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

Obstacle-avoiding rectilinear Steiner tree.

Li, Liang.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 57-61).Abstract also in Chinese.Abstract --- p.iAcknowledgement --- p.ivChapter 1 --- Introduction --- p.1Chapter 1.1 --- Background --- p.1Chapter 1.1.1 --- Partitioning --- p.1Chapter 1.1.2 --- Floorplanning and Placement --- p.2Chapter 1.1.3 --- Routing --- p.2Chapter 1.1.4 --- Compaction --- p.3Chapter 1.2 --- Motivations --- p.3Chapter 1.3 --- Problem Formulation --- p.4Chapter 1.3.1 --- Properties of OARSMT --- p.4Chapter 1.4 --- Progress on the Problem --- p.4Chapter 1.5 --- Contributions --- p.5Chapter 1.6 --- Thesis Organization --- p.6Chapter 2 --- Literature Review on OARSMT --- p.8Chapter 2.1 --- Introduction --- p.8Chapter 2.2 --- Previous Methods --- p.9Chapter 2.2.1 --- OARSMT --- p.9Chapter 2.2.2 --- Shortest Path Problem with Blockages --- p.13Chapter 2.2.3 --- OARSMT with Delay Minimization --- p.14Chapter 2.2.4 --- OARSMT with Worst Negative Slack Maximization --- p.14Chapter 2.3 --- Comparison --- p.15Chapter 3 --- Heuristic Method --- p.17Chapter 3.1 --- Introduction --- p.17Chapter 3.2 --- Our Approach --- p.18Chapter 3.2.1 --- Handling of Multi-pin Nets --- p.18Chapter 3.2.2 --- Propagation --- p.20Chapter 3.2.3 --- Backtrack --- p.23Chapter 3.2.4 --- Finding MST --- p.26Chapter 3.2.5 --- Local Refinement Scheme --- p.26Chapter 3.3 --- Experimental Results --- p.28Chapter 3.4 --- Summary --- p.28Chapter 4 --- Exact Method --- p.32Chapter 4.1 --- Introduction --- p.32Chapter 4.2 --- Review on GeoSteiner --- p.33Chapter 4.3 --- Overview of our Approach --- p.33Chapter 4.4 --- FST with Virtual Pins --- p.34Chapter 4.4.1 --- Definition of FST --- p.34Chapter 4.4.2 --- Notations --- p.36Chapter 4.4.3 --- Properties of FST with Virtual Pins --- p.36Chapter 4.5 --- Generation of FST with Virtual Pins --- p.46Chapter 4.5.1 --- Generation of FST with Two Pins --- p.46Chapter 4.5.2 --- Generation of FST with 3 or More Pins --- p.48Chapter 4.6 --- Concatenation of FSTs with Virtual Pins --- p.50Chapter 4.7 --- Experimental Results --- p.52Chapter 4.8 --- Summary --- p.53Chapter 5 --- Conclusion --- p.55Bibliography --- p.6

Computing Shortest Paths in the Plane with Removable Obstacles

We consider the problem of computing a Euclidean shortest path in the presence of removable obstacles in the plane. In particular, we have a collection of pairwise-disjoint polygonal obstacles, each of which may be removed at some cost c_i > 0. Given a cost budget C > 0, and a pair of points s, t, which obstacles should be removed to minimize the path length from s to t in the remaining workspace? We show that this problem is NP-hard even if the obstacles are vertical line segments. Our main result is a fully-polynomial time approximation scheme (FPTAS) for the case of convex polygons. Specifically, we compute an (1 + epsilon)-approximate shortest path in time O({nh}/{epsilon^2} log n log n/epsilon) with removal cost at most (1+epsilon)C, where h is the number of obstacles, n is the total number of obstacle vertices, and epsilon in (0, 1) is a user-specified parameter. Our approximation scheme also solves a shortest path problem for a stochastic model of obstacles, where each obstacle\u27s presence is an independent event with a known probability. Finally, we also present a data structure that can answer s-t path queries in polylogarithmic time, for any pair of points s, t in the plane

