Probability-Based Approach to Rectilinear Steiner Tree Problems

Transcription

1 836 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 10, NO. 6, DECEMBER 2002 Probability-Based Approach to Rectilinear Steiner Tree Problems Chunhong Chen, Member, IEEE, Jiang Zhao, Majid Ahmadi, Fellow, IEEE Abstract The rectilinear Steiner tree (RST) problem is of essential importance to the automatic interconnect optimization for VLSI design. In this paper, we present a class of probability-based approaches toward the best solutions under statistical sense show their performance in comparison with the state-of-the-art algorithm. Experiments conducted on both small- large-size problems indicate that the proposed approaches lead to promising results in terms of wire length /or CPU time. The potential advantages with our technique are also discussed for further applications. Index Terms Algorithm, probabilistic approach, rectilinear Steiner tree (RST), VLSI interconnect optimization. I. INTRODUCTION ONE OF THE key problems in VLSI interconnect design is the topology construction of signal nets with minimum cost. The Steiner tree problem is to find the tree structure that connects all pins of a signal net such that the wire length (i.e., cost) can be minimized. If all edges of the tree are restricted to the horizontal vertical directions which are a popular case in VLSI design, the problem is called the rectilinear Steiner tree (RST). In general, the RST may contain, in addition to the pins of the net, some other points (i.e., Steiner points). In particular, the RST without Steiner points is called the rectilinear minimum spanning tree (RMST) which has been well studied [1]. While the RST can generally lead to better results than the RMST in terms of wire length, it has been shown that the RST problem is NP-complete [2]. Numerous heuristics have been studied toward the optimal or near-optimal solutions (e.g., [3] [16]). For instance, Hanan [3] showed an optimal algorithm when the net contains no more than four pins. Cohoon [4] proposed an optimal algorithm when the pins of a net lie on the perimeter of a rectangle. Hwang [6] proved that the ratio of tree lengths between RMST RST is no worse than 3/2. An algorithm for the RST was also proposed in [5], while the results were far from the optimal solution. One of the well-known algorithms with good performance is the Batched Iterated 1-Steiner (BI1S) heuristic due to Kahng Robins [13]. Its improved version can be found in [17]. More recently, while another heuristic for the RST was presented [11], Warme [15] described an exact solution to large-size Steiner tree problems. Instead of mentioning all previous works, we can divide Manuscript received July 9, 2001; revised May 2, This work was supported in part by the Natural Sciences Engineering Research Council of Canada (NSERC) under Grant The authors are with the Department of Electrical Computer Engineering, University of Windsor, Windsor, ON N9B 3P4, Canada ( cchen@uwindsor.ca; zhaog@uwindsor.ca; ahmadi@uwindsor.ca). Digital Object Identifier /TVLSI them into the following three categories: minimal-spanning-tree (MST) based, computational-geometry based iterative-improvement based approaches. A good survey on Steiner tree problems can be found in [7] [16]. Also, the readers are referred to [10] for a comprehensive review of VLSI interconnect design. In this paper, we propose probability-based approaches for RST problems. By considering all possible topologic patterns connecting every pair of pins, we can calculate the probability of the patterns passing over individual edges. The optimal Steiner tree under statistical sense is the tree with maximum sum of the probabilities for all edges of which the tree is comprised. We combine the probabilistic model with other algorithms to improve their performance (i.e., the time complexity wire length which are major concerns in Steiner tree designs). We also develop a nondeterministic algorithm based on probabilistic model to provide the tradeoff between the running speed the quality of results. Experiment shows that the obtained tree topology is very close to the optimal RST. For large-size problems, the probability-based algorithms are significantly faster than the state-of-the-art algorithm with a slight loss of quality. For small-size problems where the running time is not a big issue, the better solutions can be obtained using our nondeterministic algorithm (see Section III) at the cost of increased running time. As will be seen later, an added feature of our probability-based technique is that it can be easily combined with many existing algorithms to improve their performance. Also, it would be straightforward to extend this technique to deal with more