A Parallel Algorithm for Constructing Obstacle-Avoiding Rectilinear Steiner Minimal Trees on Multi-Core Systems

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1 A Parallel Algorithm for Constructing Obstacle-Avoiding Rectilinear Steiner Minimal Trees on Multi-Core Systems Cheng-Yuan Chang and I-Lun Tseng Deartment of Comuter Science and Engineering Yuan Ze University, Taiwan Abstract - In the field of integrated circuit hysical design automation, the roblem of obstacle-avoiding rectilinear Steiner minimal tree (OARSMT) construction is a fundamental roblem and has attracted a lot of research attention. In this aer, a arallel algorithm for constructing OARSMTs is roosed. The algorithm is based on maze routing and double front-wave exansion. Exerimental results show that our algorithm not only generates very short wires, but also erforms efficiently on shared-memory multi-core comuter systems. Comared with the sequential execution of our arallel rogram that has been imlemented, the rogram can achieve % seed-u on average while running on a multicore workstation. Keywords: VLSI Physical Design Automation, Routing, Parallel Programming, Multi-Core, Algorithm Introduction As the semiconductor rocess technology advances, transistor feature sizes have been shrinking. As a result, an integrated circuit can accommodate many transistors and building blocks. To imlement an integrated circuit containing comlex functions, a number of building blocks are usually required. Since each building block erforms a secific set of functions (e.g., inverter, NAND, NOR, shift register, memory, and ALU), we can combine many building blocks and make roer connections between their terminals in order to build a larger circuit. Routing algorithms, which are caable of making connections between building blocks, lay a crucial role, esecially when the design is large or comlex. In the field of integrated circuit hysical design automation [, ], many routing algorithms require the function of constructing rectilinear Steiner minimal trees (RSMTs). That is because most integrated circuit layouts use rectilinear wires. Also, by minimizing the total wire length of an integrated circuit, the signal roagation delay, the ower consumtion, and the die size can all be reduced. Since an integrated circuit layout can contain building blocks and rerouted nets, it is desirable to consider regions that cannot be This research was suorted in art by the National Science Council of Taiwan under grants NSC-9--E--0 and NSC- 99--E--0. assed through by a net (or wire) as obstacles. Consequently, develoing efficient algorithms for solving the obstacleavoiding rectilinear Steiner minimal tree (OARSMT) roblem is imortant and has thus received a lot of research attention [-, -0]. Although many algorithms have been roosed in order to solve the OARSMT roblem, most of these algorithms are sequential, rather than arallel. As inexensive multi-core rocessors and comuter systems have become widely available, develoing arallel algorithms allows us to exloit the comutational ower from these shared-memory multicore systems. This article resents a arallel algorithm for constructing OARSMTs. The algorithm is based on Watanabe s Steiner tree construction algorithm [], but there are differences between these two aroaches. First, our algorithm is suitable for use on a shared-memory multi-core comuter system, whereas Watanabe s algorithm is suitable for use on a comuter system containing a two-dimensional array rocessor. Secifically, an array rocessor can have as many as tens of thousands of rocessing elements, while a sharedmemory multi-core comuter may have less than ten rocessor cores. Second, our algorithm adots two arallelized rocedures, namely PARALLEL-CONNECT() and PARALLEL-CLEANUP(), and these two rocedures can efficiently reduce the rogram execution time. The arallel algorithm roosed in this article has been imlemented in C++ with the use of OenMP []. Exerimental results show that the imlemented arallel rogram erforms efficiently on a shared-memory multi-core workstation. The remainder of this article is organized as follows. In Section, we formulate the OARSMT roblem. Our arallel algorithm for constructing OARSMTs is resented in Section. Exerimental results are given in Section. Finally, conclusions are drawn in Section. Problem Formulation In the xy lane, let T = {t,,, t m } be a set of m terminals which are to be connected by a net (or wire), and let O = {o, o,, o n } be a set of n rectilinear obstacles. A terminal can exist on the boundary or on a corner of an obstacle, but cannot exist within an obstacle. In Fig. (a), for examle, is a valid terminal since it is on the boundary of an obstacle, whereas is an invalid terminal since it is located

