On the Complexity of Computing Time Series Medians Under the Move-Split-Merge Metric

Authors Jana Holznigenkemper, Christian Komusiewicz , Nils Morawietz, Bernhard Seeger

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Author Details

Jana Holznigenkemper
  • Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany
Christian Komusiewicz
  • Institute of Computer Science, Friedrich-Schiller-Universität Jena, Germany
Nils Morawietz
  • Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany
Bernhard Seeger
  • Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany

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Jana Holznigenkemper, Christian Komusiewicz, Nils Morawietz, and Bernhard Seeger. On the Complexity of Computing Time Series Medians Under the Move-Split-Merge Metric. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 54:1-54:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We initiate a study of the complexity of MSM-Median, the problem of computing a median of a set of k real-valued time series under the move-split-merge distance. This distance measure is based on three operations: moves, which may shift a data point in a time series; splits, which replace one data point in a time series by two consecutive data points of the same value; and merges, which replace two consecutive data points of equal value by a single data point of the same value. The cost of a move operation is the difference of the data point value before and after the operation, the cost of split and merge operations is defined via a given constant c. Our main results are as follows. First, we show that MSM-Median is NP-hard and W[1]-hard with respect to k for time series with at most three distinct values. Under the Exponential Time Hypothesis (ETH) our reduction implies that a previous dynamic programming algorithm with running time |I|^𝒪(k) [Holznigenkemper et al., Data Min. Knowl. Discov. '23] is essentially optimal. Here, |I| denotes the total input size. Second, we show that MSM-Median can be solved in 2^𝒪(d/c)⋅|I|^𝒪(1) time where d is the total distance of the median to the input time series.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Time series analysis
  • Parameterized Complexity
  • Median String
  • Time Series
  • ETH


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