Timing-Constrained Global Routing with Buffered Steiner Trees

This dissertation deals with the combination of two key problems that arise in the physical design of computer chips: global routing and buffering. The task of buffering is the insertion of buffers and inverters into the chip's netlist to speed-up signal delays and to improve electrical properties of the chip. Insertion of buffers and inverters goes alongside with construction of Steiner trees that connect logical sources with possibly many logical sinks and have buffers and inverters as parts of these connections. Classical global routing focuses on packing Steiner trees within the limited routing space. Buffering and global routing have been solved separately in the past. In this thesis we overcome the limitations of the classical approaches by considering the buffering problem as a global, multi-objective problem. We study its theoretical aspects and propose algorithms which we implement in the tool BonnRouteBuffer for timing-constrained global routing with buffered Steiner trees. At its core, we propose a new theoretically founded framework to model timing constraints inherently within global routing. As most important sub-task we have to compute a buffered Steiner tree for a single net minimizing the sum of prices for delays, routing congestion, placement congestion, power consumption, and net length. For this sub-task we present a fully polynomial time approximation scheme to compute an almost-cheapest Steiner tree with a given routing topology and prove that an exact algorithm cannot exist unless P=NP. For topology computation we present a bicriteria approximation algorithm that bounds both the geometric length and the worst slack of the topology. To improve the practical results we present many heuristic modifications, speed-up- and post-optimization techniques for buffered Steiner trees. We conduct experiments on challenging real-world test cases provided by our cooperation partner IBM to demonstrate the quality of our tool. Our new algorithm could produce better solutions with respect to both timing and routability. After post-processing with gate sizing and Vt-assignment, we can even reduce the power consumption on most instances. Overall, our results show that our tool BonnRouteBuffer for timing-constrained global routing is superior to industrial state-of-the-art tools

Fast Repeater Tree Construction

Repeaters are used during physical design of chips to improve the electrical and timing properties of interconnections. They are added along Steiner trees that connect root gates to sinks, creating repeater trees. Their construction became a crucial part of chip design. We present a new algorithm to solve the repeater tree construction problem. We first present an extensive version of the Repeater Tree Problem. Our problem formulation encapsulates most of the constraints that have been studied so far. We also consider several aspects for the first time, for example, slew dependent required arrival times at repeater tree sinks. The employed technology, the properties of available repeaters and metal wires, the shape of the chip, the temperature, the voltages, and many other factors highly influence the results of repeater tree construction. To take all this into account, we extensively preprocess the environment to extract parameters for our algorithms. We first present an algorithm for Steiner tree creation and prove that our algorithm is able to create timing-efficient as well as cost-efficient trees. Our algorithm is based on a delay model that accurately describes the timing that one can achieve after repeater insertion upfront. Next, we deal with the problem of adding repeaters to a given Steiner tree. The predominantly used algorithms to solve this problem use dynamic programming. However, they have several drawbacks. Firstly, potential repeater positions along the Steiner tree have to be chosen upfront. Secondly, the algorithms strictly follow the given Steiner tree and miss optimization opportunities. Finally, dynamic programming causes high running times. We present our new buffer insertion algorithm, Fast Buffering, that overcomes these limitations. It is able to produce results with similar quality to a dynamic programming approach but a much better running time. In addition, we also present improvements to the dynamic programming approach that allows us to push the quality at the expense of a high running time. We have implemented our algorithms as part of the BonnTools physical design optimization suite developed at the Research Institute for Discrete Mathematics in cooperation with IBM. Our implementation deals with all tedious details of a grown real-world chip optimization environment. We have created extensive experimental results on challenging real-world test cases provided by our cooperation partner. Our algorithm can solve about 5.7 million instances per hour

Timing-Constrained Global Routing with RC-Aware Steiner Trees and Routing Based Optimization