2 within the obstacle. Invalid terminals cannot aear in an instance of the OARSMT roblem. For the figures in this article, obstacles are denoted by light-gray regions. With regard to hysical ositions of line segments of a wire, a line segment can overla with the boundary of an obstacle as shown in Fig. (b), but cannot ass through the interior of an obstacle as shown in Fig. (c). Since a wire is comosed of rectilinear line segments, the total wire length of a routing result is comuted as the sum of the lengths of all line segments. Fig. (d) illustrates a valid routing result and its total wire length is. The OARSMT roblem is defined below. Definition (The OARSMT Problem): Given a set T of terminals and a set O of rectilinear obstacles in the xy lane, we are requested to construct a net (or wire) which consists of rectilinear line segments and connects all the terminals in T, while ossibly through some Steiner oints. In addition, line segments of the net cannot ass through the interior of each obstacle in O, and the total wire length of the routing result must be minimal. The Algorithm Our aroach for constructing an OARSMT is based on Watanabe s algorithm [], which mainly consists of two reetitive hases (known as double front-wave exansion). The first hase is called the first front-wave roagation; a set of start oints P S is selected, and then we roagate from these oints in order to find another terminal t t T, which is the closest to P S. During the roagation rocess, in addition, the roagation wave cannot ass through the interior of each obstacle. Also, the shortest distance between a oint in P S and each visited grid oint must be recorded; the numbers reresenting these distances are called distance numbers. The second hase of Watanabe s algorithm is named the second front-wave roagation; another roagation wave starts from the terminal t t to reach either a oint or oints in P S. An overlaing region between the first and the second hases can thus be found. Moreover, Steiner oints can also be determined. Therefore, an OARSMT can be obtained after connecting all the Steiner oints and terminals. Our arallel algorithm for constructing an OARSMT from given sets of T and O is described in Fig.. First, a start terminal (t s ) must be selected from T. Although an arbitrary terminal can be selected from T as the start terminal, our imlementation of the algorithm selects the terminal which has the least value in terms of the sum of x and y coordinates. The set V is used for storing all the terminals that have been visited during double front-wave exansion. Since the algorithm starts by roagating from the start terminal t s, it is assigned to V in line. The set S, which can be imlemented as a task queue, is used for storing grid oints from which the next iteration of double front-wave exansion will start; these grid oints may include terminals and/or Steiner oints. t o t o (a) (b) (c) (d) Figure. Valid and invalid OARSMT examles and routing results. Algorithm PARALLEL-OARSMT-CONSTRUCTION( T, O ) Inut: T /* all the terminals which are to be connected */ O /* all the rectilinear obstacles */ Outut: An OARSMT which connects all the terminals in T using only horizontal and/or vertical line segments while these line segments cannot ass through the interior of each obstacle.. t s select a start terminal from set T. V t s. S t s. while ( V < T ) {. t i PROPAGATE_ST( S ). PROPAGATE_ND( t i ). R the intersection region between the first and the second front-wave roagation hases.. Insert t i into V. 9. if ( NULL AND T ) 0. PARALLEL-CONNECT(). PARALLEL-CLEANUP(). S. S all the grid oints from which the first front-wave roagation in the next iteration will start.. }. FINAL-CONNECT(). return the comleted OARSMT Figure. The roosed arallel algorithm for OARSMT construction. The while loo in line checks whether all the terminals have been visited or not. Lines will be executed if at least one terminal in T has not been visited. The