Multilevel Full-Chip Gridless Routing With Applications to Optical-Proximity Correction

Multilevel Full-Chip Gridless Routing With

Applications to Optical-Proximity Correction

Tai-Chen Chen, Student Member, IEEE, and Yao-Wen Chang, Member, IEEE

Abstract—To handle modern routing with nanometer effects,

we need to consider designs with variable wire/via widths and spacings, for which gridless-routing approaches are desirable due to its great flexibility. In this paper, we introduce a gridless-routing model that can obtain design-rule-correct paths and avoid redundant wires. Besides, we propose an enhanced model for the gridless-routing model to reduce the solution space and the run-time. Based on the enhanced gridless-routing model, we present the first multilevel full-chip gridless detailed router (called MGR). The router integrates global routing, detailed routing, and con-gestion estimation together at each level of multilevel routing. It can handle designs with nonuniform wire/via widths and spacings and consider routability and optical-proximity correction. Exper-imental results show that MGR achieves the best routing solutions in smaller running times than previous works, based on a set of commonly used benchmarks (with uniform and nonuniform wire widths) and a set of real industrial benchmarks (with a versatile set of design rules).

Index Terms—Design for manufacturing (DFM), gridless

rout-ing, multilevel optimization, optical-proximity correction (OPC), physical design, routing.

I. INTRODUCTION

R

ESEARCH in very large-scale integration (VLSI) routing has received much attention in the literature. Routing is typically a very complex combinatorial problem. In order to make it manageable, the routing problem is usually solved using the two-stage approach of global routing followed by detailed routing. Global routing first partitions the routing area into tiles and decides tile-to-tile paths for all nets, while detailed routing assigns actual tracks and vias for nets. Many routing algorithms adopt a flat framework of finding paths for all nets. Those algorithms can be classified into sequential and concurrent approaches. Early sequential-routing algorithms in-clude maze-searching [28], [42] and line-searching approaches [18], which route net-by-net. Most concurrent algorithms ap-ply network-flow or linear-assignment formulation [2], [37] to route a set of nets at one time.

Manuscript received October 8, 2005; revised April 1, 2006. This work was supported in part by SpringSoft, Inc., and the National Science Council of Taiwan, under Grants NSC 93-2215-E-002-009 and NSC 93-2215-E-002-029. This paper was presented in part at the 2005 ACM/IEEE Asia and South Pacific Design Automation Conference [7]. This paper was recommended by Associate Editor T. Yoshimura.

T.-C. Chen is with the Graduate Institute of Electronics Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. (e-mail: tcchen@ eda.ee.ntu.edu.tw).

Y.-W. Chang is with the Department of Electrical Engineering and Graduate Institute of Electronics Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCAD.2006.884492

The major problem of the flat frameworks lies in their scalability for handling larger designs. As technology advances, technology nodes are getting smaller and circuit sizes are get-ting larger. To cope with the increasing complexity, researchers proposed to use hierarchical approaches to handle the problem: Marek-Sadowska in [35] proposed a hierarchical global router based on linear assignment; Heisterman and Lengauer in [17] presented a hierarchical integer-linear programming approach for global routing; Wang and Kuh in [44] proposed a hierar-chical (α, β)∗algorithm for timing-driven multilayer MCM/IC routing; Chang et al. in [5] applied linear assignment to develop a hierarchical, concurrent global, and detailed router for field programmable gate arrays.

The two-level hierarchical routing framework, however, is still limited in handling the dramatically growing complexity and maintaining high-solution quality at the same time in current and future IC designs, which may contain billions of transistors in a single chip. As pointed out in [12], for a 0.07-µm

process technology, a 2.5 × 2.5 cm2 chip may contain over 360 000 horizontal and vertical routing tracks. To handle such high design complexity, the two-level hierarchical approach becomes insufficient. Therefore, it is desired to employ more levels of routing for larger IC designs.

A. Multilevel Routing

The multilevel framework has attracted much attention in the literature recently. It employs a two-stage technique: coarsening followed by uncoarsening. The coarsening stage iteratively groups a set of circuit components (e.g., circuit nodes, cells, modules, routing tiles, etc.) based on a predefined cost metric until the number of components being considered is smaller than a threshold. Then, the uncoarsening stage iteratively un-groups a set of previously clustered circuit components and refines the solution by using a combinatorial optimization technique (e.g., simulated annealing, local refinement, etc.). The multilevel framework has been successfully applied to VLSI physical design. For example, the famous multilevel partitioners ML [3], HPM [10], and hMETIS [26], the multi-level floorplanner/placerM B∗− tree [29], and the multilevel

placers mPL [4] and APlace [24], [25] all show the promise of the multilevel framework for large-scale circuit partitioning, floorplanning, and placement.

A framework similar to multilevel routing was presented in [16], [30], and [34]. Lin et al. in [30] and Hayashi and Tsukiyama in [16] presented hybrid hierarchical global routers for multilayer VLSIs, in which both bottom–up (coarsening) and top–down (uncoarsening) techniques were used for global

Fig. 1. (a) Routing example.s and t are the source and target of a routing wire, respectively. The two rectangles represent obstacles. (b) Uniform-grid model. The

black and white circles denote the routable and unroutable nodes, respectively. (c) Tile-based model. (d) Nonuniform-grid model. (e) Implicit nonuniform-grid model. (f) Nonuniform-grid graph model. The gray areas denote the obstacle zones constructed by expanding obstacles according to design rules.

