A preliminary study of a diagonal channel-routing model

The layout of two-terminal nets in a VLSI channel is realized in a new diagonal channel-routing model (DCRM), where the tracks are segments respectively displayed at +45 ° and –45 ° on the two layers of the channel. A new definition of channel density is introduced, and a lower bound to the channel width is derived by the application of an algorithm, whose complexity is evaluated as a function of the channel density, and other parameters of the problem.A simple linear-time algorithm is proposed, which produces an optimal layout (i.e., it requires a channel of minimum width) if the length of the longest net equals the lower bound for the channel width. In any case, the number of vias is at most one for each net. Some particular solutions are proposed for problems with long nets.Specific problems are much easier in DCRM than in the classical Manhattan model. For example, any shift-by-i can be realized in DCRM in a channel of widthi.This work has been supported by Consiglio Nazionale delle Ricerche of Italy under a research grant.     

Channel routing in a Diagonal Overlap model

In this paper we study the channel routing problem (CRP) in a new routing model, called Overlap Diagonal Model (ODM), where the grid consists of right and left tracks displayed at +45° and ?45°. For the unrestrictedoverlap, we present an algorithm which achieves \(w = \left[ {\frac{d}{2}} \right] + 1\) , while for the restricted-overlap, we havew=d+1, whered is the channel density,w is the channel width.     

A preliminary study of a diagonal channel-routing model

The layout of two-terminal nets in a VLSI channel is realized in a new diagonal channel-routing model (DCRM), where the tracks are segments respectively displayed at +45 ° and ?45 ° on the two layers of the channel. A new definition of channel density is introduced, and a lower bound to the channel width is derived by the application of an algorithm, whose complexity is evaluated as a function of the channel density, and other parameters of the problem. A simple linear-time algorithm is proposed, which produces an optimal layout (i.e., it requires a channel of minimum width) if the length of the longest net equals the lower bound for the channel width. In any case, the number of vias is at most one for each net. Some particular solutions are proposed for problems with long nets. Specific problems are much easier in DCRM than in the classical Manhattan model. For example, any shift-by-i can be realized in DCRM in a channel of widthi.     

Finding All Minimal Shapes in a Routing Channel

We present an algorithm to find a set of all optimal solutions for a channel placement problem. A channel consists of two rows of horizontal line segments (representing components). Each line segment contains some terminals with fixed positions. Sets of terminals, called nets, are to be connected. The relative ordering of line segments in each row is fixed. The line segments can be shifted left or right, which will affect the width needed for routing and the length of the channel. We want to find the tradeoff between channel length and routing width. Since the channel routing problem is NP-complete, we use a lower bound on routing width, called density. The density of a placement is the maximum number of nets crossing each vertical cut. We can increase the total length to minimize the channel density, or minimize the total length by increasing the channel density. The pair (density, total length) is called the shape of a placement. A shape is minimal if a decrease in density would cause an increase in total length, and vice versa. Our algorithm computes all the minimal shapes in O (N ⁴ ) time, where N is the number of nets. This is the first known algorithm for this problem whose running time is polynomial in the number of nets and independent of the length of the channel. Received January 18, 1996; revised December 2, 1996.     

Multiterminal global routing: A deterministic approximation scheme

We consider the problem of routing multiterminal nets in a two-dimensional gate-array. Given a gate-array and a set of nets to be routed, we wish to find a routing that uses as little channel space as possible. We present a deterministic approximation algorithm that uses close to the minimum possible channel space. We cast the routing problem as a new form of zero-one multicommodity flow, an integer-programming problem. We solve this integer program approximately by first solving its linear-program relaxation and then rounding any fractions that appear in the solution to the linear program. The running time of the rounding algorithm is exponential in the number of terminals in a net but polynomial in the number of nets and the size of the array. The algorithm is thus best suited to cases where the number of terminals on each net is small.This work was done while the authors were with the Computer Science Division, University of California at Berkeley. The work of Prabhakar Raghavan was supported by an IBM Doctoral Fellowship, and the work of Clark Thompson was supported by a California State MICRO grant (AT&T Foundation).     

Optimal algorithms for adjacent side routing

We consider the switchbox routing problem of two-terminal nets in the case when all thek nets lie on two adjacent sides of the rectangle. Our routing model is the standard two-layer model. We develop an optimal algorithm that routes all the nets whenever a routing exists. The routing obtained uses the fewest possible number of vias. A more general version of this problem (adjacent staircase) is also optimally solved.This research was supported in part by NSA Contract No. MDA-904-85H-0015, NSF Grant No. DCR-86-00378, and by NSF Engineering Research Centers Program NSFD CDR 88003012.     

