Localization of Electrical Flows

  title={Localization of Electrical Flows},
  author={Aaron Schild and Satish Rao and Nikhil Srivastava},
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. 

Fully dynamic spectral vertex sparsifiers and applications

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

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

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

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

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

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

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

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

  • Huan LiAaron Schild
  • Computer Science, Mathematics
    2018 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

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.



Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs

This work introduces a new approach to computing an approximately maximum s-t flow in a capacitated, undirected graph and develops the fastest known algorithm for computing approximately maximums-t flows.

A new approach to computing maximum flows using electrical flows

An algorithm which computes a (1-ε)-approximately maximum st-flow in an undirected uncapacitated graph in time O(1/ε√m/F⋅ m log2 n) where F is the flow value and the minimizer is related to an approximate blocking flow is shown.

Electric routing and concurrent flow cutting

Faster Generation of Random Spanning Trees

A new approach to the problem that integrates discrete random walk-based techniques with continuous linear algebraic methods is introduced, and the use of electrical networks and sparse linear system solvers in conjunction with random walks and combinatorial partitioning techniques is used.

Minimizing Congestion in General Networks

This work introduces a framework for solving online problems that aim to minimize the congestion in general topology networks and achieves a competitive ratio of O(log/sup 3/ n) with respect to the congestion of the network links.

Minimizing average latency in oblivious routing

This work considers the problem of minimizing average latency cost while obliviously routing traffic in a network with linear latency functions and shows that for the case when all routing requests are directed to a single target, there is a routing scheme with competitive ratio O(log n), where n denotes the number of nodes in the network.

Mixing times and e p bounds for oblivious routing

We study the task of uniformly minimizing all the ep norms of the vector of edge loads in an undirected graph while obliviously routing a multicommodity flow. Let G be an undirected graph having m

Fast Generation of Random Spanning Trees and the Effective Resistance Metric

A new algorithm is presented for generating a uniformly random spanning tree in an undirected graph that establishes a new connection between the effective resistance metric and the cut structure of the underlying graph.

Faster approximate multicommodity flow using quadratically coupled flows

This paper gives algorithms that find 1-ε approximate solutions to the maximum concurrent flow problem and maximum weighted multicommodity flow problem in time O(m4/3poly(k,ε-1)).

Sampling random spanning trees faster than matrix multiplication

An algorithm is presented that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in (n5/3 m1/3) time, based on Gaussian elimination, and the fact that effective resistance is preserved in the graph resulting from eliminating a subset of vertices (called a Schur complement).