first frontwave roagation hase is imlemented in the rocedure PROPAGATE_ST(); the rocedure returns a terminal (t i ) which is the terminal found in the hase and is the closest terminal to grid oints in S. Note that distance numbers increase as the roagation roceeds during this hase. The second front-wave roagation hase is imlemented in the rocedure PROPAGATE_ND(). The rocedure returns, which is either a newly found Steiner oint or NULL. Note that distance numbers decrease as the roagation roceeds during this hase. In line, a region R, which is the intersection of regions between the first and the second roagation hases, can be obtained. In line, the terminal that has just been discovered during the first front-wave roagation hase is inserted into V. In lines 9 0, if Steiner oint exists and it is not an element in T, the wire connections between and terminals in a region which was formed reviously will be made. We have found that the connection rocess can be arallelized; it has been imlemented in the rocedure PARALLEL-CONNECT(). The rocedure can efficiently seed u the connection rocess and will be described in detail in subsection.. t o t o

3 Since distance numbers are recorded during double frontwave exansion, arts of these numbers must be cleared in rearation for the next double-wave exansion iteration. Because the rocess of clearing distance numbers does not have any data deendency, the rocess can be arallelized; the arallelized rocess has been imlemented in the rocedure PARALLEL-CLEANUP() and will be described in detail in subsection.. In lines of our arallel algorithm, the set S is reset and then its content is udated in rearation for the next iteration of double-wave exansion. After all iterations of double-wave exansion have been comleted, there can still exist terminals that remain unconnected. The rocedure FINAL-CONNECT() will comlete the routing for these unconnected terminals. Fig. illustrates an examle of constructing an OARSMT by using our algorithm. As shown in Fig. (a), an instance of the OARSMT roblem, which is to be solved, contains four terminals and three obstacles. If t is the start terminal, the first front-wave roagation hase will generate distance numbers, as shown in Fig. (b), after is reached. The second front-wave roagation hase, which starts from and then reaches t, will generate the result as illustrated in Fig. (c). Note that the second roagation hase will only visit grid oints that have been visited reviously in the first roagation hase. Therefore, the overlaing region (region A) between the first and the second front-wave roagation hases can be obtained, as shown in Fig. (d). After the first iteration of the while loo finishes, set S contains all the grid oints which are on the boundary of region A; these oints include terminals t and. During the second iteration of the while loo, terminal t will be visited and then region B will be generated. A Steiner oint, which is at the intersection between regions A and B, can be obtained; the oint is named in Fig. (e). The rocedure PARALLEL-CONNECT() can then be invoked to connect and t as well as to connect and, as illustrated in Fig. (f). After that, the third iteration of the while loo will visit, and region C in Fig. (g) will be formed. Next, Steiner oint will be obtained, and it can then be connected to and t. Finally, the unconnected terminals (i.e., only in this examle) can be connected after invoking the rocedure FINAL-CONNECT(). The final routing result is shown in Fig. (h).. Parallel-Connect In Watanabe s PAR- algorithm [], a arameter named exansion distance (denoted by D ex ) is used to control the quality of routing during front-wave roagation hases. When D ex =, the PAR- algorithm is equivalent to Lee s algorithm [], and a wire whose length is the shortest between two given oints can be obtained. On the other hand, when D ex =, a wire which has the minimum number of bends can be generated, although the wire length may not be the shortest. The rocedure PARALLEL-CONNECT() is used to connect a Steiner oint with other oints, including terminals and/or t o o (a) t 0 t t o o t (c) o o o o o A B (e) o o C o t t t t other Steiner oints, within a region. The technique of frontwave roagation is also used in the rocedure. However, the rocedure is arallelized and arameter D ex is set to infinity. Therefore, connections