routing. Marek-Sadowska in [34] proposed a global router based on the outer most loop approach. The approach is similar to the coarsening stage of multilevel routing. Cong et al. in [12] proposed a pioneering routability-driven multilevel global-routing approach for large-scale full-chip global-routing. Cong et al. later proposed an enhanced multilevel-routing system, named MARS, which incorporates resource reservation, a graph-based Steiner tree heuristic, and a history-based multiiteration scheme to improve the quality of the multilevel-routing algorithm in [13] and [14]. Their final results of the multilevel global routing are tile-to-tile paths for all nets. The results are then fed into a nonmultilevel gridless detailed router, called DUNE [9], [11], to find the exact connection for each net (therefore, MARS is in fact a multilevel global router, but not a detailed router). Lin and Chang in [6] and [31] proposed a multilevel approach for full-chip grid-based routing, which considers both routabil-ity and performance. This framework integrates grid-based global routing, detailed routing, and resource estimation to-gether at each level, leading to more accurate routing-resource estimation during coarsening and, thus, facilitating the solu-tion refinement during uncoarsening. Their experimental re-sults show the best routability among the previous works for grid-based routing. Recently, Ho et al. in [19]–[21] presented another multilevel framework for full-chip grid-based routing considering crosstalk and antenna effects, respectively. The framework incorporates an intermediate stage of layer/track assignment between the coarsening stage and the uncoarsen-ing stage. The coarsenuncoarsen-ing stage performs only global routuncoarsen-ing while global and detailed routing are integrated together at the uncoarsening stage.

B. Gridless Detailed Routing

In the detailed-routing stage, seeking design-rule-correct paths in the routing region is a major concern. Traditional detailed routings use uniform-grid models to simplify the prob-lem, as shown in Fig. 1(b). However, uniform-grid models need very fine grids to handle nonuniform wire/via widths and spac-ings, implying larger searching time and storage requirements. Therefore, the grid-based approach is not effective to handle

modern routing problems with nanometer electrical effects, such as optical-proximity correction (OPC) and phase-shift mask. To cope with these nanometer electrical effects, gridless-routing models are desirable due to their great flexibility.

Several gridless-detailed-routing models have been investi-gated and can be classified into two types: tile-based models [32], [33], [36], [40], [43], [47] and connection-graph models [8], [23], [38], [45], [48]. As shown in Fig. 1(c), the tile-based model partitions the routing region into tiles along the boundaries of obstacles and represents the routing region by a data structure such as corner stitching [39]. Therefore, the routing problem is reduced to searching a tile-to-tile path among these tiles. Although searching a tile-to-tile path is fast, manipulating tiles such as insertion and deletion is a complex process. Furthermore, it needs postprocessing to obtain a final design-rule-correct route for the tile-to-tile path. Moreover, it is not easy to apply the tile-based models to multilayer routing with more complex design rules [12].

As shown in Fig. 1(d), Ohtsuki in [38] proposed a nonuniform-grid model, which was constructed by extend-ing lines through the boundaries of all obstacles until they intersect with other obstacles or boundaries of the routing region. Because the preconstruction and representation of the nonuniform-grid model are costly, previous works [8], [23], [45] tried to simplify the nonuniform-grid model by vari-ous techniques. However, those techniques are still costly for large-scale designs. As shown in Fig. 1(e), Zheng et al. in [48] presented an implicit nonuniform-grid model, which does not construct a connection graph explicitly and charac-terizes the search space in an on-the-fly fashion during routing. Schiele et al. in [41] constructed the obstacle zones from the obstacles by taking design rules into account. The area outside of the obstacle zones is available for placing the center lines of wires and midpoints of contacts. As shown in Fig. 1(f), Cong et al. in [9] and [11] integrated the concepts of obstacles zones [41] and the implicit nonuniform-grid model [48] to build their nonuniform-grid-graph (NGG) model. Although the NGG model has the advantages of fast implicit connection graph construction, routing based on the NGG model may incur design-rule-incorrect paths. Furthermore, routing based on the

Fig. 2. (a) Optical-proximity effects (courtesy of Synopsys). Three major OPC techniques: (b) Serif; (c) Hammerhead; and (d) Line Biasing.

NGG model may result in redundant wires even using point-to-path maze routing.

C. OPC Technologies

As the process technology continues to advance, the min-imum feature size of circuit patterns becomes significantly smaller than the lithographic wavelength. As a result, it is very hard to obtain the exact image we desire on the wafer. Resolution enhancement techniques, such as OPC, are needed to achieve acceptable process accuracy.

Applying optics simulation, we can clearly see this variation as shown in Fig. 2(a). These variations can be classified into mainly three types: corner rounding, line-end shortening, and linewidth shrinking, as shown in Fig. 2(b)–(d). (Here, a line is a horizontal/vertical segment of a net or a via.) For each type of variations, we can add pattern features to compensate for the distortions and acquire what we really need on the resist. We can add serifs at corners to make the angles sharper, add hammerheads at line ends to compensate for their shortenings, and add line biasings along line sides to compensate for their shrinkings.

OPC might incur a much larger number of pattern features, implying larger memory requirements to record these features and higher mask costs. More than a five times increases in data volume and several days of additional CPU runtime are common side effects of OPC insertion in current designs [15]. If a router can generate the configurations that require fewer shots to OPC, we can reduce the data volume for OPC. Therefore, it is desired to consider the optical effects to reduce the number of pattern features.

As a relatively new issue, there is not much work in the literature on routing with the OPC consideration. Huang and Wong in [22] presented a pioneering work on OPC-friendly maze routing based on the Lagrangian relation formulation. However, the router is grid based and considers only two-pin connections. Furthermore, it uses the flat framework and, thus, cannot handle the problem sizes of thousands of nets well. Recently, Wu et al. in [46] presented an enhanced maze routing to solve two OPC-aware maze-routing problems. However, the goals of both problems are to find the single-routing path of the kth net when a routing region with k − 1 routed paths

of two-pin nets are given. Although their method can find a routing path for the kth net under different constraints and

objective functions, it does not consider multipin nets and the net ordering problems.

