the divergence among random walks and the time of accessing the device memory. The techniques accelerating the extraction of multi-dielectric structures and for extracting multiple nets concurrently are also discussed. Numerical results are presented with several simple structures and larger ones from real VLSI circuits or MEMS. The results validate the accuracy of the presented techniques and demonstrate several tens times speedup due to the GPU-based parallel computing.

A number of knowledge graph (KG)-based recommendation algorithms have been introduced; KGs enable users and items and their attributes to be treated in an integrated way and structural information to be captured through graphs. There are many variations of the KG based recommendation algorithms. Among them, KG embedding is often used, but doing this does not take advantage of the meta-path-level proximity between users and items. This paper presents a flexible framework combining random walk and KG embedding methods. The random walk model is formulated on the basis of the similarity between nodes revealed by the KG embedding. This enables the metapath level proximity of users and items to be efficiently utilized. Comparison testing demonstrated that the proposed framework performs better than random- walk-only methods and KG-embedding-only methods, and slightly better than the existing method we have extended.

We present a divide-and-conquer algorithm to improve the three-dimensional (3-D) boundary element method (BEM) for capacitance extraction. We divide large interconnect structures into small sections, set new boundary conditions using the border for each section, solve each section, and then combine the results to derive the capacitance. The target application is critical nets, clock trees, or packages where 3-D accuracy is required. Our algorithm is a significant improvement over the traditional BEMs and their enhancements, such as the "window" method, where conductors far away are dropped, and the "shield" method where conductors hidden behind other conductors are dropped. Experimental results show that our algorithm is a magnitude faster than the traditional BEM and the window+shield method, for medium to large structures. The error of the capacitance computed by the new algorithm is within 2% for self capacitance and 7% for coupling capacitance, compared with the results obtained by solving the entire system using BEM. Furthermore, our algorithms gives accurate distributed RC, where none of the previous 3-D BEM algorithms and their enhancements can.

Efficient sampling of complex network with modified random walk strategies

<p><span>Due to network dynamics in self-organizing networks the resource discovery effort increases. To discover objects in unstructured peer-to-peer network, peers rely on traditional methods like flooding, random walk and probabilistic forwarding methods. With inadequate knowledge of paths, the peers have to flood the query message which creates incredible network traffic and overhead. Many of the previous works based on random walk were done in wired network. In this context random walk was better than flooding. But under MANETs random walk approach behaved differently increasing the overhead, due to frequent link failures incurred by mobility. Decentralized applications based on peer-to-peer computing are best candidates to run over such dynamic network. Issues of P2P service discovery in wired networks have been well addressed in several earlier works. This article evaluates the performance of random walk based resource discovery protocol over P2P Mobile Adhoc Network (MP2P) and suggests an improved scheme to suit MANET. Our version reduces the network overhead, lowers the battery power consumption, minimizes the query delay while providing equally good success rate. The protocol is validated through extensive NS-2 simulations. It is clear from the results that our proposed scheme is an alternative to the existing ones for such highly dynamic mobile network scenario.</span></p>

In this paper, we explore different Markovian random walk strategies on networks with transition probabilities between nodes defined in terms of functions of the Laplacian matrix. We generalize random walk strategies with local information in the Laplacian matrix, that describes the connections of a network, to a dynamic determined by functions of this matrix. The resulting processes are non-local allowing transitions of the random walker from one node to nodes beyond its nearest neighbors. We find that only two types of Laplacian functions are admissible with distinct behaviors for long-range steps in the infinite network limit: type (i) functions generate Brownian motions, type (ii) functions Lévy flights. For this asymptotic long-range step behavior only the lowest non-vanishing order of the Laplacian function is relevant, namely first order for type (i), and fractional order for type (ii) functions.In the first part, we discuss spectral properties of the Laplacian matrix and a series of relations that are maintained by a particular type of functions that allow to define random walks on any type of undirected connected networks. Once described general properties, we explore characteristics of random walk strategies that emerge from particular cases with functions defined in terms of exponentials, logarithms and powers of the Laplacian as well as relations of these dynamics with non-local strategies like Lévy flights and fractional transport. Finally, we analyze the global capacity of these random walk strategies to explore networks like lattices and trees and different types of random and complex networks.