will be made within the region and have the minimum number of bends. As a result, final routing results generated by our algorithm can have minimal total wire lengths and minimal number of bends. An examle for illustrating detailed stes of PARALLEL- CONNECT() is shown in Fig.. In this examle, we assume that Steiner oint has been found, and we need to connect to t as well as connect to, as shown in Fig. (a). Also, connections must be made within region A. After the frontwave roagation hase, which starts from, with D ex set to is finished, distance numbers for visited grid oints are shown in Fig. (b). In the backtracking hase, terminal t must be connected to and terminal must also be connected to o t 0 t t t o o o t A (b) (d) (f) o o o o o o B o (g) (h) Figure. An examle of constructing an OARSMT by using our algorithm. o t t t t

4 . Since t and must be located on different corners of region A and must be on the boundary of region A, the two connection rocesses are indeendent and can thus be arallelized. Therefore, the connection between t and can be made by one thread while the connection between and can be made by another thread. Because there are two ossible routes for the connection between t and, one of the two ossible connection results, shown in Fig. (c) and Fig. (d), will be generated. After all the terminals have been visited during iterations of double front-wave exansion, there may still exist terminals that have not been connected. These terminals will be connected via rocedure FINAL-CONNECT().. Parallel-CleanU After each iteration of the first and the second front-wave roagation hases, an intersection region R can be obtained, as mentioned reviously. However, distance numbers for grid oints which are not located in the intersection region must be reset to zero before the next iteration of double front-wave exansion starts. The rocess of resetting distance numbers of grid oints is named the clean-u rocess and can be arallelized. Fig. (a) illustrates an examle where region A has been generated. After the execution of the clean-u rocess, distance numbers for grid oints outside region A will be reset, as shown in Fig. (b). Another occasion that requires the invocation of the clean-u rocess is when wire connections within a region have been made, and the region must be removed in rearation for the next iteration of double front-wave exansion. For examle, in Fig. (c), region A has been generated, followed by the generation of region B. Also, Steiner oint has been obtained. After terminals t and have been connected to, therefore, distance numbers for grid oints after erforming double front-wave exansion as well as distance numbers for grid oints inside region A must be reset. However, distance numbers for grid oints inside regions which have not been removed must be ket intact. The result after executing the clean-u rocess is illustrated in Fig. (d). Since the two occasions for erforming the clean-u rocess can be merged, we imlemented the clean-u rocess in rocedure PARALLEL-CLEANUP(). The rocedure can be arallelized because there is no data deendency while resetting distance numbers for individual grid oints. Exerimental Results The algorithm resented in Fig. has been imlemented in C++ with the use of OenMP []. Exeriments were carried out on a Linux workstation containing two Quad-Core Xeon E0.GHz rocessors and GB of RAM. Table I shows statistics information for each of benchmark testcases; the information includes the number of terminals (denoted by #Pin) and the number of obstacles (denoted by #Obs). Among these benchmark testcases, five were industrial t o o A (a) o o B o t testcases (ind to ind) from Synosys, eleven testcases (rc0 to rc) were used in [, ], and five randomly-generated testcases (rt0 to rt0) were used in [, ]. Table I also shows the execution times of our rogram for solving these benchmark testcases. On average, % seed-u can be achieved while executing the arallel rogram using threads for solving these benchmark testcases. Table II resents comarisons of total wire lengths for routing results generated by different OARSMT algorithms; the testcases listed in the table are the same as the ones listed in Table I. Additionally, Table III shows comarisons of total wire lengths for routing results generated by different o o t Thread Thread B B t o o Thread t Thread (c) (d) Figure. An examle for illustrating detailed stes of PARALLEL- CONNECT(). o o o (b) A t 0 o t 0 t o (a) o A B B t 0 o t o t t (c) (d) Figure. Examles for illustrating functionalities of rocedure PARALLEL- CLEANUP(). o o A (b) o o o B o o t t t