D. Our Contribution

In this paper, we propose the first multilevel full-chip grid-less detailed router. The four main features of the proposed method are as follows: 1) A gridless-routing model and its enhanced model that can obtain design-rule-correct paths and avoid redundant wires; 2) the first multilevel gridless router that integrates gridless global and detailed routings; 3) a multi-level gridless router that can handle designs with nonuniform wire/via widths and spacing; and 4) a routability-driven and OPC-aware multilevel gridless router that can optimize routing-completion rates and reduce OPC-pattern feature requirements. Experimental results show that our multilevel gridless router (called MGR) achieves 100% routing-completion rates with less running times than previous works based on a set of commonly used Microelectronics Center of North Carolina (MCNC) benchmarks. Furthermore, MGR can handle designs with nonuniform wire widths well and obtain better routing solutions (still maintain 100% routing completion for all cir-cuits) than the state-of-the-art multilevel gridless-routing sys-tem (multilevel gridless global routing + flat gridless detailed routing) [14]. In particular, MGR is the first router to complete the routing for the set of commonly used MCNC benchmarks of nonuniform wire sizes and to route the real industrial Faraday benchmarks with a versatile set of design rules. Moreover, our OPC-aware MGR archives an average reduction of 9% pattern features and still maintains 100% routability for the 11 MCNC-benchmark circuits.

The rest of this paper is organized as follows. Section II presents the global, detailed, and multilevel-routing mod-els. Section III presents our framework for routability-driven and OPC-aware routings. Experimental results are shown in Section IV. Finally, we give concluding remarks in Section V.

II. PRELIMINARIES

Routing in modern ICs is a very complex process, and we can hardly obtain high-quality solutions directly. Therefore, the routing problem is usually solved using the two-stage approach of global routing followed by detailed routing. Global routing first partitions the routing area into tiles and decides tile-to-tile paths for all nets, while detailed routing assigns actual tracks and vias for nets.

A. Modeling of Global Routing

Our global-routing algorithm is based on a graph-search technique guided by the congestion associated with routing regions and topologies. The router assigns higher costs to route nets through congested areas to balance the net distribution among routing regions.

Before we can apply the graph-search technique to multilevel routing, we first need to model the routing resource as a graph such that the graph topology can represent the chip structure. Fig. 3 illustrates the graph modeling. For the modeling, we first partition a chip into an array of rectangular subregions. These subregions are called global-routing cells (GRCs). A node in the routing graph represents a GRC in the chip, and an edge

Fig. 3. Modeling of global routing: (a) partitioned layout and (b) routing graph.

denotes the boundary between two adjacent GRCs. Each edge is assigned a capacity according to the width/height of a GRC. The routing graph is used to represent the routing area and is called a multilevel-routing graph, denoted byGk, wherek is the level ID. A global router finds GRC-to-GRC paths for all nets onG0 to guide the detailed router. The goal of global routing

is to route as many nets as possible, while meeting the capacity constraint of each edge and any other constraint, if specified. Note that, because of the gridless nature of our routing problem, the cost of routing a net is associated with the wire width and spacing.

B. Modeling of Detailed Routing

Seeking design-rule-correct and high-quality paths are two major concerns for detailed routing, which rely greatly on an appropriate detailed routing model. In the following, we first review the recently proposed NGG model and point out its defi-ciencies of incurring design-rule-incorrect paths and redundant wires. We then present the implicit-triple-line-graph (ITLG) model to remedy those deficiencies. We further propose the enhanced-ITLG (EITLG) model to enhance the ITLG model by reducing its induced solution space and, thus, running time.

1) NGG Model: Cong et al. in [9] and [11] integrated the

concepts of obstacle zones [41] and the implicit nonuniform-grid model [48] to build their NGG model. For each obstacle, its obstacle zone is constructed by expanding the obstacle for a range, which is the sum of the obstacle spacing and the half width of the routing wire. As shown in Fig. 4(a), the expanded range (gray area) is the sum ofdsandwi/2, wheredsandwi are the obstacle spacing to satisfy the design rules and the width of the routing wire, respectively. With the boundaries of each obstacle zone, twox-coordinates (the left and right boundaries)

and two y-coordinates (the top and bottom boundaries) are

obtained. Thex-coordinates and y-coordinates of all obstacle

zones and the sources and target t of the routing wire are stored

into two sets, ICG_xand ICG_y, separately. Based on ICG_xand ICG_y, an implicit connection graph is constructed as shown in Fig. 4(b). A vertical (horizontal) dashed lines in the im-plicit connection graph is generated through eachx-coordinate

(y-coordinate) in ICGx(ICGy). A node in the implicit connec-tion graph denotes an intersecconnec-tion of a horizontal and a vertical dashed lines. There are two types of nodes: routable nodes and unroutable nodes. A routable node allows a routing path to pass through it without violating the design rules; it is unroutable, otherwise. As shown in Fig. 4(b)–(e), the respective black and

white circles are the routable and unroutable nodes. To seek a design-rule-correct path from the source s to the target t,

therefore, we only need to check if there exists a feasible path along which all nodes are routable.

The NGG model has the advantage of fast implicit connection-graph construction. The NGG model performs rout-ing by checkrout-ing if a feasible path from the source to the target exists. Even though all nodes along this path are routable nodes, it may still incur design-rule-incorrect paths. Fig. 4(b) shows the implicit connection graph constructed by the NGG model for the simple example of Fig. 4(a). Since all nodes are routable nodes (black circles), a shortest path that connects two routable nodes between s and t can be found [see Fig. 4(c)].

The resulting path is not a legal solution, however, since the obstacle and the routing wire do not have enough spacings to satisfy the design rules.

For multiterminal nets, Cong et al. in [14] used the NGG model to construct an implicit connection graph and applied the

A∗ point-to-path maze-searching algorithm to find a path from the source to the target. Nevertheless, the resulting path may incur redundant wires, implying that we need postprocessing to remove the redundant wires and more runtime to obtain a final solution. As the routing example shown in Fig. 5(a), the targett

connects with the prerouted wire. Using the implicit connection graph constructed by the NGG model will result in the final path shown in Fig. 5(c), which incurs a redundant wire.