general Steiner tree problems with the obstacles or blockages [8]. In the next section, we first describe some background together with the probabilistic model. In Section III, we present the Steiner tree construction algorithm other probabilitybased algorithms. Then, the results from extensive experiments are given in Section IV, where comparison with previous works is emphasized in terms of both wire length CPU time. Finally, Section V concludes the paper. II. PROBABILISTIC MODEL Here, we introduce some preliminaries propose a probabilistic model for RSTs. A. Grid Graph Consider a set of points (or pins), in a plane, where the location of is denoted by ( ). Assuming for (More discussions will be given later if this is not the case), we can construct a grid /02$ IEEE

2 CHEN et al.: PROBABILITY-BASED APPROACH TO RST PROBLEMS 837 Fig. 1. The grid graph for a set of three points. Fig. 3. Probabilistic analysis for the segments through which the shortest paths between points (p p ) pass. Fig. 2. An optimal Steiner tree of Fig. 1. graph which consists of the intersections (or, segments) of horizontal vertical lines through all points. It was shown [3] that only those segments within the smallest rectangle enclosing all points need to be considered in obtaining the RST. An optimal RST is a subset of segments,, such that is a tree for given points the total wire length over all segments in is minimum. Fig. 1 illustrates the grid graph for a set of three points,. An optimal Steiner tree of Fig. 1 is shown in Fig. 2, where is a Steiner point. If we number the columns rows of the grid graph, the symbol can be used to represent the horizontal segment which lies on row between columns. Similarly, we use to represent the vertical segment which lies on column between rows. For instance, the segments in Fig. 1 are denoted by, respectively. Note that the rows are numbered from the bottom to the top in the graph the columns are numbered from the left to the right, as shown in Fig. 1. For convenience of further discussion, we have the following definitions. Definition 1: Given two points,, the value is called the horizontal grid-distance between them, where are the column numbers of, respectively. Similarly, the vertical grid-distance between them is defined to be, where are the row numbers of, respectively. Definition 2: If for the two points,, then is called the th horizontal physical-length of the grid graph, where. If, then is called the th vertical physicallength of the graph, where. B. Probabilistic Model Consider two points, in the grid graph as shown in Fig. 3. Without loss of generality, we assume. Let. The horizontal vertical grid-distances between are, respectively. Let be the number of all possible shortest paths from to. The number of those paths which pass through the segment (i.e., in Fig. 3) only depends on is denoted by. The number of those paths which pass through the segment (i.e., in Fig. 3) is also a function of, denoted by. Obviously, we have. In particular, for any positive integer, wehave. From a statistical point of view, the probability of a shortest path between the two points passing through (or, )isgiven by (or, ). From Fig. 3, can be written as or (1) (2) (3) (4) Theorem 1: If we define for any integer, then can be computed recursively as follows: where,,, are defined as earlier.

3 838 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 10, NO. 6, DECEMBER 2002 Proof: The second part of the theorem is straightforward we prove the first part only. Adding (3) (4) gives TABLE I F (m; n) FUNCTION (5) or (6) From (1) (5), we have (7) TABLE II G(m; n) FUNCTION Since, (7) can be rewritten as (8) From (6) (8), we have (9) Furthermore, from Fig. 3, we can express as follows: Fig. 4. The graphic representations for F (m; n) log F (m; n). In other words, we have proved the following theorem: Theorem 2: The recursive expressions for are given by (12) More generally, we have (13) (10) The values of for, are shown in Table I Table II, respectively. Fig. 4 shows the graphic representations for. Similarly, can be expressed as (11) C. Probability Matrix To calculate the probabilities of segments in Fig. 3, we take the specific horizontal segment in the figure, where,. Among all shortest paths

