Real and Complex Dynamics of Iterative MethodsView this Special Issue
Accounting for Recent Changes of Gain in Dealing with Ties in Iterative Methods for Circuit Partitioning
In iterative methods for partitioning circuits, there is often a choice among several modules which will all produce the largest available reduction in cut size if they are moved between subsets in the partition. This choice, which is usually made by popping modules off a stack, has been shown to have a considerable impact on performance. By considering the most recent change in the potential reduction in cut size associated with moving each module between subsets, the performance of this LIFO (last-in first-out) approach can be significantly improved.
Let be a circuit, where is a set of modules and is a set of nets, each of which connects two or more modules. A balanced bipartition divides the circuit into two disjoint subsets, each containing the same number of modules. Finding a balanced bipartition in which as many nets as possible connect modules in the same partition is an important problem, with applications that include the design automated of VLSI chips and multichip boards.
Many heuristics for circuit bipartitioning have been proposed [1–4], among which iterative improvement algorithms are perhaps the most successful. An iterative improvement algorithm can be deployed as a freestanding heuristic, as a framework for further refinement, or as a local optimizer forming part of a hybrid with metaheuristic methods [5–7]. These techniques rely on the quality of the iterative improvement algorithm. All such algorithms must start with an initial partition and move successive modules between subsets, using greedy decisions designed to reduce the number of nets connecting the subsets. The Kernighan-Lin algorithm (KL) [8, 9] is an example of a technique which swaps pairs of modules, and the Fiduccia-Mattheyses algorithm (FM)  is a faster version which moves one module at a time; there are several variants [3, 4, 11, 12].
Hagen et al.  identified the way in which an iterative improvement algorithm deals with the situation in which several modules have an equal claim to be moved between subsets as a significant denominator of its performance. Their experiments suggested that stack-based (LIFO) management of module moves reduced the number of connections between subsets most rapidly. In this paper, we build on this work by introducing an improved way of choosing candidate moves into an implementation of the FM algorithm for bipartitioning. We show that the proposed strategy outperforms existing strategies.
2. The Fiduccia-Mattheyses Iterative Method
The FM algorithm  starts with two randomly chosen subsets. It moves a module at a time from one subset to the other in an attempt to minimize the number of nets connecting both subsets, which is the cut size. Each module is associated with a gain which represents the reduction in cut size which would result from moving it to the other subset. Thus, , where is the set of nets connecting to other modules exclusively in the subset to which currently belongs and is the set of nets which connect exclusively to modules in the other subset. The module to move must be chosen on the basis of its gain and also the balance criterion between subsets; otherwise, we are likely to obtain a trivial solution in which the module with the fewest connections occupies one subset, and all the other modules occupy the other. The FM algorithm operates in a series of passes. After a module has been moved, it is locked to prevent further movements, until the end of the pass, which occurs when all modules have been locked. At the end of the pass, the algorithm backtracks to the situation obtained after the move that produced the fewest cut size. All the modules are now unlocked again and another pass starts. This process continues until a pass produces no reduction in cut size. Algorithm 1 shows the pseudocode of the FM algorithm.
3. The Significance of Ties between Modules
Heuristics such as KL and FM involve a chain of moves, for each of which the module with the highest gain is selected. A tie occurs if more than one module has the same highest gain. Hagen et al.  observed several ties while running FM for circuit partitioning; and Kim and Moon  and Yoon and Kim  also noticed ties occurring while running KL for graph bipartitioning. We found ties occurring regularly when running FM for circuit partitioning, and their number is tabulated, for several test circuits, in the second column of Table 2. This suggests that the way in which ties are dealt with is not a detail of the implementation but can drive the search along different paths.