2) ITLG Model: To find a design-rule-correct path and

avoid redundant wires, we introduce the ITLG model. For each obstacle, as usual, its obstacle zone is constructed by expanding the obstacle for a range, which is the sum of the obstacle spacing and the half width of the routing wire. In addition to the coordinates along the boundaries of an obstacle zone, the ITLG model also stores the coordinate of the center of the obstacle zone into ICG_x and ICG_y. For each obstacle zone, therefore, threex-coordinates (the left boundary, the right boundary, and

the center) and three y-coordinates (the top boundary, the

bottom boundary, and the center) are obtained. As mentioned in Section II-B1, the x-coordinates and y-coordinates of all

obstacle zones and the sources and target t of the routing wire

are stored into two sets, ICGxand ICGy, separately. Based on ICGx and ICGy, we can obtain an implicit connection graph with vertical and horizontal dashed lines, as shown in Fig. 4(d). To find a design-rule-correct path from the sources to the target t, therefore, we only need to check if there exists a feasible path

along which all nodes are routable.

For each obstacle zone, we refer to a perpendicular line (cen-ter line) as the dashed line that passes the cen(cen-ter of the obstacle zone and perpendicular (parallel) to the routing direction of the obstacle zone. The routing direction of an obstacle zone is hor-izontal (vertical), if the obstacle zone is located in a horhor-izontal (vertical) routing layer. The ITLG model uses the perpendicular and center lines to find a design-rule-correct path and to avoid redundant wires, respectively. We elaborate on the use of the two lines as follows: 1) Perpendicular line: The perpendicular lines are used to avoid design-rule-incorrect paths. Adding the perpendicular line of an obstacle zone introduces intersections of the perpendicular line and the lines, which are orthogonal to the perpendicular line in the implicit connection graph. Since

Fig. 4. (a) Routing example. The gray area denotes the obstacle zone constructed by expanding a range, which is the sum of the wire spacing and the half width of the routing wire.dsandwiare wire/via spacing that satisfies the design rules and the width of the routing wire, respectively.s and t are the source and target of

the routing wire, respectively. (b) Implicit connection graph constructed by the NGG model. The black circles denote the routable nodes. (c) Design-rule-incorrect path, for which the obstacle and the routing wire do not have enough spacings. (d) Implicit connection graph constructed by our ITLG model. The black and white circles denote the routable and unroutable nodes, respectively. (e) Design-rule-correct path found through five routable nodes.

Fig. 5. (a) Routing example. The gray areas denote the obstacle zones constructed by expanding a range, which is the sum of the wire spacing and the half width of the routing wire. The horizontal (vertical) edge of the prerouted wire is in a horizontal (vertical) routing layer. The target of the routing wiret connects with the

prerouted wire.dsandwiare the obstacle spacing that satisfies the design rules and the width of the routing wire, respectively.s and t are the source and target

of the routing wire, respectively. (b) Implicit connection graph constructed by the NGG model. The black circles denote the routable nodes. Sincet connects with

the prerouted wire, the nodes in the obstacle are routable nodes. (c) Path is found through four routable nodes and incurs a redundant wire between the prerouted wire and the routing wire. (d) Implicit connection graph constructed by our ITLG model. The gray circles denote the touch nodes. (e) Path is found through two routable nodes and incurs no redundant wire between the prerouted wire and the routing wire.

the nodes (intersections) are located inside the obstacle zone, they are unroutable nodes. These unroutable nodes can be used to avoid a path from crossing the obstacle zone directly. As shown in Fig. 4(d), the white circle is introduced by the perpen-dicular line and is an unroutable node. With the unroutable node in mind, we can find the design-rule-correct, detour path shown in Fig. 4(e) and avoid the design-rule-incorrect path shown in Fig. 4(c). 2) Center line: The center lines are used to avoid redundant wires. Adding the center line of an obstacle zone

introduces intersections of the center line and the lines, which are orthogonal to the center line in the implicit connection graph. Nodes (intersections) located inside target obstacles are touch nodes, which are also routable nodes. Target obstacles denote obstacles (prerouted wires), which are connected with the target of the routing wire. If the source of the routing wire connects with one of these touch nodes, the source and the target are connected. Therefore, we can find a path from the source to one of the touch nodes to avoid redundant wires. As

Fig. 6. (a) Routing example. The gray areas denote the obstacle zones constructed by expanding a range which is the sum of the wire spacing and the half width of the routing wire. The target of the routing wire connects with the prerouted wire in the left side.dsandwiare wire/via spacing that satisfies the design rules and the width of the routing wire, respectively.s and t are the respective source and target of the routing wire. (b) Implicit connection graph constructed by our

ITLG model. The total number of nodes is 9× 10. (c) Implicit connection graph constructed by our EITLG model. The total number of nodes is reduced to 7× 8. The black, white, and gray circles denote the routable, unroutable, and touch nodes, respectively. Because the target of the routing wire connects with the prerouted wire in the left side, the nodes located above the obstacle zones in the left (right) side are routable (unroutable) nodes.

shown in Fig. 5(d), touch nodes are located inside the prerouted wire. Therefore, we can find a shorter path with nonredundant wires shown in Fig. 5(e).

3) EITLG Model: Although the ITLG model can avoid

design-rule-incorrect paths and redundant wires, the induced solution space (the total number of nodes) of the ITLG model may be more than twice of that of the NGG model, implying larger runtime and storage requirements. Therefore, we shall propose an enhanced model, called the EITLG model, to reduce the solution space of the ITLG model.

We show how to reduce the solution space of the ITLG model as follows: 1) Perpendicular line: To avoid a design-rule-incorrect path, the ITLG model uses the perpendicular lines of all obstacle zones to identify unroutable nodes located inside the zones to avoid a path from crossing the zone directly. If there exists a line parallel to the perpendicular line of an obstacle zone and passing through the zone, this line can introduce unroutable nodes inside the zone, too. Since both of this line and the perpendicular line can introduce unroutable nodes inside the obstacle zone, we do not need to construct the perpendicular line. As shown in Fig. 6, each obstacle zone contains lines perpendicular to the routing direction of the zone and passing through the zone. Therefore, the EITLG model does not construct the perpendicular line for each obstacle zone, as shown in Fig. 6(c), reducing significant solution space for detailed routing. 2) Center line: To avoid redundant wires, the ITLG model uses center lines of all obstacle zones to generate nodes. Since only the nodes located inside target obstacles are touch nodes, we only need to construct the center lines of the target obstacles (obstacle zones). As shown in Fig. 6, only the two obstacles in the left side connect with the target. Therefore, the EITLG model does not construct the center line of the obstacle in the right side of Fig. 6(c).

