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Dynamic Maintenance of Monotone Dynamic Programs and Applications

Authors Monika Henzinger , Stefan Neumann , Harald Räcke , Stefan Schmid

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Author Details

Monika Henzinger
  • Institute of Science and Technology Austria, Klosterneuburg, Austria
Stefan Neumann
  • KTH Royal Institute of Technology, Stockholm, Sweden
Harald Räcke
  • TU München, Germany
Stefan Schmid
  • TU Berlin, Germany
  • Fraunhofer SIT, Darmstadt, Germany

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Monika Henzinger, Stefan Neumann, Harald Räcke, and Stefan Schmid. Dynamic Maintenance of Monotone Dynamic Programs and Applications. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 36:1-36:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


Dynamic programming (DP) is one of the fundamental paradigms in algorithm design. However, many DP algorithms have to fill in large DP tables, represented by two-dimensional arrays, which causes at least quadratic running times and space usages. This has led to the development of improved algorithms for special cases when the DPs satisfy additional properties like, e.g., the Monge property or total monotonicity. In this paper, we consider a new condition which assumes (among some other technical assumptions) that the rows of the DP table are monotone. Under this assumption, we introduce a novel data structure for computing (1+ε)-approximate DP solutions in near-linear time and space in the static setting, and with polylogarithmic update times when the DP entries change dynamically. To the best of our knowledge, our new condition is incomparable to previous conditions and is the first which allows to derive dynamic algorithms based on existing DPs. Instead of using two-dimensional arrays to store the DP tables, we store the rows of the DP tables using monotone piecewise constant functions. This allows us to store length-n DP table rows with entries in [0,W] using only polylog(n,W) bits, and to perform operations, such as (min,+)-convolution or rounding, on these functions in polylogarithmic time. We further present several applications of our data structure. For bicriteria versions of k-balanced graph partitioning and simultaneous source location, we obtain the first dynamic algorithms with subpolynomial update times, as well as the first static algorithms using only near-linear time and space. Additionally, we obtain the currently fastest algorithm for fully dynamic knapsack.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic programming
  • Theory of computation → Dynamic graph algorithms
  • Theory of computation → Packing and covering problems
  • Dynamic programming
  • dynamic algorithms
  • data structures


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