In this thesis we consider the global routing problem, which arises as one of the major subproblems in the physical design step in VLSI design. In global routing, we are given a three-dimensional grid graph G with edge capacities representing available routing space, and we have to connect a set of nets in G without overusing any edge capacities. Here, each net consists of a set of pins corresponding to vertices of G, where one pin is the sender of signals, while all other pins are receivers. Traditionally, next to obeying all edge capacity constraints, the objective has been to minimize wire length and possibly via (edges in z-direction) count, and timing constraints on the chip were only modeled indirectly. We present a new approach, where timing constraints are modeled directly during global routing: In joint work with Stephan Held, Dirk Mueller, Daniel Rotter, Vera Traub and Jens Vygen, we extend the modeling of global routing as a Min-Max Resource Sharing Problem to also incorporate timing constraints. For measuring signal delays we use the well-established Elmore delay model. One of the key subproblems here is the computation of Steiner trees minimizing a weighted sum of routing space usages and signal delays. For k pins, this problem is NP-hard to approximate within o(log k), and even the special case k = 2 is NP-hard, as was shown by Haehnle and Rotter. We present a fast approximation algorithm with strong approximation bounds for the case k = 2. For k > 2 we use a multi-stage approach based on modifying the topology of a short Steiner tree and using our algorithm for the two-pin case for computing new connections. Moreover, we present a layer assignment algorithm that assigns z-coordinates to the edges of a given two-dimensional tree. We also discuss the topic of routing based optimization. Here, the starting point is a complete routing, and timing optimization tools make changes that require incremental adaptations of the underlying routing. We investigate several aspects of this problem and derive a new routing flow that includes our timing-aware global router and routing based optimization steps. We evaluate our results from this thesis in practice on industrial 14nm microprocessor designs from IBM. Our theoretical results are validated in practice by a strong performance of our timing-aware global routing framework and our new routing flow, yielding significant improvements over the traditional global routing method and the previously used routing flow. Therefore, we conclude that our approaches and results from this thesis are not only theoretically sound but also give compelling results in practice

Performance and power optimization in VLSI physical design

As VLSI technology enters the nanoscale regime, a great amount of efforts have been made to reduce interconnect delay. Among them, buffer insertion stands out as an effective technique for timing optimization. A dramatic rise in on-chip buffer density has been witnessed. For example, in two recent IBM ASIC designs, 25% gates are buffers. In this thesis, three buffer insertion algorithms are presented for the procedure of performance and power optimization. The second chapter focuses on improving circuit performance under inductance effect. The new algorithm works under the dynamic programming framework and runs in provably linear time for multiple buffer types due to two novel techniques: restrictive cost bucketing and efficient delay update. The experimental results demonstrate that our linear time algorithm consistently outperforms all known RLC buffering algorithms in terms of both solution quality and runtime. That is, the new algorithm uses fewer buffers, runs in shorter time and the buffered tree has better timing. The third chapter presents a method to guarantee a high fidelity signal transmission in global bus. It proposes a new redundant via insertion technique to reduce via variation and signal distortion in twisted differential line. In addition, a new buffer insertion technique is proposed to synchronize the transmitted signals, thus further improving the effectiveness of the twisted differential line. Experimental results demonstrate a 6GHz signal can be transmitted with high fidelity using the new approaches. In contrast, only a 100MHz signal can be reliably transmitted using a single-end bus with power/ground shielding. Compared to conventional twisted differential line structure, our new techniques can reduce the magnitude of noise by 45% as witnessed in our simulation. The fourth chapter proposes a buffer insertion and gate sizing algorithm for million plus gates. The algorithm takes a combinational circuit as input instead of individual nets and greatly reduces the buffer and gate cost of the entire circuit. The algorithm has two main features: 1) A circuit partition technique based on the criticality of the primary inputs, which provides the scalability for the algorithm, and 2) A linear programming formulation of non-linear delay versus cost tradeoff, which formulates the simultaneous buffer insertion and gate sizing into linear programming problem. Experimental results on ISCAS85 circuits show that even without the circuit partition technique, the new algorithm achieves 17X speedup compared with path based algorithm. In the meantime, the new algorithm saves 16.0% buffer cost, 4.9% gate cost, 5.8% total cost and results in less circuit delay

