We consider metrical task systems on tree metrics and present an $O(\mathrm{depth} \times \log n)$-competitive randomized algorithm based on the mirror descent framework introduced in our prior work on the $k$-server problem. For the special case of hierarchically separated trees (HSTs), we use mirror descent to refine the standard approach based on gluing unfair metrical task systems. This yields an $O(\log n)$-competitive algorithm for HSTs, thus removing an extraneous $\log\log n$ in the bound of Fiat and Mendel (2003). Combined with well-known HST embedding theorems, this also gives an $O((\log n)^2)$-competitive randomized algorithm for every $n$-point metric space.


  1. online algorithms
  2. convex optimization
  3. finite metric space

MSC codes

  1. 68W27

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Information & Authors


Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 909 - 923
ISSN (online): 1095-7111


Submitted: 9 January 2019
Accepted: 16 February 2021
Published online: 27 May 2021


  1. online algorithms
  2. convex optimization
  3. finite metric space

MSC codes

  1. 68W27



Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : CCF-1616297, CCF-1407779
National Science Foundation https://doi.org/10.13039/100000001 : CCF-1740551, CCF-1749609
Simons Foundation https://doi.org/10.13039/100000893

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