# Localization of Electrical Flows

@inproceedings{Schild2018LocalizationOE, title={Localization of Electrical Flows}, author={Aaron Schild and Satish Rao and Nikhil Srivastava}, booktitle={SODA}, year={2018} }

We show that in any graph, the average length of a flow path in an electrical flow between the endpoints of a random edge is $O(\log^2 n)$. This is a consequence of a more general result which shows that the spectral norm of the entrywise absolute value of the transfer impedance matrix of a graph is $O(\log^2 n)$. This result implies a simple oblivious routing scheme based on electrical flows in the case of transitive graphs.

## 10 Citations

### Fully dynamic spectral vertex sparsifiers and applications

- Computer Science, MathematicsSTOC
- 2019

The key ingredients in these results are the intepretation of Schur complement as a sum of random walks, and a suitable choice of terminals based on the behavior of these random walks to make sure the majority of walks are local, even when the graph itself is highly connected.

### An almost-linear time algorithm for uniform random spanning tree generation

- Mathematics, Computer ScienceSTOC
- 2018

An m1+o(1)βo( 1)-time algorithm for generating uniformly random spanning trees in weighted graphs with max-to-min weight ratio β is given and it is shown that most random walk steps occur far away from an unvisited vertex.

### Robust Routing Using Electrical Flows

- Computer ScienceSIGSPATIAL/GIS
- 2021

This work proposes a novel method to produce alternative routes that is fundamentally different from the aforementioned approaches and borrows concepts from electrical flows and their decompositions, showing that it is as fast as the plateau method while also recovering much of the headroom towards the quality of the penalty method.

### Minor Sparsifiers and the Distributed Laplacian Paradigm

- Computer Science2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

This work studies distributed algorithms built around minor-based vertex sparsifiers, gives the first algorithm in the CONGEST model for solving linear systems in graph Laplacian matrices to high accuracy and presents a nontrivial distributed implementation of their construction.

### Accelerated Distributed Laplacian Solvers via Shortcuts

- Computer ScienceArXiv
- 2021

This work refine the analysis of the distributed Laplacian solver recently established by Forster, Goranci, Liu, Peng, Sun, and Ye (FOCS ’21), via the Ghaffari-Haeupler framework (SODA ’16) of low-congestion shortcuts and considers a hybrid communication model which enhances CONGEST with very limited global power in the form of the recently introduced node-capacitated clique.

### Fully Dynamic Electrical Flows: Sparse Maxflow Faster Than Goldberg-Rao

- Computer Science2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

The algorithm revolves around dynamically maintaining the augmenting electrical flows at the core of the interior point method based algorithm from [Mądry JACM '16]. This entails designing data structures that, in limited settings, return edges with large electric energy in a graph undergoing resistance updates.

### Dynamic Graph Algorithms and Graph Sparsification: New Techniques and Connections

- Computer ScienceArXiv
- 2019

This thesis develops new algorithmic techniques from both dynamic and sparsification perspective for a multitude of graph-based optimization problems which lie at the core of Spectral Graph Theory, Graph Partitioning, and Metric Embeddings and introduces novel reduction techniques that show unexpected connections between seemingly different areas such as dynamic graph algorithms and graph sparsifiers.

### Almost Universally Optimal Distributed Laplacian Solvers via Low-Congestion Shortcuts

- Computer Science
- 2021

A hybrid communication model which enhances CONGEST with limited global power in the form of the node-capacitated clique (NCC) model is considered, and the existence of a Laplacian solver with round complexity no(1) log(1/ε) is shown.

### Spectral Subspace Sparsification

- Computer Science, Mathematics2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
- 2018

A new approach to spectral sparsification that approximates the quadratic form of the pseudoinverse of a graph Laplacian restricted to a subspace and yields sparsifiers that are reweighted minors of the input graph, giving a near-optimal answer to a variant of the Steiner point removal problem.

### Closing the Gap Between Cache-oblivious and Cache-adaptive Analysis

- Computer Science, MathematicsSPAA
- 2020

The gap between cache-oblivious and cache-adaptive analysis is closed by showing how to make a smoothed analysis of cache- Adaptive algorithms via random reshuffling of memory fluctuations, and suggesting that cache- obliviousness is a solid foundation for achieving cache- adaptivity when the memory profile is not overly tailored to the algorithm structure.

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