Algorithm ICG construction, shown in Fig. 7, constructs ICGx and ICGy for routing a wire from the source s to the targett based on the EITLG model. The time complexity of

this algorithm isO(n lg n) by implementing the two sets ICGx and ICG_ywith two sorted arrays, wheren is the total number

of obstacle zones.

Fig. 7. Algorithm to construct ICGxand ICGyfor routing a wire from the sources to the target t based on the EITLG model.

C. Modeling of Multilevel Routing

Fig. 8 shows our multilevel framework. As illustrated in Fig. 8, G0 corresponds to the routing graph of the level 0

of the multilevel-coarsening stage. At each level, our global router first finds routing paths for the local nets (or local 2-pin connections) (those nets [connections] that entirely sit inside a tile) and, then, the detailed router is used to determine the exact wiring. The congestion estimation is performed after the detailed routing finishes routing a net. After the global routing,

Fig. 8. Multilevel-framework flow.

detailed routing, and congestion estimation are performed, we merge four adjacent tiles of G0 into a larger tile for use at the next level (i.e., level 1 here). Coarsening continues until the number of tiles at a level, say the kth level, is below a

threshold. After finishing coarsening, the uncoarsening stage tries to refine the routing solution starting from the last level

k where coarsening stops. During uncoarsening, the unroutable

nets during coarsening are considered, and maze routing and rip-up and reroute are performed to refine the routing solution. Then, we proceed to the next level (levelk − 1) of uncoarsening

by expanding each tiles to four finer tiles. The process continues up to level 0 when the final routing solution is obtained.

III. MULTILEVELROUTINGFRAMEWORK

Although our multilevel routing framework is inspired by the work [6] and [31], our routing model is totally different from that of [6] and [31]. The works [6] and [31] are for grid-based routing and, thus, they can apply a graph-searching technique (e.g., breadth-first search or depth-first search) on predefined grids. However, this paper is for gridless routing. Since the solution space of gridless routing is significantly larger than that of grid-based routing, we apply a different routing model for gridless routing, as discussed in Section II-B. Besides, our congestion estimation is significantly different from that of [6] and [31].

MGR tends to route wider nets first, since a wider net requires more routing resource. Beside, MGR tends to route shorter nets first, since we route local nets at each level of coarsening. It is obvious that the local nets at the lower level (say, level 0) are usually shorter than those at the higher level (say, levelk). Naturally, a shorter net enjoys less freedom while

searching for a path to route it. This fact holds even during rip-up and reroute. Thus, this observation implicitly suggests that a shorter net has a higher priority than a longer net as far as routability is concerned. Kastner et al. in [27] also suggested this finding. Although this net ordering scheme may not be the optimal solution for some routing problems (for example, when timing is considered, routing the most critical net first often leads to better timing performance), it is still a reasonable alternative.

A. Multilevel Routing

In the following, we present our framework for multilevel-gridless routing and summarize it in Fig. 9.

Fig. 9. Multilevel gridless-routing algorithm.

Given a netlist, we first run the minimum spanning tree (MST) algorithm to construct the topology for each net and, then, decompose each net into two-pin connections, with each connection corresponding to an edge of the MST. Our multi-level framework starts from coarsening the finest tiles of multi-level 0. At each level, tiles are processed one by one, and only local nets (connections) are routed. At each level, the two-stage routing approach of global routing followed by detailed routing is applied.

The global routing is based on the approach used in the Pattern Router [27] and first routes local nets (con-nections) on the tiles of level 0. Let the multilevel rout-ing graph of level i be Gi= (Vi, Ei). Let Re={e ∈ Ei| e is the edge chosen for routing}. We apply the cost function α :

Ei → to guide the routing

α(Re) = max

e∈Rece (1)

whereceis the congestion of edgee and is defined by

ce= de

pe

where deandpeare the density and capacity associated with

e, respectively. Pattern routing uses an L- or Z-shaped route

between two points. Therefore, the wire length is minimum. We measure the routing congestion based on the channel density defined by the sum of wire spacing and wire width for gridless routing (note that the definition is different from the case in grid-based routing, for which channel density is defined as the maximum number of parallel nets passing through a routing channel). If pattern routing fails, we give up routing the connection. We refer to a failed net (failed connection) as that causes an overflow. The failed nets (connections) will be reconsidered (refined) at the uncoarsening stage.

After the global routing is completed, we perform detailed routing with the guidance of the global-routing results and find a real path in the chip. Our detailed router is based on Dijkstra’s shortest path algorithm and supports local refine-ment. After the detailed routing finishes routing a net, the channel density associated with an edge of a multilevel graph is updated accordingly. This is called congestion estimation. There are at least two advantages by using this approach. First, routing-congestion estimation is more accurate than that per-forming global routing alone, since we can precisely evaluate the routing region. Second, we can obtain a good initial solution for the following refinement very effectively, since pattern routing enjoys very low time complexity and uses fewer routing resources due to its simple L- and Z-shaped routing patterns.

The uncoarsening stage starts to refine each local failed net (connection), left from the coarsening stage. The global router is now changed to the maze router with the same cost function in the coarsening stage. Uncoarsening continues until the first levelG0 is reached and the final solution is found. Note that the global maze routing, here, serves as an elaborate rip-up and reroute processor, in contrast to the simple L- and Z-shaped routing during coarsening (for rip-up and reroute in our multilevel-routing algorithm, we mean the maze routing at the uncoarsening stage; it is only applied to global routing for better efficiency and quality tradeoff). This two-stage approach of global and local refinement of detailed routing gives our overall refinement scheme.

B. OPC-Aware Multilevel Routing

In modern nanometer-process technologies, such as 90-nm technology and beyond, most of the metal layers need OPC to control the linewidth and length variations. Considering OPC in the routing stage, we can maximize the effects of the correction and, thus, reduce the number of OPC-pattern features during masking.

