Approximating the median under the Ulam metric

Diptarka Chakraborty, Debarati Das, Robert Krauthgamer

Research output: Chapter in Book/Report/Conference proceedingConference contribution

8 Scopus citations


We study approximation algorithms for variants of the median string problem, which asks for a string that minimizes the sum of edit distances from a given set of m strings of length n. Only the straightforward 2-approximation is known for this NP-hard problem. This problem is motivated e.g. by computational biology, and belongs to the class of median problems (over different metric spaces), which are fundamental tasks in data analysis. Our main result is for the Ulam metric, where all strings are permutations over [n] and each edit operation moves a symbol (deletion plus insertion). We devise for this problem an algorithms that breaks the 2-approximation barrier, i.e., computes a (2 − δ)-approximate median permutation for some constant δ > 0 in time Õ(nm2 + n3). We further use these techniques to achieve a (2 − δ) approximation for the median string problem in the special case where the median is restricted to length n and the optimal objective is large Ω(mn). We also design an approximation algorithm for the following probabilistic model of the Ulam median: the input consists of m perturbations of an (unknown) permutation x, each generated by moving every symbol to a random position with probability (a parameter) ε > 0. Our algorithm computes with high probability a (1 + o(1/ε))-approximate median permutation in time O(mn2 + n3).

Original languageEnglish (US)
Title of host publicationACM-SIAM Symposium on Discrete Algorithms, SODA 2021
EditorsDaniel Marx
PublisherAssociation for Computing Machinery
Number of pages15
ISBN (Electronic)9781611976465
StatePublished - 2021
Event32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021 - Alexandria, Virtual, United States
Duration: Jan 10 2021Jan 13 2021

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms


Conference32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021
Country/TerritoryUnited States
CityAlexandria, Virtual

All Science Journal Classification (ASJC) codes

  • Software
  • General Mathematics


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