5 TABLE II. Comarisons of Total Wire Lengths for Different OARSMT Algorithms Testcase Lin [] Lin [] Long [] Li [] Liu [9] Liu [0] Ajwani [] Our Program ind ind ind ind ind 9 0 rc rc rc rc rc rc rc rc rc rc rc 09 r rt rt rt rt Normalized OARSMT algorithms when all the obstacles in each of the benchmark testcases have been removed. As can be seen in both tables, our aroach is very cometitive in terms of minimizing total wire lengths. Conclusion We roosed a arallel algorithm which is caable of constructing obstacle-avoiding rectilinear Steiner minimal trees (OARSMTs) in the gridded xy lane. The algorithm is based on maze routing and double front-wave exansion. Although our aroach does not use refinement techniques, exerimental results have shown that it is very cometitive in terms of minimizing total wire lengths. Moreover, % seed-u can be achieved on average while executing the arallel rogram using threads on a multi-core comuter system. [] References [] Chung-Wei Lin, Szu-Yu Chen, Chi-Feng Li, Yao-Wen Chang, and Chia-Lin Yang, Efficient Obstacle-Avoiding Rectilinear Steiner Tree Construction, In Proceedings of International Symosium on Physical Design,. -, 00. [] Jieyi Long, Hai Zhou, and S. O. Memik, EBOARST: An Efficient Edge-Based Obstacle-Avoiding Rectilinear Steiner Tree Construction Algorithm, IEEE Transactions on Comuter-Aided Design of Integrated Circuits and Systems, vol., no.,. 9-, 00. [] Chung-Wei Lin, Szu-Yu Chen, Chi-Feng Li, Yao-Wen Chang, and Chia-Lin Yang, Obstacle-Avoiding TABLE I. EXECUTION TIMES OF OUR PROGRAM FOR SOLVING BENCHMARK TESTCASES Testcase #Pin #Obs Sequential (sec.) Parallel - threads (sec.) Seed-U (%) ind ind ind ind ind rc rc rc rc rc rc rc rc rc rc rc rt rt rt rt rt Average. Rectilinear Steiner Tree Construction Based on Sanning Grahs, IEEE Transactions on Comuter-Aided Design of Integrated Circuits and Systems, vol., no.,. -, 00. [] Liang Li and Evangeline F.Y. Young, Obstacle- Avoiding Rectilinear Steiner Tree Construction, In Proceedings of International Conference on Comuter- Aided Design,. -, 00.

6 TABLE III. Comarisons of Total Wire lengths for Different OARSMT Algorithms When Testcases Do Not Contain Obstacles Testcase #Pin #Obs Long [] Li [] Chu [] Ajwani [] Our Program ind ind ind ind ind rc rc rc rc rc rc rc rc rc rc rc rt 0 0 rt rt rt rt Normalized [] Naveed A. Sherwani, Algorithms for VLSI Physical Design Automation, Kluwer Academic Publishers, 999. [] Sabih H. Gerez, Algorithms for VLSI Design Automation, Wiley, 99. [] Gaurav Ajwani, Chris Chu, and Mak Wai-Kei, FOARS: FLUTE Based Obstacle-Avoiding Rectilinear Steiner Tree Construction, IEEE Transactions on Comuter- Aided Design of Integrated Circuits and Systems, vol. 0, no.,. 9-0, 0. [] Tao Huang and Evangeline F.Y. Young, Obstacle- Avoiding Rectilinear Steiner Minimum Tree construction: An Otimal Aroach, In Proceedings of International Conference on Comuter-Aided Design,. 0-, 00. [9] Chih-Hung Liu, Shih-Yi Yuan, Sy-Yen Kuo, and Yao- Hsin Chou, An O(n log n) Path-Based Obstacle- Avoiding Algorithm for Rectilinear Steiner Tree Construction, In Proceedings of Design Automation Conference,. -9, 009. [0] Chih-Hung Liu, Shih-Yi Yuan, Sy-Yen Kuo, and Jung- Hung Weng, Obstacle-Avoiding Rectilinear Steiner Tree Construction Based on Steiner Point Selection, In Proceedings of International Conference on Comuter- Aided Design,. -, 009. [] Chris Chu and Yiu-Chung Wong, FLUTE: Fast Looku Table Based Rectilinear Steiner Minimal Tree Algorithm for VLSI Design, IEEE Transactions on Comuter-Aided Design of Integrated Circuits and Systems, vol., no.,. 0-, 00. [] Takumi Watanabe, Hitoshi Kitazawa, and Yoshi Sugiyama, A Parallel Adatable Routing Algorithm and Its Imlementation on a Two-Dimensional Array Processor, IEEE Transactions on Comuter-Aided Design of Integrated Circuits and Systems, vol., no.,. -0, 9. [] Barbara Chaman, Gabriele Jost, and Ruud van der Pas, Using OenMP: Portable Shared Memory Parallel Programming, The MIT Press, 00. [] C. Y. Lee, An Algorithm for Path Connections and Its Alications, IRE Transactions on Electronic Comuters, vol. EC-0, no.,. -, Setember, 9.