4 CHEN et al.: PROBABILITY-BASED APPROACH TO RST PROBLEMS 839 from to, the number of paths passing through this segment is given by Thus, the probability of a shortest path passing through this segment is Similarly, the probability of a shortest path between passing through the specific vertical segment (see Fig. 3) is given by (14) (15) where. If we account for the shortest paths for all pairs of points in the grid graph, two probability matrices, (denoted by ), can be used to represent the probabilities of all horizontal vertical segments, respectively, through which the shortest paths would pass. The element in (or in ) corresponds to the horizontal (or vertical) segment (or ) in the graph. is an matrix is an matrix. The contribution of each pair of points to the matrices is determined by the (14) (15). Intuitively, the greater value of an element implies higher probability that the corresponding segment is to be chosen in obtaining a shortest path (or Steiner tree). An optimal tree under statistical sense is the tree in which the sum of probabilities of the segments involved is maximum. III. ALGORITHMS Based on the above probabilistic model, we present several algorithms below for Steiner tree construction: pure probabilistic, MST combined nondeterministic algorithms. The basic idea of the pure probabilistic algorithm is to select, step by step, the segments with higher probability from the probability matrices until a Steiner tree is constructed. As its name implies, the Minimal-Spanning-Tree (MST) combined algorithm is to join the probabilistic model with MST only consider the ( ) MST edges (compared to pairs of points in the pure probabilistic algorithm) for calculating the probability matrices. The segment selection for Steiner tree construction is the same as in the pure probabilistic algorithm, i.e., choosing the segments one by one in decreasing order of their corresponding probabilities. Differing from these two heuristics that are deterministic, nondeterministic algorithm constructs the tree more romly by generating a group of segments whose appearing probabilities are equal to the corresponding values in the probability matrices. For a given problem, the results may vary each time the algorithm is run on it. The more time the algorithm is executed, the higher probability there is to get the optimal result. A. Pure Probabilistic Algorithm The pseudocode of pure probability algorithm is given. Pure Probabilistic Algorithm 1. Compute,, : Given, compute the row number column number for compute the horizontal vertical physical-lengths, i.e., for ; 2. Compute : Compute for ; 3. Obtain : Obtain the probability matrices,, using (14) (15) for all pairs of points normalize all elements of :, 4. Get all points in connected: Select the vertical horizontal segments one by one in the decreasing order of their corresponding probabilities in. Ignore a specific segment if selecting it would lead to a cycle that contradicts the tree definition; 5. Delete redundant segments: Delete degree-one-segments that are not connected to any point in ; 6. Obtain a set of Steiner points calculate wire length: Obtain, which is a set of degree-tree degree-four-points (except those in ) calculate the total wire length of the tree. Most of the above algorithm is selfexplanatory except Step 3 where the matrix normalization is necessary since the segments with same probability need to be treated differently, depending on their physical lengths. The shorter segment is selected first in the tree construction so that the total wire length can be reduced. If there are points with the same -coordinate, i.e.,, which implies that no horizontal segments in the th column are required, then we delete the th column of (instead of dividing it by as shown in Step 3). Similarly, the th row of will be deleted if some points have the same -coordinate, i.e.,, meaning that no vertical segments in the -th row are needed. The time complexity of the earlier algorithm is due to Step 3 which is computationally expensive. To improve the efficiency, we introduce the MST combined algorithm later. B. MST Combined Algorithm The MST combined algorithm modifies Steps 2 3 in the pure probabilistic algorithm. First, it uses Kruskal s algorithm [12] to construct the minimal spanning tree for obtain a ;

5 840 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 10, NO. 6, DECEMBER 2002 set of ( ) edges,. Then, the maximal vertical grid-distance maximal horizontal grid-distance are calculated, where are the maximum of, respectively, for any edge. This is followed by computation of for. Finally, the probability matrices,, are obtained using (14) (15) for all ( ) pairs of points in performing the matrix normalization as shown before. The rest of the algorithm is the same as in the pure probabilistic algorithm. It should be noted that while the MST construction requires time, only ( ) pairs of points need to be considered. This reduces the time complexity from to. In addition, our experiments show that are normally less than 10, indicating an efficient computation of even for large design problems. Therefore, the MST combined algorithm is much faster than the pure probabilistic algorithm. C. Nondeterministic Algorithm The