GAT-Steiner: Rectilinear Steiner Minimal Tree Prediction Using GNNs

TL;DR

GAT-Steiner tackles the NP-hard Rectilinear Steiner Minimum Tree problem by predicting Steiner points with a Graph Attention Network on Hanan-grid representations, enabling parallel GPU inference. The approach achieves near-perfect Steiner-point accuracy (random nets ~$99.992\%$, ISPD19 ~$99.909\%$) with sub-percent increases in wire length on suboptimal nets ($0.420\%$ random, $0.480\%$ ISPD19) and substantial runtime speedups over traditional solvers (average ~$9.363\×$ over GeoSteiner, up to ~$24.305\×$ for smaller nets). It uses disjoint batching to process multiple nets concurrently, and includes a lightweight non-Steiner refinement to mitigate degree-2 mispredictions without compromising efficiency. Overall, GAT-Steiner demonstrates that learned GNN models can robustly and efficiently accelerate RSMT prediction in VLSI routing pipelines, with practical benefits for large-scale design flows.

Abstract

The Rectilinear Steiner Minimum Tree (RSMT) problem is a fundamental problem in VLSI placement and routing and is known to be NP-hard. Traditional RSMT algorithms spend a significant amount of time on finding Steiner points to reduce the total wire length or use heuristics to approximate producing sub-optimal results. We show that Graph Neural Networks (GNNs) can be used to predict optimal Steiner points in RSMTs with high accuracy and can be parallelized on GPUs. In this paper, we propose GAT-Steiner, a graph attention network model that correctly predicts 99.846% of the nets in the ISPD19 benchmark with an average increase in wire length of only 0.480% on suboptimal wire length nets. On randomly generated benchmarks, GAT-Steiner correctly predicts 99.942% with an average increase in wire length of only 0.420% on suboptimal wire length nets.

Figures (7)

• Figure 1: GAT message passing of nodes on a Hanan grid to update a single node's feature vector, $\overrightarrow{h}_5$. Three attention heads are shown (green, red, blue) and three layers are stacked on top of each other. Node features can be coordinates, node type, etc. and are aggregated with multiple-levels.
• Figure 2: The model flow, starting with Hanan grids of nets; adjacency matrices and feature vectors are constructed. Then, feature embeddings are found using 2 GAT layers using multiple attention heads, with ELU and sigmoid activation functions respectively. Steiner point probability for each node is found and points above the threshold are selected as Steiner points. Finally, nets are routed by finding the MST of pins and Steiner nodes.
• Figure 3: A net with a mispredicted degree-2 "Steiner" node (which isn't by definition a Steiner node) that increases the total wire length by 6.34%.
• Figure 4: GAT-Steiner makes an average error of less than 1% in wire length on suboptimal nets across all datasets and has the smallest maximum suboptimal wire length outlier. The red line shows the average wire length increase.
• Figure 5: GAT-Steiner's accuracy remains above 99% as problem size scales up while other heuristics' accuracy decline.