# Clustering Permutations: New Techniques with Streaming Applications

arXiv: https://arxiv.org/abs/2212.01821 · HTML: https://mlsanity.com/paper/2212.01821/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2022.

## Built on by (most cited first, 2 of 2)

- [Universal Weak Coreset](https://mlsanity.com/paper/2305.16890/) (2023)
- [A Generic Framework for Fair Consensus Clustering in Streams](https://mlsanity.com/paper/2602.11500/) (2026)

## Abstract (arXiv)

We study the classical metric k-median clustering problem over a set of input rankings (i.e., permutations), which has myriad applications, from social-choice theory to web search and databases. A folklore algorithm provides a 2-approximate solution in polynomial time for all k=O(1), and works irrespective of the underlying distance measure, so long it is a metric; however, going below the 2-factor is a notorious challenge. We consider the Ulam distance, a variant of the well-known edit-distance metric, where strings are restricted to be permutations. For this metric, Chakraborty, Das, and Krauthgamer [SODA, 2021] provided a (2-δ)-approximation algorithm for k=1, where δ≈ 2-40. Our primary contribution is a new algorithmic framework for clustering a set of permutations. Our first result is a 1.999-approximation algorithm for the metric k-median problem under the Ulam metric, that runs in time (k log (nd))O(k)n d3 for an input consisting of n permutations over [d]. In fact, our framework is powerful enough to extend this result to the streaming model (where the n input permutations arrive one by one) using only polylogarithmic (in n) space. Additionally, we show that similar results can be obtained even in the presence of outliers, which is presumably a more difficult problem.