# Dynamic Streaming Spectral Sparsification in Nearly Linear Time and Space

@article{Kapralov2019DynamicSS, title={Dynamic Streaming Spectral Sparsification in Nearly Linear Time and Space}, author={Mikhail Kapralov and Navid Nouri and Aaron Sidford and Jakab Tardos}, journal={ArXiv}, year={2019}, volume={abs/1903.12150} }

In this paper we consider the problem of computing spectral approximations to graphs in the single pass dynamic streaming model. We provide a linear sketching based solution that given a stream of edge insertions and deletions to a $n$-node undirected graph, uses $\tilde O(n)$ space, processes each update in $\tilde O(1)$ time, and with high probability recovers a spectral sparsifier in $\tilde O(n)$ time. Prior to our work, state of the art results either used near optimal $\tilde O(n)$ space…

## 15 Citations

### Faster Spectral Sparsification in Dynamic Streams

- Computer ScienceArXiv
- 2019

A novel method for `bucketing' vertices of the input graph into clusters that allows fast recovery of edges of high effective resistance: the buckets are formed by performing ball-carving on the inputgraph using (an approximation to) its effective resistance metric.

### Graph Spanners by Sketching in Dynamic Streams and the Simultaneous Communication Model

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This paper introduces several new graph sketching techniques for approximating the shortest path metric of the input graph, and gives new spanner construction algorithms for any number of passes, simultaneously improving upon all prior work on this problem.

### Online Spectral Approximation in Random Order Streams

- Computer Science, MathematicsArXiv
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An online algorithm is proposed that avoids an additive error with the same time and space complexities as the algorithm of Cohen et al., and provides a better upper bound on the approximation size when a given matrix has small rank.

### On Constructing Spanners from Random Gaussian Projections

- Computer ScienceArXiv
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A strong lower bound is proved for a large family of graph sketching algorithms that encompasses prior work on spanners and many (but importantly not also all) related cut-based problems mentioned above.

### Breaking the n-Pass Barrier: A Streaming Algorithm for Maximum Weight Bipartite Matching

- Computer Science, MathematicsArXiv
- 2020

This is the first work that implements the SDD solver and IPM in the streaming model in $\widetilde{O}(n)$ spaces for graph matrix and combines a number of techniques from different fields such as the interior point method, symmetric diagonally dominant system solving, the isolation lemma, and LP duality.

### Sparsification of Balanced Directed Graphs

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- 2020

The upper bounds work for general weighted graphs, while the lower bounds even hold for unweighted graphs with no parallel edges.

### Space-Efficient Interior Point Method, with applications to Linear Programming and Maximum Weight Bipartite Matching

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To the best of the knowledge, this is the first LP solver in streaming that has no space dependence on m, and a space-efficient interior point method that optimizes solely on the dual program is developed.

### Solving tall dense linear programs in nearly linear time

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- 2020

Interestingly, the running time of this robust, primal-dual O(√d)-iteration interior point method is near optimal for solving dense linear programs among algorithms that do not use fast matrix multiplication.

### Sparsification of Directed Graphs via Cut Balance

- Computer ScienceICALP
- 2021

An interesting application of digraph sparsification via cut balance is shown by using it to give a very short proof of a celebrated maximum flow result of Karger and Levine (STOC 2002).

### Graph Spanners in the Message-Passing Model

- Computer Science, MathematicsITCS
- 2020

This work presents the first tradeoffs for total communication versus the quality of the spanners computed, for two or more sites, as well as for additive and multiplicative notions of distortion.

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