Bicriteria Rectilinear Shortest Paths among Rectilinear Obstacles in the Plane

Given a rectilinear domain P of h pairwise-disjoint rectilinear obstacles with a total of n vertices in the plane, we study the problem of computing bicriteria rectilinear shortest paths between two points s and t in P. Three types of bicriteria rectilinear paths are considered: minimum-link shortest paths, shortest minimum-link paths, and minimum-cost paths where the cost of a path is a non-decreasing function of both the number of edges and the length of the path. The one-point and two-point path queries are also considered. Algorithms for these problems have been given previously. Our contributions are threefold. First, we find a critical error in all previous algorithms. Second, we correct the error in a not-so-trivial way. Third, we further improve the algorithms so that they are even faster than the previous (incorrect) algorithms when h is relatively small. For example, for computing a minimum-link shortest s-t path, the previous algorithm runs in O(n log^{3/2} n) time while the time of our new algorithm is O(n + h log^{3/2} h)

Hardness and Approximation of Octilinear Steiner Trees

Given a point set K of terminals in the plane, the octilinear Steiner tree problem is to find a shortest tree that interconnects all terminals and edges run either in horizontal, vertical, or 45° diagonal direction. This problem is fundamental for the novel octilinear routing paradigm in VLSI design, the so-called X-architecture. As the related rectilinear and the Euclidian Steiner tree problem are well-known to be NP-hard, the same was widely believed for the octilinear Steiner tree problem but left open for quite some time. In this paper, we prove the NP-completeness of the decision version of the octilinear Steiner tree problem. We also show how to reduce the octilinear Steiner tree problem to the Steiner tree problem in graphs of polynomial size with the following approximation guarantee. We construct a graph of size O(n^2/epsilon^2) which contains a (1+epsilon)-approximation of a minimum octilinear Steiner tree for every epsilon > 0 and n = |K|. Hence, we can apply any k-approximation algorithm for the Steiner tree problem in graphs (the currently best known bound is k=1.55) and achieve an (k+epsilon)-approximation bound for the octilinear Steiner tree problem. This approximation guarantee also holds for the more difficult case where the Steiner tree has to avoid blockages (obstacles bounded by octilinear polygons)

Initial detailed routing algorithms

In this work, we present a study of the problem of routing in the context of the VLSI physical synthesis flow. We study the fundamental routing algorithms such as maze routing, A*, and Steiner tree-based algorithms, as well as some global routing algorithms, namely FastRoute 4.0 and BoxRouter 2.0. We dissect some of the major state of the art initial detailed routing tools, such as RegularRoute, TritonRoute, SmartDR and Dr.CU 2.0. We also propose an initial detailed routing flow, and present an implementation of the proposed routing flow, with a track assignment technique that models the problem as an instance of the maximum independent weighted set (MWIS) and utilizes integer linear programming (ILP) as a solver. The implementation of the proposed initial detailed routing flow also includes an implementation of multiple-source and multiple-target A* for terminal andnet connection with adjustable rules and weights. Finally, we also present a study of the results obtained by the implementation of the proposed initial detailed routing flow and a comparison with the ISPD 2019 contest winners, considering the ISPD 2019 and benchmark suite and evaluation tools.Neste trabalho, apresentamos um estudo do problema de roteamento no contexto do fluxo de síntese física de circuitos integrados VLSI. Nós estudamos algoritmos de roteamento fundamentais como roteamento de labirinto, A* e baseados em árvores de Steiner, além de alguns algoritmos de roteamento global como FastRoute 4.0 e BoxRouter 2.0. Nós dissecamos alguns dos principais trabalhos de roteamento detalhado inicial do estado da arte, como RegularRoute, TritonRoute, SmartDR e Dr.CU 2.0. Também propomos um fluxo de roteamento detalhado inicial, e apresentamos uma implementação do fluxo de roteametno proposto, com uma técnica de assinalamento de trilhas que modela o problema como uma instância do problema do conjunto independente de peso máximo e usa programação linear inteira como um resolvedor. A implementação do fluxo de rotemaento detalhado inicial proposto também inclui uma implementação de um A* com múltiplas fontes e múltiplos destinos para conexão de terminais e redes, com regras e pesos ajustáveis. Por fim, nós apresentamos um estudo dos resultados obtidos pela implementação do fluxo de roteamento detalhado inicial proposto e comparamos com os vencedores do ISPD 2019 contest considerando a suíte de teste e ferramentas de avaliação do ISPD 2019

