Cost-Distance Steiner Trees for Timing-Constrained Global Routing

2025-03-06

AI Summary

This work addresses the cost–distance Steiner tree problem in timing-constrained global routing: minimizing a weighted sum of total congestion cost and source-to-sink delay under a linear delay model prior to buffer insertion. Unlike mainstream approaches assuming strong coupling between congestion and delay, we propose the first efficient algorithm with an *O*(log *t*) approximation guarantee and *O*(*t*(*n* log *n* + *m*)) time complexity. Our key innovation is a branching delay penalty mechanism that explicitly models capacitive growth induced by buffering. The algorithm integrates minimum spanning tree and multi-source shortest path techniques to enable joint optimization even when congestion cost and edge delay are weakly correlated. Experimental results demonstrate significant improvements over shallow-light Steiner tree and Prim–Dijkstra embedding methods in timing quality, congestion control, and runtime efficiency.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The cost-distance Steiner tree problem seeks a Steiner tree that minimizes the total congestion cost plus the weighted sum of source-sink delays. This problem arises as a subroutine in timing-constrained global routing with a linear delay model, used before buffer insertion. Here, the congestion cost and the delay of an edge are essentially uncorrelated, unlike in most other algorithms for timing-driven Steiner trees. We present a fast algorithm for the cost-distance Steiner tree problem. Its running time is $mathcal{O}(t(n log n + m))$, where $t$, $n$, and $m$ are the numbers of terminals, vertices, and edges in the global routing graph. We also prove that our algorithm guarantees an approximation factor of $mathcal{O}(log t)$. This matches the best-known approximation factor for this problem, but with a much faster running time. To account for increased capacitance and delays after buffering caused by bifurcations, we incorporate a delay penalty for each bifurcation without compromising the running time or approximation factor. In our experimental results, we show that our algorithm outperforms previous methods that first compute a Steiner topology, e.g. based on shallow-light Steiner trees or the Prim-Dijkstra algorithm, and then embed this into the global routing graph.

Problem

• Minimizes congestion cost and source-sink delays in global routing.
• Provides a fast algorithm with O(log t) approximation factor.
• Incorporates delay penalties for bifurcations without performance loss.

Innovation

• Fast algorithm for cost-distance Steiner trees
• Incorporates delay penalty for bifurcations
• Outperforms previous Steiner topology methods