TY - GEN

T1 - Approximating the median under the Ulam metric

AU - Chakraborty, Diptarka

AU - Das, Debarati

AU - Krauthgamer, Robert

N1 - Funding Information:
Work partially supported by NUS ODPRT Grant, WBS No. R-252-000-A94-133.
Publisher Copyright:
Copyright © 2021 by SIAM

PY - 2021

Y1 - 2021

N2 - 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).

AB - 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).

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M3 - Conference contribution

AN - SCOPUS:85105271417

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 761

EP - 775

BT - ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

A2 - Marx, Daniel

PB - Association for Computing Machinery

T2 - 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2021

Y2 - 10 January 2021 through 13 January 2021

ER -