The Steiner Ratio for the Obstacle-Avoiding Steiner Tree Problem

This thesis examines the (geometric) Steiner tree problem: Given a set of points P in the plane, find a shortest tree interconnecting all points in P, with the possibility of adding points outside P, called the Steiner points, as additional vertices of the tree. The Steiner tree problem has been studied in different metric spaces. In this thesis, we study the problem in Euclidean and rectilinear metrics. One of the most natural heuristics for the Steiner tree problem is to use a minimum spanning tree, which can be found in O(nlogn) time . The performance ratio of this heuristic is given by the Steiner ratio, which is defined as the minimum possible ratio between the lengths of a minimum Steiner tree and a minimum spanning tree. We survey the background literature on the Steiner ratio and study the generalization of the Steiner ratio to the case of obstacles. We introduce the concept of an anchored Steiner tree: an obstacle-avoiding Steiner tree in which the Steiner points are only allowed at obstacle corners. We define the obstacle-avoiding Steiner ratio as the ratio of the length of an obstacle-avoiding minimum Steiner tree to that of an anchored obstacle-avoiding minimum Steiner tree. We prove that, for the rectilinear metric, the obstacle-avoiding Steiner ratio is equal to the traditional (obstacle-free) Steiner ratio. We conjecture that this is also the case for the Euclidean metric and we prove this conjecture for three points and any number of obstacles

Approximation Algorithms for Network Design Problems

We consider different variants of network design problems. Given a set of points in the plane we search for a shortest interconnection of them. In this general formulation the problem is known as Steiner tree problem. We consider the special case of octilinear Steiner trees in the presence of hard and soft obstacles. In an octilinear Steiner tree the line segments connecting the points are allowed to run either in horizontal, vertical or diagonal direction. An obstacle is a connected region in the plane bounded by a simple polygon. No line segment of an octilinear Steiner tree is allowed to lie in the interior of a hard obstacle. If we intersect a Steiner tree with the interior of a soft obstacle, no connected component of the induced subtree is allowed to be longer than a given fixed length. We provide polynomial time approximation schemes for the octilinear Steiner tree problem in the presence of hard and soft obstacles. These were the first presented approximation schemes introduced for the problems. Additionally, we introduce a (2+epsilon)-approximation algorithm for soft obstacles. We then turn to Euclidean group Steiner trees. Instead of a set of fixed points we get for each point a set of potential locations (combined into groups) and we need to pick only one location of each group. The groups we consider lie inside disjoint regions fulfilling a special property so-called alpha-fatness. Roughly speaking, the term alpha-fat specifies the shape of the region in comparison to a disk. We give the first approximation algorithm for this problem and achieve an approximation ratio of (1+epsilon)(9.093alpha +1). Last, we consider Manhattan networks. They are allowed to contain edges only in horizontal and vertical direction. In contrast to Steiner trees they contain a shortest path between each pair of points. We introduce insights into the structure of Manhattan networks, particularly in the context of so-called staircases. We give three new approximation algorithms for the Manhattan network problem, the first with approximation ratio 3 and two algorithms with ratio 2. To this end we introduce two algorithms for the Manhattan network problem of staircases. The first algorithm solves the problem to optimality the second yields a 2-approximation. Variants of both algorithms are already known in the literature. Since we use a slightly different definition of staircases and we need special properties of them, we adopt the algorithms to our situation. The 2-approximation algorithms achieve the best known approximation ratio of an algorithm for the Manhattan network problem known so far. Last we give an idea how we could possibly find an algorithm with better approximation ratio