PEERS: A Parallel and Exact Effective Resistance Solver via Implicit Inversion and Augmented Symbolic Analysis

2026-06-30

AI Summary

This work addresses the long-standing trade-off among accuracy, memory usage, and computational efficiency in computing effective resistances in large-scale power grids. While approximate methods suffer from insufficient accuracy, exact approaches incur prohibitive O(n²) memory overhead or redundant queries. The paper introduces PEERS, a solver that uniquely unifies theoretically optimal parallel span complexity O(n^α) with strict O(nnz(L)) space complexity. Its core innovation lies in an implicit inverse computation model based on Cholesky factors, integrated with a state-inheriting depth-first search and a dynamic query-update mechanism, enabling efficient computation of all edge effective resistances in a single parallel sweep. Experiments demonstrate that PEERS achieves an average 83.3× speedup over state-of-the-art parallel solvers under identical memory constraints, processes million-node circuits in 18.8 seconds, and scales to networks with up to 17 million nodes.

Abstract

High-precision effective resistance computation is a cornerstone of Electronic Design Automation (EDA) sign-off, yet it remains a fundamental bottleneck in large-scale power grid analysis, spectral sparsification, and circuit reliability. Existing approaches face a prohibitive "precision-memory impasse": approximate methods lack the stringent accuracy required for high-stakes industrial sign-off, while exact methods either suffer from redundant query overheads or trigger $O(n^2)$ memory explosions. To resolve this, we propose PEERS, a Parallel and Exact Effective Resistance Solver powered by an implicit inverse computing model of the Cholesky factor. By integrating a state-inherited augmented depth-first search (DFS) with a dynamic query update mechanism, PEERS eliminates numerical redundancy and evaluates all-edge resistance queries in a single parallel sweep. We provide a rigorous Work-Span analysis, proving that for graphs satisfying an $O(n^α)$ separator theorem, PEERS achieves a theoretically optimal parallel span of $O(n^α)$ while strictly maintaining $O(nnz(L))$ space complexity. Numerical evaluations on industrial benchmarks demonstrate that PEERS achieves an average speedup of 83.3x over state-of-the-art parallel solvers under identical memory constraints. Notably, PEERS processes a 1-million-node industrial graph in just 18.8 seconds and scales to 17 million nodes in under an hour, providing the first computationally feasible path for exact all-edge resistance analysis in multi-million-gate designs.

Problem

• effective resistance
• precision-memory tradeoff
• large-scale power grid analysis
• circuit reliability
• spectral sparsification

Innovation

• implicit inversion
• augmented DFS
• parallel exact solver
• effective resistance
• Cholesky factor