# Sublinear Update Time Randomized Algorithms for Dynamic Graph Regression

## TL;DR

This paper introduces the first sublinear time randomized algorithms for dynamic graph regression, enabling faster updates in data science applications by leveraging advanced sketching techniques.

## Contribution

It proposes novel sublinear update time algorithms for dynamic graph regression using subsampled randomized Hadamard transform and CountSketch, improving efficiency over existing methods.

## Key findings

- Supports edge insertion and deletion with $O(rd)$ update time
- Supports node operations with $O(qd)$ update time
- Achieves $1 ext{-} ext{approx}$ solutions with sublinear complexity

## Abstract

A well-known problem in data science and machine learning is {\em linear regression}, which is recently extended to dynamic graphs. Existing exact algorithms for updating the solution of dynamic graph regression require at least a linear time (in terms of $n$: the size of the graph). However, this time complexity might be intractable in practice. In the current paper, we utilize {\em subsampled randomized Hadamard transform} and \textsf{CountSketch} to propose the first sublinear update time randomized algorithms for regression of general dynamic graphs. Suppose that we are given a $n\times d$ matrix embedding $\mathbf M$ of the graph, where $d \ll n$ and $\mathbf M$ has certain properties. Let $r$ be the number of samples required by subsampled randomized Hadamard transform for a $1\pm \epsilon$ approximation, which is a sublinear of $n$. Our first algorithm supports edge insertion and edge deletion and updates the approximate solution in $O(rd)$ time. Our second algorithm is based on \textsf{CountSketch} and supports edge insertion, edge deletion, node insertion and node deletion. It updates the approximate solution in $O(qd)$ time, where $q=O\left(\frac{d^2}{\epsilon^2} \log^6(d/\epsilon) \right)$.