There have been a number of studies on the handling of ties in the circuit partitioning problem. Krishnamurthy  pointed out that the arbitrary handling of ties can cause the FM  algorithm to perform poorly. He introduced a gain vector, which is effectively a form of look-ahead, and observed an improvement in performance. However, the gain vector requires a lot of memory , and ties may still occur. Hagen et al.  observed that stacking sets of modules with the same gain yield considerably better solutions than queueing or random ordering, in both the FM and Krishnamurthy algorithms. These authors suggest that a stack allows modules with the same gain to move one after another. Thus, they proposed a variant on Krishnamurthy’s gain vector which includes locality information and found that this improves performance. We also note that there have been some studies of mechanisms in algorithms for other graph partitioning problems [11, 14].
4. Favoring the Movement of Modules Related to Recently Moved Modules
A typical iterative improvement algorithm uses the gain bucket structure illustrated in Figure 1 and Algorithm 2, in which modules with the same gain are placed in the same bucket. If more than one module is in the bucket corresponding to the maximum gain, a tie occurs.
(a) LIFO: buckets are stacks
(b) FIFO: buckets are queues
After a module has been moved, the iterative improvement algorithm updates the gains of the modules to which it is connected. Their gains may increase, reduce, or stay the same, as shown in Figure 2. We can use these updates to influence the order in which we deal with the modules which all have the maximum gain. Existing strategies simply consider the modules to be stacked or queued within a bucket. Motivated by the conjecture that it should be advantageous to move modules related to other modules that have recently been moved, we modify the corresponding LIFO (last-in first-out) and FIFO (first-in first-out) paradigms by considering changes in gain. We call these variants, explained in Figure 3 and Algorithm 3, VLIFO and VFIFO. The difference between LIFO (or FIFO) and VLIFO (or VFIFO) lies in the way in which updated modules are added to the stack (or queue) that constitutes a bucket: modules with increased gain are pushed onto the top of the stack (or the head of the queue), modules with reduced gain are inserted at the bottom of the stack (or the tail of the queue), and modules with unchanged gain remain in the same places.
(a) Module with maximum gain
(b) Updated gains
(a) VLIFO: buckets are modified stacks
(b) VFIFO: buckets are modified queues
We tested the proposed algorithm on 10 benchmark circuit graphs, including seven ACM/SIGDA benchmarks. Table 1 shows the numbers of nodes, nets, and pins (connections to modules) in each circuit. In all of our experiments, we assume that the modules have unit area and constrain the partition sizes to differ by at most one module. Although a less accurate balance is usually acceptable in practical circuit partitioning, a tight constraint facilitates comparison with other partitioning techniques.
Table 2 shows the performance of five ways of dealing with ties: VFIFO, FIFO, Random, LIFO, and VLIFO. All the methods apart from Random took similar times to run; Random was slower. In line with results reported previously , LIFO significantly outperformed FIFO and Random on all the circuits; and Random performed better than FIFO. VLIFO gave the best results, and VFIFO gave the worst.
These results suggest that links between consecutively moved modules affect performance. We investigated this further by measuring the proportion of modules selected for moving which have just had their gain values reduced or increased (see Table 3 and Figure 4). The results, tabulated in Table 3, confirm that performance is strongly linked to the proportion of modules selected which have just had their gain increased.
6. Concluding Remarks
We have described an improved way of dealing with ties when modules are moved within a circuit partitioning algorithm based on iterative improvement algorithm. We modified the LIFO strategy, usually considered to be the best, by considering the most recent change in gain for each module. The application of this technique to multiway partitioning  is a promising avenue for future study.
A preliminary version of this paper appeared in the Proceedings of the International Conference on Computer Design, pp. 30–34, 2008.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by the Gachon University research fund of 2015 (GCU-2015-0030). The author would like to thank Professor Byung-Ro Moon for his helpful comments and suggestions, which improved the quality of this paper.
J.-P. Kim, Y.-H. Kim, and B.-R. Moon, “A hybrid genetic approach for circuit bipartitioning,” in Genetic and Evolutionary Computation—GECCO 2004: Genetic and Evolutionary Computation Conference, Seattle, WA, USA, June 26–30, 2004. Proceedings, Part II, vol. 3103 of Lecture Notes in Computer Science, pp. 1054–1064, Springer, Berlin, Germany, 2004.View at: Publisher Site | Google Scholar