There are generally two major approaches to OPC: model-based and rule-model-based approaches. The model-model-based method applies optics simulation to add OPC pattern features to fix the OPC problem. It is typically more accurate, but is much more complicated and time-consuming. In contrast, the rule-based method adds the OPC-pattern features rule-based on some predefined design rules. This approach is inevitably less accu-rate, but is much simpler and more efficient. Our router adopts the rule-based approach since it is obviously not feasible to incorporate the very time-consuming model-based approach into a multilevel-routing framework.

In the following, we demonstrate how to consider the OPC rules during routing by incorporating a set of major OPC design rules into the cost function of our router. Note that it is not our intention here to elaborate on all OPC design rules or the accuracy of the rules. Though not presented here, nevertheless, it is not hard to incorporate other (more accurate) OPC design rules into the cost function of our router.

1) OPC Cost Function: The OPC effect of a line is

re-lated to its neighboring configuration. Since the neighboring configuration of a line is not fixed in the routing stage (not all lines are routed), it is very hard to evaluate the OPC cost with unfixed neighboring configuration. Therefore, we propose a combined estimation of actual and estimated OPC cost to calculate the OPC cost for a line considering the routed and unrouted neighboring lines. We define the OPC cost for a linee by

cost(e) = costa(e) + coste(e). (2) The combined cost consists of an actual cost cost_a(e) (for real neighboring configuration) and an estimated cost cost_e(e) (for the worst case neighboring configuration). At first, the OPC cost for a line is estimated by the worst case neighboring configuration alone. After a connection is routed successfully, the real neighboring configuration will be updated dynamically. Therefore, our OPC cost is based on the OPC effect incurred by both the already routed nets and the estimated unrouted nets. As routing proceeds, we have more and more accurate OPC effect for routing succeeding nets. We describe how to calculate these two cost as follows: 1) Actual cost: We calculate the actual cost for a line based on the OPC effect caused by the neighboring routed lines. In addition, the optical interference is limited within a region of several wavelengths [22], [46]. Therefore, only neighboring routed lines within the effective region (spacing) are considered. Here, the effective region is defined by the foundry. Consequently, we define the actual cost for a linee to be the total number of pattern features as follows:

cost_a(e) = lo/LL + wo/WL (3) where lo and wo are the overlapping length and width with the routed neighboring lines within the effective region (spac-ing), respectively. Here,LL andWL(the length and unit-width for adding a pair of line biasings) are parameters related to the process technology and are defined by the foundry. 2) Estimated cost: Evaluating the OPC cost for a line without considering the OPC effect caused by the neighboring unrouted lines may be inaccurate. Since we do not know the final layout of the neighboring configuration, we consider the worst case neighboring configuration for the cost estimation. In other words, we assume that a line segment is fully surrounded by adjacent lines. Therefore, the OPC cost for the line is proportional to its length and width. As shown in Fig. 2(b), for a line, we need to add four serifs at the corners to increase the fidelity of images. As the length of a line increases, the ends of the line are shortened. As shown in Fig. 2(c), therefore, we need to add two hammerheads at the line ends for a long line (a line is said to be a long line if its length is longer than or equal toLT, whereLT is the threshold length for a long line

Fig. 10. Algorithm to computefeature(v).

and is defined by the foundry). Besides, the overlapping length of a line with neighboring lines may increase as the length of the line increases; further, a wider line is easier to be affected by neighboring lines than a narrower one. These phenomena make the sides of a line shrink more seriously. Therefore, as shown in Fig. 2(d), we need some line biasings in the line sides to correct the optical-proximity effects for a line. The total number of line biasings for a line is determined by the length and width of the line. According to the above modeling, we define the estimated cost for a linee whose length and width are le and

we, respectively, to be the total number of pattern features as follows:

cost_e(e) =

4 + f (le, we), when le< LT 6 + f (le, we), otherwise wheref is a step function and is defined as follows:

f (le, we) = 2× (le/LL + we/WL) . (4) Therefore, the total OPC cost for a connection is the sum of the OPC costs for lines that belong to the connection.

2) OPC Cost Minimization: We apply the following

algo-rithm, called simultaneous path length and OPC cost Mini-mization (SPOM), to perform Dijkstra’s shortest path algorithm to find a shortest path with the minimum number of pattern features. It associates each basic routing node u (a node in

an implicit connection graph) with two labels: dist(u) and

feature(u), where dist(u) is the distance of the shortest path from source s to u and feature(u) is the minimum

num-ber of pattern features along the shortest path from s to u.

Initialize dist(u) = ∞, feature(u) = ∞, ∀u = s, dist(s) = 0,

and feature(s) = 0. The computation of label dists is the same

as original Dijkstra’s shortest path algorithm. Letu be a basic

routing node on the wavefront and v be a neighboring basic

routing node of u. The predecessor routing node of u is the

region from which the wavefront was propagated for obtaining the minimum feature(u). The propagation direction of u is the

direction from the predecessor routing node of u to u. The

computation of feature(v) is shown in Fig. 10, where w(u , v)

ando(u , v) are the distance and the number of additional pattern

features between nodesu and v, respectively. Besides, r denotes

that in the last routing node, we insert a via along the shortest path froms to u and is initialized to s.

The basic idea is to compare the distance label dists first

and then compare the pattern-feature number label features.

The value feature(v) of a neighboring routing node v with

dist(v) < dist(u) stays unchanged because the path from s through u to v is not the shortest path between s and v. Note

that it is possible that there may exist several shortest paths with different numbers of pattern features. It is clear that Algorithm SPOM guarantees to find a shortest path with the minimum number of pattern features, if such a path exists.