Shortest Paths and Steiner Trees in VLSI Routing

Routing is one of the major steps in very-large-scale integration (VLSI) design. Its task is to find disjoint wire connections between sets of points on a chip, subject to numerous constraints. This problem is solved in a two-stage approach, which consists of so-called global and detailed routing steps. For each set of metal components to be connected, global routing reduces the search space by computing corridors in which detailed routing sequentially determines the desired connections as shortest paths. In this thesis, we present new theoretical results on Steiner trees and shortest paths, the two main mathematical concepts in routing. In the practical part, we give computational results of BonnRoute, a VLSI routing tool developed at the Research Institute for Discrete Mathematics at the University of Bonn. Interconnect signal delays are becoming increasingly important in modern chip designs. Therefore, the length of paths or direct delay measures should be taken into account when constructing rectilinear Steiner trees. We consider the problem of finding a rectilinear Steiner minimum tree (RSMT) that --- as a secondary objective --- minimizes a signal delay related objective. Given a source we derive some structural properties of RSMTs for which the weighted sum of path lengths from the source to the other terminals is minimized. Also, we present an exact algorithm for constructing RSMTs with weighted sum of path lengths as secondary objective, and a heuristic for various secondary objectives. Computational results for industrial designs are presented. We further consider the problem of finding a shortest rectilinear Steiner tree in the plane in the presence of rectilinear obstacles. The Steiner tree is allowed to run over obstacles; however, if it intersects an obstacle, then no connected component of the induced subtree must be longer than a given fixed length. This kind of length restriction is motivated by its application in VLSI routing where a large Steiner tree requires the insertion of repeaters which must not be placed on top of obstacles. We show that there are optimal length-restricted Steiner trees with a special structure. In particular, we prove that a certain graph (called augmented Hanan grid) always contains an optimal solution. Based on this structural result, we give an approximation scheme for the special case that all obstacles are of rectangular shape or are represented by at most a constant number of edges. Turning to the shortest paths problem, we present a new generic framework for Dijkstra's algorithm for finding shortest paths in digraphs with non-negative integral edge lengths. Instead of labeling individual vertices, we label subgraphs which partition the given graph. Much better running times can be achieved if the number of involved subgraphs is small compared to the order of the original graph and the shortest path problems restricted to these subgraphs is computationally easy. As an application we consider the VLSI routing problem, where we need to find millions of shortest paths in partial grid graphs with billions of vertices. Here, the algorithm can be applied twice, once in a coarse abstraction (where the labeled subgraphs are rectangles), and once in a detailed model (where the labeled subgraphs are intervals). Using the result of the first algorithm to speed up the second one via goal-oriented techniques leads to considerably reduced running time. We illustrate this with the routing program BonnRoute on leading-edge industrial chips. Finally, we present computational results of BonnRoute obtained on real-world VLSI chips. BonnRoute fulfills all requirements of modern VLSI routing and has been used by IBM and its customers over many years to produce more than one thousand different chips. To demonstrate the strength of BonnRoute as a state-of-the-art industrial routing tool, we show that it performs excellently on all traditional quality measures such as wire length and number of vias, but also on further criteria of equal importance in the every-day work of the designer

High-performance and Low-power Clock Network Synthesis in the Presence of Variation.

