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Consider an electrical network modeled as a planar graph G, where each edge represents a 1Ω resistor. How quickly can we compute the exact effective resistance between two vertices in G? Equivalently, how quickly can we compute the exact current flowing along each edge if we attach a 1V battery to two vertices in G?

Kirchhoff's well-known voltage and current laws reduce this problem to solving a system of linear equations with one variable per edge. More recent results — described explicitly by Klein and Randić (1993) but implicit in earlier work of Doyle and Snell (1984) — reduce the problem to solving a linear system with one variable per vertex, representing that node's potential; the matrix for this linear system is the Laplacian matrix of the graph.

Either linear system can be solved exactly in $O(n^{3/2})$ time using nested dissection and planar separators [Lipton Rose Tarjan 1979]. Is this the fastest algorithm known?

Recent seminal results of Spielman, Teng, and others imply that the Laplacian system in arbitrary graphs can be solved approximately in near-linear time. See [Koutis Miller Peng 2010] for the current best running time, and this amazing article by Erica Klarreich at the Simons Foundation for a high-level overview. But I'm specifically interested in exact algorithms for planar graphs.

Assume a model of computation that supports exact real arithmetic in constant time.

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the Klarreich article mentions applications in (optimizing) max flow near the end and is already out of date due to the recent Orlin $O(m \cdot n)$ breakthrough, which is apparently not related to the Laplacian direction of attack. see also this recent tcs.se question, Are any of the state of the art Maximum Flow algorithms practical? –  vzn Feb 14 '13 at 19:25
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Using nested dissections you can even solve a linear system (based on a planar graph) in $O(\sqrt{n^\omega})$. This was for example noted in a paper I have with Günter Rote and Ares Ribó and in this paper by Alon and Yuster.

The former paper also contains an approach how to compute the pairwise resistances between vertices on a a common face in $O(\sqrt{n^\omega})$. This paper by Kenyon might also contain helpful ideas.

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