Two-layer channel routing with vertical unit-length overlap

We show that anyn-net 2-terminal channel routing problem of densityd can be wired on a two-layer grid of widthw =d +O(d ^2/3) when vertical wire segments are allowed to overlap for a distance of length 1. This is a considerable asymptotic improvement over the best known, and optimal, channel width of 2d-1 for models in which no vertical overlap is allowed [RBM, PL]. Our result also improves the 3d/2+O(1) channel width achieved by a recent algorithm [G] for the same vertical overlap model. The algorithm presented in this paper produces at most 4 overlaps of unit length between any two nets, usesO(n) contacts, and can be implemented to run inO(nd ^2/3) time. We also generalize the algorithm to multi-terminal channel routing problems for which our algorithm uses a width ofw = 2d +O(d ^2/3).This work was supported by the Office of Naval Research under Contract N00014-84-K-0502 and by the National Science Foundation under Grant DMC-84-13496.     

基于图论模型的一类集成电路布线算法

针对具有曼哈顿模型的一类通道布线,提出了一个依据图论模型的最优轨道高度布线算法。算法根据通道上结点的水平约束图和垂直约束图,依次安排好每一个结点的布线轨道,进而通过通孔可以把所有的结点在2层轨道上布线完成。通过计算分析,该算法相对以前的算法能够达到更优的布线高度,并且其复杂性保持不变。     

Multiterminal global routing: A deterministic approximation scheme

We consider the problem of routing multiterminal nets in a two-dimensional gate-array. Given a gate-array and a set of nets to be routed, we wish to find a routing that uses as little channel space as possible. We present a deterministic approximation algorithm that uses close to the minimum possible channel space. We cast the routing problem as a new form of zero-one multicommodity flow, an integer-programming problem. We solve this integer program approximately by first solving its linear-program relaxation and then rounding any fractions that appear in the solution to the linear program. The running time of the rounding algorithm is exponential in the number of terminals in a net but polynomial in the number of nets and the size of the array. The algorithm is thus best suited to cases where the number of terminals on each net is small.     

Hierarchical channel router

The channel routing problem is a special case of the wire routing problem when interconnections have to be performed within a rectangular strip having no obstructions between terminals located on opposite sides of the rectangle.We present here a new channel routing algorithm, based on reduction of the problem to the case of a ($\text{2 × n}$) grid and on consistent utilization of a ‘divide-and-conquer’ approach. For the current implementation of the algorithm, the running time is proportional to $\text{N}{}^1\text{n}\text{log}\text{(m)}$ where $\text{N}$ is the number of nets, $\text{n}$ the length of the channel (number of columns) and $\text{m}$ the width of the channel (number of tracks).Traditional technological restrictions are assumed, ie, net terminals are located on vertical grid lines, two wiring layers are available for interconnections. One layer is used exclusively for vertical segments and another for horizontal. Vias are introduced for each layer change.This algorithm consistently outperforms several known routers in quality of wiring. We tested the algorithm on several benchmark problems. One of them, Deutsch's ‘difficult example’, was routed with only 19 horizontal wiring tracks (the absolute minimum for this case), whereas all other known routers required 20 or more tracks.     

Two algorithms for three-layer channel routing

A channel router is an important design aid in the design automation of VLSI circuit layout. Many algorithms have been developed based on various wiring models with routing done on two layers. With the recent advances in VLSI process technology, it is possible to have three independent layers for interconnection. In this paper two algorithms are presented for three-layer channel routing. The first assumes a very simple wiring model. This enables the routing problem to be solved optimally in a time of $\text{O}\text{(}\text{n}\text{log}\text{n}\text{)}$. The second algorithm is for a different wiring model and has an upper bound of $\text{O(}\text{n}{}^2\text{)}$ for its execution time. It uses fewer horizontal tracks than the first algorithm. For the second model the channel width is not bounded by the channel density.     

Channel routing in knock-knee mode: Simplified algorithms and proofs

We give unified and simplified algorithms and proofs for three results on channel routing in knock-knee mode. LetP be a channel routing problem with densityd _max.

1. [Rivest/Baratz/Miller, Preparata/Lipski]. If all nets inP are two-terminal nets thend _max tracks suffice.
2. [Preparata/Sarrafzadeh]. If all nets inP are two- or three-terminal nets then [3d _max/2] tracks suffice.
3. [Sarrafzadeh/Preparata]. 2d _max-1 tracks always suffice.

In all three cases a solution can be found in linear time; this is an improvement in case (b).     

Optimal multilayer channel routing with overlap

This paper presents algorithms for multiterminal net channel routing where multiple interconnect layers are available. Major improvements are possible if wires are able to overlap, and our generalized main algorithm allows overlap, but only on everyKth (K ≥ 2) layer. Our algorithm will, for a problem with densityd onL layers,L ≥K + 3,provably use at most three tracks more than optimal: ?(d + 1)/?L/K?? + 2 tracks, compared with the lower bound of ?d/?L/K??. Our algorithm is simple, has few vias, tends to minimize wire length, and could be used if different layers have different grid sizes. Finally, we extend our algorithm in order to obtain improved results for adjacent (K = 1) overlap: ?(d + 2)/?2L/3?? + 5 forL ≥ 7.     

