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Clustering Permutations: New Techniques with Streaming Applications

Authors Diptarka Chakraborty, Debarati Das, Robert Krauthgamer

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Author Details

Diptarka Chakraborty
  • National University of Singapore, Singapore
Debarati Das
  • Pennsylvania State University, University Park, PA, USA
Robert Krauthgamer
  • Weizmann Institute of Science, Rehovot, Israel

Funding

  • Chakraborty, Diptarka: Work partially supported by an MoE AcRF Tier 2 grant (WBS No. A-8000416-00-00) and an NUS ODPRT grant (WBS No. A-0008078-00-00).
  • Krauthgamer, Robert: Work partially supported by ONR Award N00014-18-1-2364, the Israel Science Foundation grant #1086/18, and a Minerva Foundation grant, and by the Israeli Council for Higher Education (CHE) via the Weizmann Data Science Research Center.

Cite As Get BibTex

Diptarka Chakraborty, Debarati Das, and Robert Krauthgamer. Clustering Permutations: New Techniques with Streaming Applications. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 31:1-31:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ITCS.2023.31

Abstract

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)} nd³ 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Clustering
  • Approximation Algorithms
  • Ulam Distance
  • Rank Aggregation
  • Streaming

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