Semiconductor technology scaling requires continuous evolution of all aspects of physical design of integrated circuits. Among the major design steps, clock-network synthesis has been greatly affected by technology scaling, rendering existing methodologies inadequate. Clock routing was previously sufficient for smaller ICs, but design difficulty and structural complexity have greatly increased as interconnect delay and clock frequency increased in the 1990s. Since a clock network directly influences IC performance and often consumes a substantial portion of total power, both academia and industry developed synthesis methodologies to achieve low skew, low power and robustness from PVT variations. Nevertheless, clock network synthesis under tight constraints is currently the least automated step in physical design and requires significant manual intervention, undermining turn-around-time. The need for multi-objective optimization over a large parameter space and the increasing impact of process variation make clock network synthesis particularly challenging. Our work identifies new objectives, constraints and concerns in the clock-network synthesis for systems-on-chips and microprocessors. To address them, we generate novel clock-network structures and propose changes in traditional physical-design flows. We develop new modeling techniques and algorithms for clock power optimization subject to tight skew constraints in the presence of process variations. In particular, we offer SPICE-accurate optimizations of clock networks, coordinated to reduce nominal skew below 5 ps, satisfy slew constraints and trade-off skew, insertion delay and power, while tolerating variations. To broaden the scope of clock-network-synthesis optimizations, we propose new techniques and a methodology to reduce dynamic power consumption by 6.8%-11.6% for large IC designs with macro blocks by integrating clock network synthesis within global placement. We also present a novel non-tree topology that is 2.3x more power-efficient than mesh structures. We fuse several clock trees to create large-scale redundancy in a clock network to bridge the gap between tree-like and mesh-like topologies. Integrated optimization techniques for high-quality clock networks described in this dissertation strong empirical results in experiments with recent industry-released benchmarks in the presence of process variation. Our software implementations were recognized with the first-place awards at the ISPD 2009 and ISPD 2010 Clock-Network Synthesis Contests organized by IBM Research and Intel Research.Ph.D.Electrical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/89711/1/ejdjsy_1.pd

Optimization Schemes for Variability-Driven VLSI Design Automation

Today's IC design is facing several challenges due to increasing circuit complexity and decreasing feature size, as it pushes to extend Moore's law into nano-scale dimensions. Apart from the challenges that arise directly as a result of feature scaling (e.g., increasing leakage power, reliability issues), imperfections in the manufacturing process have recently turned into a major design hurdle, due to the variations they cause in the device and interconnect parameters from their target values. From an IC design automation perspective, a shift in paradigm, from deterministic to probabilistic, is needed to handle the unpredictable nature of these fabrication variations. In such a probabilistic paradigm, the varying circuit parameters such as leakage power or delay should be accurately modeled, and their correlations due to common sources of variations or physical location on the chip should be well captured. Moreover, variability-driven (probabilistic) design automation needs to efficiently generate a high quality solution. A particular challenge in variability-driven design automation is to define optimality measures among candidate solutions, which allow for inferior solutions to be removed from the solution space thus reducing the run-time complexity. In this dissertation, the superiority probability is introduced as such an optimality measure, and two methods are proposed to compute this probability: an accurate Conditional Monte Carlo simulation method, and an efficient moment-matching approximation method. The effectiveness of using the superiority probability is shown in the context of two important design automation applications: 1) the buffer insertion problem, 2) the dual-Vth leakage optimization problem. Another important task in variability-driven design automation is to develop optimization techniques that can provably generate the optimal solution in an efficient way. In this dissertation, the application of the gate sizing problem is explored to optimally reduce the loss due to fabrication variations in the presence of a timing constraint. The presented formulation, in contrast with the existing variability-driven approaches which are all based on heuristics, is provably optimal. Moreover, unlike existing approaches, it is independent of any assumption on the source and nature of variations

Synthesis of Clock Trees with Useful Skew based on Sparse-Graph Algorithms

Computer-aided design (CAD) for very large scale integration (VLSI) involve

Circuit delay optimization by buffering the logic gates

Avec la miniaturisation actuelle, les circuits démontrent de plus en plus l'importance des délais d'interconnexion. Afin de réduire ce délai, l'insertion de tampons doit être effectuée durant la synthèse logique et la synthèse physique. Cette activité d'optimisation est souvent basée sur la programmation dynamique. Dans ce mémoire, la technique branch-and-bound est utilisé et le problème pour le cas spécifique d'arbres de tampons équilibrés est résolu, où toutes les charges ont un temps requis et une capacité identique. Une analyse mathématique est faite pour tenir compte d'une variété de questions de conception telles que la topologie, la bibliothèque de tampons et le changement de phase en présence d'inverseur. En combinant la programmation dynamique et les techniques branch-and-bound, une méthode hybride est présentée qui améliore le temps d'exécution tout en conservant une utilisation de mémoire raisonnable. Les concepts mathématiques et algorithmiques fondamentaux utilisés dans ce mémoire peuvent être employés pour généraliser la méthode proposée pour un ensemble de charges avec des capacités et des temps requis différents