Sequence Pair Representation Research Articles

Handling soft modules in general nonslicing floorplan using Lagrangian relaxation

In the early stage of floorplan design, many modules have large flexibilities in shape (soft modules). Handling soft modules in general nonslicing floorplan is a complicated problem. Many previous works have attempted to tackle this problem using heuristics or numerical methods, but none of them can solve it optimally and efficiently. In this paper, we show how this problem can be solved optimally by geometric programming using the Lagrangian relaxation technique. The resulting Lagrangian relaxation subproblem is so simple that the optimal size of each module can be computed in linear time. We implemented this method in a simulated annealing framework based on the sequence pair representation. The geometric program is invoked in every iteration of the annealing process to compute the optimal size of each module to give the best packing. The execution time is much faster (at least 15 times faster for data sets with more than 50 modules) than that of the most updated previous work by Murata and Kuh (1998). For a benchmark data with 49 modules, we take 3.7 h in total for the whole annealing process using a 600-MHz Pentium III processor while the convex programming approach described by Murata and Koh needs seven days using a 250-MHz DEC Alpha. Our technique will also be applicable to other floorplanning algorithms that use constraint graphs to find module positions in the final packing.

Floorplanning using a tree representation

We present an ordered tree (O tree) structure to represent nonslicing floorplans. The O tree uses only n(2+[lg n]) bits for a floorplan of n rectangular blocks. We define an admissible placement as a compacted placement in both x and y directions. For each admissible placement, we can find an O-tree representation. We show that the number of possible O-tree combinations is O(n!2/sup 2n-2//n/sup l.5/). This is very concise compared to a sequence pair representation that has O((n!)/sup 2/) combinations. The approximate ratio of sequence pair and O-tree combinations is O(n/sup 2/(n/4e)/sup n/). The complexity of O tree is even smaller than a binary tree structure for slicing floorplan that has O(n!2/sup 5n-3//n/sup 1.5/) combinations. Given an O tree, it takes only linear time to construct the placement and its constraint graph. We have developed a deterministic floorplanning algorithm utilizing the structure of O tree. Empirical results on MCNC (www.mcnc.org) benchmarks show promising performance with average 16% improvement in wire length and 1% less dead space over previous central processing unit (CPU) intensive cluster refinement method.

Symmetry within the sequence-pair representation in the context of placement for analog design

This paper addresses the problem of device-level placement for analog layout, focusing mainly on symmetry-related aspects. Different from most of the existent analog placement approaches, employing basically simulated annealing optimization algorithms operating on flat (absolute) spatial representations, our model uses a more recent topological representation called sequence-pair, which has the advantage of not being restricted to slicing floorplan topologies. In this paper, we explain how specific features essential to analog placement, such as the ability to deal with complex symmetry constraints (for instance, an arbitrary number of symmetry groups of cells), can be easily handled by employing the sequence-pair representation. Several analog examples substantiate the effectiveness of our placement tool, which is already in use in an industrial environment.