nondeterministic algorithm only modifies the segment selection rule, i.e., Step 4 in the pure probabilistic algorithm, by generating the segments based on their probability values. Assuming that the sum of all entries in the two probability matrices (i.e., ) is, the probability that a segment is to be generated each time is equal to its corresponding value in the matrix divided by. The segments with higher values in the matrices are more likely to be chosen from a probabilistic point of view. The other steps involved are the same as in the pure probabilistic algorithm. Once a Steiner tree is constructed, we say that one pass is completed. The wire length for each pass is recorded compared with the best result from the previous passes. The best result is updated if a shorter wire length is achieved. This process repeats for a given number of passes. Therefore, the algorithm provides the tradeoff between the running time result quality. IV. EXPERIMENT AND DISCUSSION We implemented the proposed algorithms carried out the experiments with Steiner tree construction for problems of various sizes. In order to evaluate the performance, we also implemented the BI1S algorithm [17] which is one of the heuristics with good performance. The programs were run on a SunBlade 1000 workstation. As the first set of experiments, Figs. 5 7 show the results for several simple examples using the pure probabilistic algorithm presented in Section III. While the optimal RST is unknown in general, the effectiveness of the algorithm can still be evidenced by inspection of these small-size problems. Particularly, for the case of Fig. 7 which was taken from [5], the result due to the algorithm of [5] is shown in Fig. 7(a) with the total wire length of 32, compared to the length of only 30 by our algorithm as shown in Fig. 7(b). In the following, we use Fig. 7 to demonstrate the procedure of pure probabilistic algorithm. From Step 1 of the algorithm, we obtain four vectors that denote the row number, column number, horizontal physical-length vertical physical-length. They are respectively:,, Fig. 5. The result on an example with N = 5 using pure probabilistic algotithm. Fig. 6. The result on an example with N =11 using pure probabilistic algorithm. (Note that replacing R C with R can lead to a better result.) Fig. 7. Comparison of the results by (a) the algorithm from [5] (b) pure probabilistic algorithm on an example with N =6.. From Step 2, both are calculated, as shown in Tables I II. After performing Step 3, we have the following probability matrices: Then, Steps 4 through 6 of the algorithm generate the Steiner tree

6 CHEN et al.: PROBABILITY-BASED APPROACH TO RST PROBLEMS 841 TABLE III COMPARISON OF DIFFERENT ALGORITHMS ON THREE SMALL-SIZE EXAMPLES as shown in Fig. 7(b), which turns out to be an optimal RST. While the proposed algorithm produces the promising results, it is generally not optimal. In Fig. 6, for instance, a better solution could be found by replacing the segments with the segment (shown as the dotted line). For comparison, the experimental data given by different algorithms are shown in Table III, where WL represents the wire length, CPU represents the CPU time (in seconds), MST corresponds to the Minimal Spanning Tree resulting from [12]. From this table, it can be seen that the performance of the proposed algorithms is comparable to the BI1S algorithm in terms of wire length, while the nondeterministic algorithm is the slowest. As the second set of experiments, we used a large number of test instances that were generated by romly selecting the coordinates for a set of points from integers ranging between The instance sizes (i.e., the number of points or terminals) are chosen from 3 to 200. The instances are divided into two groups: group 1 in which the instance sizes are less than 20 group 2 in which the instance sizes range from 20 to 200 with increments of ten. For each instance size, we generated 30 instances romly. The result quality from the proposed algorithms is evaluated using the ratio of RST wire length to MST wire length. The execution time is computed for comparison based on average CPU time needed for each size. For group 1, we tested four algorithms: pure probabilistic, MST combined, nondeterministic BI1S algorithms. The results are shown in Fig. 8 (for wire length comparison) Fig. 9 (for CPU time comparison). From Fig. 8, while the pure probabilistic MST combined algorithms get less improvement over the minimal spanning tree than the BI1S does, the nondeterministic algorithm is comparable to the BI1S in terms of wire length. Since each point on the curves in Fig. 8 represents an average of the results over 30 instances, it is clear that the nondeterministic algorithm can outperform