# A Non-iterative Parallelizable Eigenbasis Algorithm for Johnson Graphs

## TL;DR

This paper introduces a non-iterative, highly parallelizable algorithm for efficiently generating eigenbases of Johnson graphs, significantly reducing computation time with exact results under ideal precision.

## Contribution

It presents the first non-iterative, parallelizable method for eigenbasis computation of Johnson graphs, achieving near-linear parallel time complexity.

## Key findings

- Algorithm runs in $O(n)$ parallel time with unlimited processors.
- Produces exact eigenbases in finite-precision models.
- Enables efficient projection computations onto eigenspaces.

## Abstract

We present a new $O(k^2 \binom{n}{k}^2)$ method for generating an orthogonal basis of eigenvectors for the Johnson graph $J(n,k)$. Unlike standard methods for computing a full eigenbasis of sparse symmetric matrices, the algorithm presented here is non-iterative, and produces exact results under an infinite-precision computation model. In addition, our method is highly parallelizable; given access to unlimited parallel processors, the eigenbasis can be constructed in only $O(n)$ time given n and k. We also present an algorithm for computing projections onto the eigenspaces of $J(n,k)$ in parallel time $O(n)$.