IV. EXPERIMENTALRESULTS

We implemented MGR in the C++ language on a 1-GHz Sun Blade-2000 workstation with 8-GB memory. Our routing package is available at http://eda.ee.ntu.edu.tw/research.htm. We used two sets of benchmarks, the MCNC benchmarks (provided by the authors of [14]) and the Faraday benchmarks introduced in [1], for our comparative study on routing. The MCNC benchmarks are considered the largest benchmarks commonly used in academia, while the Faraday benchmarks are real industrial designs with many more nets and more complex design rules than the MCNC benchmarks.

Tables I and II list the set of benchmarks. In these tables, “Circuit” gives the names of the circuits, “Size (µm2)” gives the layout dimensions in square micrometers, “#Layers” denotes the number of routing layers used, “#Nets” gives the total number of nets, “#Connections” gives the number of two-pin connections after net decomposition, “#Pins” gives the number of pins, “Wire/Via Width (µm)” gives the design rules for

wire/via width, and “Wire/Via Spacing (µm)” gives the design

rules for wire/via spacing. Table III lists the design rules for the Faraday benchmarks with six metal layers and five via layers, including widths and spacings.

A. Comparison of MGR Based on the ITLG and EITLG Models

Table IV gives the comparison of MGR based on the ITLG and EITLG models. In the table, “#Failed Connections” denotes the number of failed connections, “Comp. Rates” gives the routing completion rates, “#Nodes” denotes the total number of nodes, “Mem. (MB)” denotes the storage requirements in megabytes, and “Time (sec)” represents the runtimes in seconds.

The experimental results show that MGR based on the EITLG model is much more efficient. As shown in the table, MGR based on the EITLG model achieves equal routing solu-tions to MGR based on the ITLG model with 1.90× runtime speedup, 4.65 × reduction in the total number of nodes, and 1.41× reduction in the storage requirements.

B. Multilevel Routing WithMCNC Benchmarks

1) MCNC Benchmarks With Uniform Nets: Table V

com-pares MGR with the multilevel grid-based router with the routability mode proposed in [6] and [31] (called MR) and the multilevel gridless-routing system (multilevel grid-less global routing + flat gridgrid-less detailed routing) in [14] (called MARS).

TABLE I

STATISTICS OF THEMCNC BENCHMARKS. NOTE: WECORRECTSOME“SIZE,” “#CONNECTIONS,”AND“#PINS”INTHISTABLE, SINCETHOSE

INFORMATION IN[6], [14],AND[31] AREINCORRECT

TABLE II

STATISTICS OF THEFARADAYBENCHMARKS

TABLE III

DESIGNRULES OF THEFARADAYBENCHMARKS. LAYERSMETAL1, METAL2, METAL3, METAL4, METAL5,ANDMETAL6 AREROUTINGLAYERSWHILE

LAYERSV1, V2, V3, VL,ANDVQ ARE VIALAYERS(-:VIALAYERSDONOTHAVETHOSEINFORMATION)

TABLE IV

COMPARISON OF(A) MGR BASED ON THEITLG MODEL AND(B) MGR BASED ON THEEITLG MODEL

As shown in the Table V, MARS, MGR, and MR all achieve 100% routing completion for the set of 11 benchmark cir-cuits, but MGR has significantly better runtime efficiency than the other two works. For example, MGR obtains 6.38 times speedup over MR and about 2.16 times over MARS (note that it is hard to make a fair comparison between MARS and MGR, because MARS and MGR ran on different machines. Nevertheless, they both ran on Sun workstations. Therefore, we try our best to make a fair comparison by normalizing the runtime based on their clock rates).

2) MCNC Benchmarks With Nonuniform Nets: We also

per-formed experiments on the MCNC benchmarks of nonuniform wire widths. We modify the original MCNC benchmarks of uni-form wire sizes to generate a set of benchmarks of nonuniuni-form

wire sizes by using the following rules, which were proposed by [14]. The longest 10% nets are widened to twice the original width, while the next 10% are widened to 150% the original width. However, because the benchmarks S5378-S38584 are standard-cell designs, widening any pin violates the design rules for via spacing. Therefore, it is unreasonable and incorrect to test these six modified benchmarks. Since MR is a grid-based router and the routing results of MR with nonuniform wire widths violate the design rules, we do not compare MGR with MR.

Table VI gives the routability comparison of MGR with MARS. In the table, “#Total Subnets” denotes the total number of two-pin connections seen by the detailed router of MARS, since the detailed router of MARS segments long two-pin

TABLE V

COMPARISONAMONG(A) MR: MULTILEVELGRID-BASEDROUTINGWITH THEROUTABILITYMODE[6], [31], (B) MARS: MULTILEVELGRIDLESS

GLOBALROUTING+ FLATGRIDLESSDETAILEDROUTING[14],AND(C) MGR. NOTE: (A)AND(C) RAN ON A1-GHZSUNBLADE-2000 WITH8-GB MEMORY AND(B) RAN ON A440-MHZSUNULTRA-10 WITH384-MB MEMORY(∗:FORFAIRCOMPARISONS,

WENORMALIZE THERUNTIMES OFMARSBY THEFACTOR440/1000)

TABLE VI

COMPARISON OF(A) MARS: MULTILEVELGRIDLESSGLOBALROUTING+ FLATGRIDLESSDETAILEDROUTING[14]AND(B) MGR. NOTE: (A) RAN ON A440-MHZSUNULTRA-10 WITH384-MB MEMORY AND(B) RAN ON A1-GHZSUNBLADE-2000 WITH8-GB MEMORY(NOTE

THATBECAUSE THEBENCHMARKCIRCUITSS5378-S38584 VIOLATE THEDESIGNRULES OF VIASPACING, WEDIDNOTLISTTHESE

CASES INTHISTABLE). (∗:FORFAIRCOMPARISONS, WENORMALIZE THERUNTIMES OFMARSBY THEFACTOR440/1000)

Fig. 11. Full-chip routing solution for “vd_Mcc2” obtained from MGR. The bounding box is the boundary of this benchmark circuit.

connections into short subnets. As shown in the table, MGR still achieves 100% routing completion for all of the five circuits with 1.15 times runtime speedup, while MARS completed rout-ing for only four circuits. Note that MGR is the first router to complete the routing for this set of benchmarks of nonuniform wire sizes. In particular, we expect that the difference will be much more significant for larger and difficult designs such as vd_Mcc2. Figs. 11 and 12 show the full-chip and partial routing solutions for “vd_Mcc2” obtained from MGR, respectively.