the BI1S for some instances. As shown in Fig. 9, the pure probabilistic MST combined algorithms are faster than the BI1S. However, the CPU time of the nondeterministic algorithm is longer than the BI1S since many Steiner trees (the selected number of passes is ten in this experiment) have to be constructed. Fortunately, running speed is not an issue for small-size instances. In summary, the probability-based algorithms have very good average performance for small-size problems. For group 2, we tested three algorithms: pure probabilistic, MST combined BI1S algorithms. The results are shown in Figs Since the nondeterministic algorithm is com- Fig. 8. Wire length comparison for different algorithms on small-size problems. Fig. 9. Execution time comparison for different algorithms on small-size problems. Fig. 10. Wire length comparison for different algorithms on large-size problems. Fig. 11. Execution time comparison for different algorithms on large-size problems. putationally expensive for large-size problems, its performance is not shown for this experiment. From Fig. 10, the pure proba-

7 842 IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 10, NO. 6, DECEMBER 2002 bilistic algorithm MST combined algorithm get an average of 2% 6% improvement over MST wire length, respectively, compared to about 11% improvement with the BI1S algorithm. This indicates that the performance of probability-based approaches is still not good enough for large problems that are of theoretical interest. Therefore, further work is needed to enhance their performance. However, the MST combined algorithm is still faster than the BI1S, as can be seen in Fig. 11. Note that in this paper, we are not claiming our algorithm is better than the state-of-the-art algorithm. Instead, these preliminary experiments are used to show the general effectiveness of probabilistic approaches to Steiner problems. There are a few unique potential advantages with the probability-based techniques: 1) The proposed algorithms represent a class of segment-oriented methods which construct Steiner trees by selecting segments one after another. The segments with high probability are first selected during the tree construction. In contrast, the BI1S typically belongs to a class of point-oriented methods which select (Steiner) points one by one to find the best tree. In this sense, the segment-oriented methods are capable of dealing with congestion-driven interconnect design [8], [9]. This is because the segments falling into the congested areas can be assigned a lower weight so that they would be less likely to be selected in the tree construction. Thus, both wire length congestion can be minimized by taking into account the wiring flexibility of each individual tree the effect among different trees. 2) A natural solution for performance/efficiency improvement is to combine the probabilistic algorithm with the BI1S. Intuitively, the points connected to a low-probability segment can be ignored during the tree construction within the BI1S. This speeds up the process without suffering from wire length penalty. Another example for possible improvement is to use a prior decomposition/hierarchy technique followed by the probability approach. 3) As mentioned above, calculation of functions in our algorithms is time consuming. Therefore, an important speedup strategy is to obtain their values only once store them into a look-up table that can be used for different Steiner tree problems, enabling a more efficient computation. To summarize, our experiments show that the probabilitybased approaches have desirable performance in terms of computational costs generate reasonably good results in terms of wire length minimization. While they are still worse than the state-of-the-art algorithm for large-size problems, we strongly believe that further research can take advantage of the above unique properties to improve the performance. V. CONCLUSION We have presented a new class of approaches to Steiner tree problems based on probabilistic analysis, with the goal of finding the best solutions under statistical sense. We have described several algorithms their performance comparison with the existing algorithms. Since the actual VLSI designs have a small number of terminals in each signal net, one can use the nondeterministic algorithm which has the comparable performance to the state-of-the-art algorithm in terms of wire length. Further research work is needed to improve the algorithms for large-size problems. We also expect to explore the potential advantages of the probabilistic technique to extend the proposed probability model to other VLSI design applications, such as timing-driven interconnect optimization. REFERENCES [1] F. K. Hwang, An O(n log N ) algorithm for rectilinear minimal spanning trees, J. ACM, vol. 26, pp , [2] M. R. Gray D. S. Johnson, The rectilinear Steiner tree problem is NP-complete, SIAM J. Appl. Math, vol. 32, pp , [3] M. Hanan, On Steiner s problem with rectilinear distance, SIAM J. Appl. Math, pp , [4] J. P. Cohoon, D. S. Richards, J. S. Salowe, An optimal Steiner tree algorithms for a net whose terminals lie on the perimeter of a rectangle, IEEE Trans. Comput.