Rectilinear Minimum Spanning Tree Research Articles

Zero-Change Netlist Transformations: A New Technique for Placement Benchmarking

In this paper, the authors introduce the concept of zero-change netlist transformations (ZCNTs) to: 1) quantify the suboptimality of existing placers on artificially constructed instances and 2) "partially" quantify the suboptimality of placers on synthesized netlists from arbitrary netlists by giving lower bounds to the suboptimality gap. Given a netlist and its placement from a placer, a class of netlist transformations that synthesizes a different netlist from the given netlist is formally defined, but yet the new netlist has the same half-perimeter wire length (HPWL) on the given placement. Furthermore, and more importantly, the optimal HPWL value of the new netlist is no less than that of the original netlist. By applying the transformations and reexecuting the placer, any deviation in HPWL as a lower bound to the gap from the optimal HPWL value of the new synthesized netlist can be interpreted. The transformations allow us to: 1) increase the cardinality of hyperedges; 2) reduce the number of hyperedges; and 3) increase the number of two-pin edges, while maintaining the placement HPWL constant. It is developed here methods that apply ZCNTs to synthesize netlists having typical netlist statistics. Furthermore, an approach to estimate the suboptimality of other metrics, such as rectilinear minimum-spanning tree (RMST) and minimum-Steiner tree, is extended. Using these transformations, the suboptimality of some of the existing academic placers (FengShui, Capo, mPL, Dragon) is studied on synthesized netlists from the IBM benchmarks with instances ranging from 10k to 210k placeable instances. The results show that current placers exhibit suboptimal behavior to ZCNTs with varying degree according to the placer. Systematic suboptimality deviations in HPWL and RMST are displayed on the synthesized netlists from IBM (version 1) benchmarks. The specific nature of the transformations points out troublesome netlist structures and possible directions for improvement in the existing placers

A Prim-based Heuristic Algorithm for Obstacle Avoiding Rectilinear Steiner Minimal Tree

Abstract Obstacle Avoiding Rectilinear Steiner Minimal Tree (OARSMT) is known to provide the shortest interconnection between n terminals in the presence of m obstacles using the rectilinear distance. We introduce a new line-based graph, which we call the Maneuver Graph that constructs a reduced set of possible paths between any two terminals in the presence of m obstacles with complexity of O(m) . Moreover, a dynamic programming algorithm is developed to recover the two-terminal shortest path over the Maneuver Graph with complexity of O(m) . Based on the Maneuver Graph we introduce a Prim-based heuristic algorithm that constructs an approximated OARMST in a way analogous to the Obstacle Avoiding Rectilinear Minimum Spanning Tree (OARMST) construction. The nearest terminal is redefined as that one with the shortest interconnection to a terminal, a Steiner or a corner point of the currently constructed sub-tree. This interconnection is established with minimum added tree length and maximum overlap with the already existing interconnections. We show that this algorithm has at worst a complexity of O(n3m2) , which is still less than those of 4-steinerization methods. We also show that the algorithm can achieve a percentage-of-improvement about 10% over OARMST which is better than that obtained by the k-steinerization methods.

A simple proof of the planar rectilinear Steiner ratio

The rectilinear Steiner ratio is the worst-case ratio of the length of a rectilinear minimum spanning tree to the length of a minimal tree. Hwang proved that the ratio for point sets in the plane is 3/2. We provide a simple proof of the 3/2-bound.

Fast vicinity-upgrade algorithm for rectilinear steiner trees

A new algorithm for rectilinear Steiner trees (RST) is presented. The proposed algorithm is based on successively constructing a vicinity structure from a rectilinear minimum spanning tree (MST) and generating a refined RST. The algorithm achieves an 8.5–10% average cost improvement over the rectilinear MST at a time complexity of O(n log n). To address specifically the linear programming approach of global routing, the algorithm is modified to generate K trees.