Artificial neural net applications in telecommunication systems

The artificial neural net development has had something of a renaissance in the last decade with an impressive range of application areas. From the viewpoint of telecommunication networks and systems, an increasing number of studies can be observed in recent literature dealing with proposed applications of neural nets in telecommunication environments, such as connection admission control in broadband networks, the control of high-speed interconnection networks, channel allocation in cellular mobile systems, adaptive routing, etc. These proposed applications largely use three main neural net classes: feed-forward nets with backpropagation learning, Hopfield feedback nets, and selforganising neural nets. In this paper, we first give an overview of neural net classes and their main properties, and then present a review of applications in telecommunication systems, where attention is devoted to numerical aspects such as the convergence property and learning speed of the proposed neural nets.     

一种通孔最小化的三层通道布线算法

本文提出的连通孔最小化的三层通道布线算法,根据线网的接点位置建立线网分层图,并对此图进行三着色,确定出初始布线集和分拆线网集,通过引进线网次序图、压缩空段长度、填充原则和逐步排障法等,将初始布线集和拆网集的线网分配到相应的走线道上.实现线网的互连.本算法已用PASCAL语言编程,并在XT/286机上实现.结果表明,该算法不仅使通孔数大大减少,而且有些例子的走线道数也较一般布线法少.     

Optimal algorithms for adjacent side routing

We consider the switchbox routing problem of two-terminal nets in the case when all thek nets lie on two adjacent sides of the rectangle. Our routing model is the standard two-layer model. We develop an optimal algorithm that routes all the nets whenever a routing exists. The routing obtained uses the fewest possible number of vias. A more general version of this problem (adjacent staircase) is also optimally solved.     

Optimal multilayer channel routing with overlap

This paper presents algorithms for multiterminal net channel routing where multiple interconnect layers are available. Major improvements are possible if wires are able to overlap, and our generalized main algorithm allows overlap, but only on everyKth (K 2) layer. Our algorithm will, for a problem with densityd onL layers,L K + 3,provably use at most three tracks more than optimal: (d + 1)/L/K + 2 tracks, compared with the lower bound of d/L/K. Our algorithm is simple, has few vias, tends to minimize wire length, and could be used if different layers have different grid sizes. Finally, we extend our algorithm in order to obtain improved results for adjacent (K = 1) overlap: (d + 2)/2L/3 + 5 forL 7.This work was supported by the Semiconductor Research Corporation under Contract 83-01-035, by a grant from the General Electric Corporation, and by a grant at the University of the Saarland.     

一种新的与线网顺序无关的随机优化总体布线算法

针对目前总体布线中仍然存在的3个关键问题;布线结果受布线顺序的影响、总体布线图中拥挤区域的不可预见性、线网连接式样受到算法的限制等,该文提出了一种新的不受线网顺序影响的总体布线算法,并实现了相应的总体布线器RINO－Router。该算法采用随机优化方法来保 证先后被拆线重布的线网有相同的通过拥挤区域的机会,并能得到GRG边的拥挤度估计值;采用高效的Steiner树改造算法构造避开拥挤区域的布线树,采用典型电路实例进行了测试,并将布线结果与基于多商品流算法的总体布线器Matula-Router进行了对比。结果表明,RINO－Router能够在短得多的运行时间内求得质量与Matula-Router相近的总体布线解。     

确定区域详细布线算法

提出了一种确定区域的详细布线算法,它能对不同设计模式进行布线。该算法能适用于任意多层布线情况,并且支持不同布线层具有的不同工艺参数,在构造布线树时,考虑芯片当前的走线拥挤度,使布线比较平均,并加快了算法运行速度、改善了布线质量,在连接两点线网时,构造基于二维迷宫布线结果的分层图,提出了一种对分层图的启发式染色算示来进行布线层分配,大大提高算法布线速度,采用拆线重布的方法来处理布线失败的线网。     

Optimal approach on net routing for VLSI physical design based on Tabu-ant colonies modeling

To get a more efficient program for net routing design in VLSI physical design, a new mixed algorithm is presented by combining ant colonies algorithm and Tabu search algorithm for improving net routing design scheme in VLSI physical design. The models by considering different structure property such as two-terminal, multiple-terminal, multi-layers and gridless net routing are developed with introducing the proper parameters matching which can be obtained by computer experiments. The results show that the new algorithm can avoid the low convergence rate in the initial stage of basic ant colonies system. The efficiency of the Tabu-ant colonies is improved about 16.667%; meantime, the Tabu-ant colonies system can also avoid the local optimal solution effectively. It builds a basis for future work in solving multiple-terminal, multiple-layers and gridless net routing problems with high efficiency.