Simulation of solute transport in discrete fracture networks using the time domain random walk method

Approximation of the Lévy–Feller advection–dispersion process by random walk and finite difference method

Thermal issues are critical in 2.5D/3D IC design, and liquid cooling provides an effective solution for heat dissipation. Widely used compact thermal models (CTMs) convert chips into circuit networks for fast thermal simulations. However, current matrix-solving acceleration methods for CTM-derived circuits are inadequate for high-speed iterative transient thermal analysis of large-scale liquid-cooled 2.5D/3D ICs during design optimization. In contrast, the random walk method can provide fast solutions for local nodes in large-scale circuit networks, but it is not applicable to the circuit networks of the CTMs with liquid cooling. In this paper, we propose LCTMwalk, a novel GPU-accelerated random walk method for transient thermal analysis of liquid-cooled 2.5D/3D ICs. To enable random walks on the liquid-cooled CTM-derived circuit network, we replace the voltage-controlled current source model with the diode model. Additionally, we improve the transient analysis by using a time-backward random walk with time-domain path reuse, accelerating the solution of temperature at local circuit nodes. Experimental results show LCTMwalk can solve million-scale cases in only 500 ms, and achieves a 14-22× speedup compared to the state-of-the-art alternating direction implicit (ADI) method with GPU. Besides, LCTMwalk exhibits good generalizability and can be applied to various 2.5D/3D IC structures with high accuracy (error<1 K compared to 3D-ICE).

We study the diffusive transport of Markovian random walks on arbitrary networks with stochastic resetting to multiple nodes. We deduce analytical expressions for the stationary occupation probability and for the mean and global first passage times. This general approach allows us to characterize the effect of resetting on the capacity of random walk strategies to reach a particular target or to explore the network. Our formalism holds for ergodic random walks and can be implemented from the spectral properties of the random walk without resetting, providing a tool to analyze the efficiency of search strategies with resetting to multiple nodes. We apply the methods developed here to the dynamics with two reset nodes and derive analytical results for normal random walks and Lévy flights on rings. We also explore the effect of resetting to multiple nodes on a comb graph, Lévy flights that visit specific locations in a continuous space, and the Google random walk strategy on regular networks.

Meta-path-based random walk strategy has attracted tremendous attention in heterogeneous network representation, which can capture network semantics with heterogeneous neighborhoods of nodes. Despite the success of meta-path-based random walk strategy in plain heterogeneous networks which contain no attributes, it remains unexplored how meta-path-based random walk strategy could be utilized on attributed heterogeneous networks to simultaneously capture structural heterogeneity and attribute proximity. Moreover, the importance of node attributes and structural relations generally varies across data sets, thus requiring careful considerations when they are incorporated into representations. To tackle these problems, we propose a novel method, Attributed Heterogeneous Network embedding based on Aggregate-path (AHNA), which generates aggregate-path-based random walks on attributed heterogeneous networks and adaptively fuses topological structures and node attributes based on the learned importance. Specifically, AHNA first converts node attributes to additional links in the network to deal with the heterogeneity of structures and attributes, which is followed by an adaptive random walk strategy to strike the importance balance between node attributes and topological structures, thereby generating high-quality representations. Extensive experiments are conducted on three real-world data sets, where AHNA outperforms state-of-the-art approaches by up to 22.7%, 2.6%, and 2.3% on link prediction, community detection, and node classification, respectively. Moreover, our qualitative analysis indicates that AHNA can capture different balances of topological structures and node attributes on various data sets and thus boost the quality of node representations.

Accurate segmentation of pulmonary nodules is an important basis for doctors to determine lung cancer. Aiming at the problem of incorrect segmentation of pulmonary nodules, especially the problem that it is difficult to separate adhesive pulmonary nodules connected with chest wall or blood vessels, an improved random walk method is proposed to segment difficult pulmonary nodules accurately in this paper. The innovation of this paper is to introduce geodesic distance to redefine the weights in random walk combining the coordinates of the nodes and seed points in the image with the space distance. The improved algorithm is used to achieve the accurate segmentation of pulmonary nodules. The computed tomography (CT) images of 17 patients with different types of pulmonary nodules were selected for segmentation experiments. The experimental results are compared with the traditional random walk method and those of several literatures. Experiments show that the proposed method has good accuracy in the segmentation of pulmonary nodule, and the accuracy can reach more than 88% with segmentation time is less than 4 seconds. The results could be used to assist doctors in the diagnosis of benign and malignant pulmonary nodules and improve clinical efficiency.