-Aided Design, vol. 9, pp , Apr [5] R. Condamoor I. G. Tollis, A new heuristic for rectilinear Steiner trees, in Proc. Int. Symp. Circuits Syst., 1990, pp [6] F. K. Hwang, On Steiner minimal trees with rectilinear distance, SIAM J. Appl. Math, pp , [7] F. K. Hwang D. S. Richards, Steiner tree problems, Networks, vol. 22, pp , [8] C. J. Alpert et al., Steiner tree optimization for buffers, blockages bays, IEEE Trans. Computer -Aided Design, vol. 20, pp , Apr [9] E. Bozorgzadeh, R. Kastner, M. Sarrafzadeh, Creating exploiting flexibility in Steiner trees, in Proc. 38th IEEE/ACM Design Automation Conf., June 2001, pp [10] J. Cong, L. He, C. K. Koh, P. H. Madden, Performance optimization of VLSI interconnect layout, Integration VLSI J., vol. 21, pp. 1 94, [11] I. I. Moiu, V. Vazirani, J. L. Ganley, A new heuristic for rectilinear Steiner trees, IEEE Trans. Computer-Aided Design, vol. 19, pp , Oct [12] J. L. Ganley J. P. Cohoon, Routing a multi-terminal critical net: Steiner tree construction in the presence of obstacles, in Proc. Int. Symp. Circuits Syst., May 1994, pp [13] A. B. Kahng G. Robins, A new class of iterative Steiner heuristics with good performance, IEEE Trans. Computer-Aided Design, vol. 11, pp , July [14] M. Borah, R. M. Owens, M. J. Irwin, An edge-based heuristic for Steiner routing, IEEE Trans. Computer-Aided Design, vol. 13, pp , Dec [15] D. M. Warme, P. Winter, M. Zachariasen, Exact solutions to large-scale plane Steiner tree problems, in Proc. 10th Annu. ACM-SIAM Symp. Discrete Algorithms (SODA 99), 1999, pp [16] D. Z. Du, J. M. Smith, J. H. Rubinstein, Advances in Steiner Trees. Boston, MA: Kluwer Academic, [17] J. Griffith, G. Robins, J. S. Salowe, T. Zhang, Closing the gap: Near-optimal Steiner trees in polynomial time, IEEE Trans. Computer- Aided Design, vol. 13, pp , Nov Chunhong Chen (M 99) received the B.S. M.S. degrees from Tianjin University, Tianjin, China, the Ph.D. degree from Fudan University, Shanghai, China, all in electrical engineering. From September 1997 to August 1998, he was a Research Associate at the Hong Kong University of Science Technology, Hong Kong. From December 1998 to January 2001, he was a Postdoctoral Fellow at Northwestern University, Evanston, IL. Since February 2001, he has been with the Department of Electrical Computer Engineering at the University of Windsor, Ontario, Canada, where he is presently an Assistant Professor. His current research interests include physical layout, logic synthesis, timing analysis, power optimization for integrated circuits. Jiang Zhao received the B.S. degree in electrical engineering from Wuhan University, China, in Currently, he is studying for the M.S. degree at the Department of Electrical Computer Engineering, University of Windsor. His recent research is on interconnect optimization VLSI design.

8 CHEN et al.: PROBABILITY-BASED APPROACH TO RST PROBLEMS 843 Majid Ahmadi (S 75 M 77 SM 84 F 02) received the B.Sc. degree in electrical engineering from Arya Mehr University, Tehran, Iran, the Ph.D. degree in electrical engineering from Imperial College of London University, London, U.K., in , respectively. He has been with the Department of Electrical Computer Engineering, University of Windsor, Windsor, ON, Canada, since 1980, currently as Professor Associate Director for Research Center for Integrated Microsystems. His research interests include digital signal processing, machine vision, pattern recognition, neural network architectures, applications, VLSI implementation. He has coauthored the book Digital Filtering in 1-D 2-Dimensions, Design Applications (New York: Plenum, 1989) has published more than 300 articles in this area. He is an Associate Editor for the Journal of Pattern Recognition, Journal of Circuits, Systems Computers, the International Journal of Computers in Electrical Engineering. Dr. Ahmadi was a recipient of an Honorable Mention Award from the Editorial Board of the Pattern Recognition Journal in 1992 the Distinctive Contributed Paper Award from the Multiple-Valued Logic Technical Committee of the IEEE Computer Society in He is the IEEE-CAS representative on the Neural Network Council Past Chair of the IEEE Circuits Systems Neural Systems Applications Technical Committee. He is a Fellow of the Institution of Electrical Engineers, U.K.