A full-scale solution to the rectilinear obstacle-avoiding Steiner problem

Abstract

Routing is one of the important steps in very/ultra large-scale integration (VLSI/ULSI) physical design. Rectilinear Steiner minimal tree (RSMT) construction is an essential part of routing. Macro cells, IP blocks, and pre-routed nets are often regarded as obstacles in the routing phase. Obstacle-avoiding RSMT (OARSMT) algorithms are useful for practical routing applications. However, OARSMT algorithms for multi-terminal net routing still cannot meet the requirements of practical applications. This paper focuses on the OARSMT problem and gives a solution to full-scale nets based on two algorithms, namely An-OARSMan and FORSTer. (1) Based on ant colony optimization (ACO), An-OARSMan can be used for common scale nets with less than 100 terminals in a circuit. An heuristic, greedy obstacle penalty distance (OP-distance), is used in the algorithm and performed on the track graph. (2) FORSTer is a three-step heuristic used for large-scale nets with more than 100 terminals in a circuit. In Step 1, it first partitions all terminals into some subsets in the presence of obstacles. In Step 2, it then connects terminals in each connected graph with one or more trees, respectively. In Step 3, it finally connects the forest consisting of trees constructed in Step 2 into a complete Steiner tree spanning all terminals while avoiding all obstacles. (3) These two graph-based algorithms have been implemented and tested on different kinds of cases. Experimental results show that An-OARSMan can handle both convex and concave polygon obstacles with short wire length. It achieves the optimal solution in the cases with no more than seven terminals. The experimental results also show that FORSTer has short running time, which is suitable for routing large-scale nets among obstacles, even for routing a net with one thousand terminals in the presence of 100 rectangular obstacles.

Introduction

Routing a net, finding a rectilinear Steiner minimal tree (RSMT) for a given terminal set, is one of the fundamental problems in very/ultra large-scale integration (VLSI/ULSI) physical design. Many algorithms have been proposed focusing on the RSMT problem. However, only a few RSMT algorithms take obstacles into consideration. In practical routing applications, macro cells, IP blocks, and pre-routed nets are often regarded as obstacles. Obstacle-avoiding RSMT (OARSMT) construction is often used as an estimation for wire length even delay throughout the process of routing. Therefore, we need efficient OARSMT algorithms in physical design. Garey and Johnson [1] proved that the RSMT problem is NP-complete. Thus, OARSMT problem is more complicated and polynomial-time algorithms can unlikely solve it exactly.

The OARSMT problem has been well studied for the case of two-terminal net. Lee et al. [2] presented maze algorithm to route two-terminal nets optimally. Some improvements on Lee's algorithm were proposed later [3], [4], [5], [6]. The line-probe routing algorithm was introduced in [7], [8]. This kind of algorithms is suitable for small-scale problems. Zheng et al. [9] proposed an algorithm, in which the searching space is restricted to a strong connection graph (implicit connection graph). The time and space complexity depend on the instance (the number of terminals and obstacle border segments). Wu et al. [10] introduced a sparse connection graph (track graph) to reduce the searching space efficiently and routed the shortest path based on the graph.

We need to route multi-terminal nets in the routing phase. However, routers often use the multi-terminal variant of the maze routing algorithm, which incurs the same space demand as that of the two-terminal variety and usually obtains solutions far from the optimal.

Ganley et al. [11] proposed an algorithm to construct the optimal three-terminal or four-terminal OARSMT. Then G3S, G4S, and B3S heuristics [11] were proposed for the cases with less than 20 terminals. Zachariasen et al. [12] gave an exact algorithm for finding an obstacle-avoiding Euclidean Steiner tree with less than 150 terminals. The O(mn) two-step heuristic [13] (m is the number of obstacles and n is the number of terminals) works well when the number of terminals is less than seven and obstacles are convex ones. There are some recent achievements [14], [15], [16]. Shi et al. [14] achieved short wire length but could only route the nets with less than 50 terminals, which could not meet the needs of routing full-scale nets. A rectilinear approach was proposed in [15], which has the limitation of handling blockages with complex shapes such as concave polygons and handling large-scale cases.

Thus, it still needs full-scale solutions to the OARSMT problem, which can handle more terminals in the present of convex and concave polygon obstacles with short length performance and high time efficiency. The major contribution of this paper is a full-scale solution based on two heuristics, namely An-OARSMan and FORSTer, to the obstacle-avoiding RSMT problem. A special data structure, track graph, is used in An-OARSMan. We present the T-reduction to delete redundant vertices and edges in the track graph. Then the searching space is reduced. An-OARSMan can obtain the optimal solution while the terminal number is less than seven and it is suitable for the common scale net routing with less than 100 terminals. FORSTer takes short running time. The full Steiner tree (FST) construction and detour method are used in the FORSTer algorithm, which makes it more practical for large-scale net routing with more than 100 terminals.

The rest of this paper is organized as follows. In Section 2, we introduce some basic definitions of the OARSMT problem, connection graphs, and the ant colony optimization (ACO). Section 3 proposes the An-OARSMan heuristic. In Section 4, the FORSTer algorithm is described in detail. Section 5 presents the experimental results and Section 6 concludes the paper. The partial of this work, i.e., the An-OARSMan heuristic, has been published in ASP-DAC 2005 [17]. This paper presents progress in large-scale extension and includes detailed analysis.