Fig. 12. Partial layout for “vd_Mcc2” obtained from MGR. We can see from the layout that the three leftmost vertical lines are of different widths.

The bounding box in Fig. 11 is the boundary of this benchmark circuit. We can see in Fig. 12 that the three leftmost vertical lines have different widths.

C. Multilevel Routing WithFaraday Benchmarks

To our best knowledge, MGR is the first academic router to route real industrial designs, the Faraday benchmarks. We compared MGR only with MR, since MR is the only academic router with source codes available to the public. As shown in Table VII, MGR achieves almost 100% routing-completion rates, while MR can only achieve about 65%–80% routing-completion rates. MR cannot achieve high routing routing-completion rates, because it works on uniform predefined grids and as-sumes the design rules (wire/via width/spacing) for all routing

TABLE VII

COMPARISON OF(A) MR: MULTILEVELGRID-BASEDROUTINGWITH THEROUTABILITYMODE[6], [31], (B) MGR. NOTE: (A)AND(B) RAN ON A

1-GHZSUNBLADE-2000 WITH8-GB MEMORY; (NR: NOROUTINGRESULTDUE TOOUT OFMEMORY)

TABLE VIII

COMPARISON OFOURROUTABILITY-DRIVEN ANDOPC-AWAREMGR

layers to be the same. Consequently, the predefined grids are of equal size for all routing layers. However, the design rules of the Faraday benchmarks are not the same for the six routing layers. To ensure that MR satisfies the design rules, we make the pitch of its predefined grids the maximum sum of the wire/via widths and spacings. This restriction makes MR consume more routing resource in lower routing layers, resulting in inferior routing solutions.

D. OPC-Aware Multilevel Routing

In the last experiment, we performed experiments on OPC-aware routing. For OPC-OPC-aware routing, no previous routers are available to us for comparative studies (the only work on OPC-aware routing in the literature is by Huang and Wong [22], which is a flat grid-based maze router that handles only hun-dreds of two-pin connections). We refer to a line as a long line if its length is five times longer than the minimum wire width. We setLL/WLto 10/2 times of the minimum wire width. Besides, we set the effective range of OPC effect to three times of the wire pitch.

The results are listed in Table VIII. In the table, “#Pattern Features” denotes the total number of pattern features. Com-pared with our routibility-driven MGR, the experimental re-sults show that our OPC-aware MGR achieves an average 9% reduction in the number of pattern features required and still maintains 100% routing completion for all circuits, with very small overheads in the runtime. The results show the effectiveness of our OPC-aware multilevel router.

V. CONCLUSION

In this paper, we have introduced a gridless-routing model, which can obtain design-rule-correct paths and avoid

redun-dant wires. Based on the gridless-routing model, we have proposed the first multilevel full-chip gridless detailed router. The router can handle designs with nonuniform wire/via widths and spacings and consider routability and OPC. Experimental results have shown that our approach achieves the best routing solutions in smaller running times than previous works, based on a set of commonly used MCNC benchmarks (with uniform and nonuniform wire widths) and the real industrial Faraday benchmarks (with a versatile set of design rules).

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Tai-Chen Chen (S’02) received the B.S. and M.S. degrees in computer and information science from National Chiao Tung University, Hsinchu, Taiwan, in 1999 and 2001, respectively. He is currently working toward the Ph.D. degree in the Graduate Institute of Electronics Engineering, National Taiwan Univer-sity, Taipei, Taiwan.

His current research interests include computer-aided design and gridless routing for nanometer elec-trical effects.

Mr. Chen is the recipient of the Best M.S. Thesis Award from the National Science Council of Taiwan in 2002.

Yao-Wen Chang (S’94–M’96) received the B.S. degree from the National Taiwan University, Taipei, Taiwan, in 1988 and the M.S. and Ph.D. degrees from the University of Texas at Austin in 1993 and 1996, respectively, all in computer science.

He is a Professor in the Department of Electrical Engineering and the Graduate Institute of Electron-ics Engineering, National Taiwan University. He is currently also a Visiting Professor at Waseda Uni-versity, Kitakyushu, Japan. He was with the IBM T.J. Watson Research Center, Yorktown Heights, NY, in the summer of 1994. From 1996 to 2001, he was on the faculty of the National Chiao Tung University, Hsinchu, Taiwan. He has coauthored more than 110 ACM/IEEE conference/journal papers and one book on routing. His current research interests are in VLSI physical design, design for manufacturing, and FPGA. He has been working closely with industry on projects in these areas. He is an Editor of the Journal of Computer and Information Science.

Dr. Chang is currently the Chair of the Design Automation and Test (DAT) Consortium of the Ministry of Education, Taiwan, a member of the Board of Governors of Taiwan IC Design Society, and a member of the IEEE Circuits and Systems Society, ACM, and ACM/SIGDA. He currently serves on the ACM/SIGDA Physical Design Technical Committee and the technical program committees of important conferences on VLSI design automation, including ASP-DAC (Topic Chair), DAC, DATE, FPT, GLSVLSI, ICCAD, ICCD, ISPD, SOCC (Topic Chair), and VLSI-DAT (Topic Chair). He was a recipient of the 2006 ACM ISPD Placement Contest Award, Best Paper Award at ICCD-1995, and eight Best Paper Nominations from DAC-2007, ISPD-2007, DAC-2005, 2004 ACM TODAES, ASP-2003, ICCAD-2002, ICCD-2001, and DAC-2000. He was also a recipient of many awards for research performance, such as the 2005 and 2006 First-Class Principal Investigator Awards and the 2004 Mr. Wu Ta You Memorial Award from National Science Council of Taiwan, the 2004 MXIC Young Chair Professorship from the MXIC Corporation, and for excellent teaching from the National Taiwan University and National Chiao Tung University.