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More on Change-Making and Related Problems

Authors Timothy M. Chan , Qizheng He

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Author Details

Timothy M. Chan
  • Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, USA
Qizheng He
  • Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, USA

Funding

  • Chan, Timothy M.: Supported in part by NSF Grant CCF-1814026.

Acknowledgements

We thank Adam Polak and Chao Xu for discussion and, in particular, for bringing the minimum word break problem to our attention.

Cite As Get BibTex

Timothy M. Chan and Qizheng He. More on Change-Making and Related Problems. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ESA.2020.29

Abstract

Given a set of n integer-valued coin types and a target value t, the well-known change-making problem asks for the minimum number of coins that sum to t, assuming an unlimited number of coins in each type. In the more general all-targets version of the problem, we want the minimum number of coins summing to j, for every j = 0,…,t. For example, the textbook dynamic programming algorithms can solve the all-targets problem in O(nt) time. Recently, Chan and He (SOSA'20) described a number of O(t polylog t)-time algorithms for the original (single-target) version of the change-making problem, but not the all-targets version. 
In this paper, we obtain a number of new results on change-making and related problems:  
- We present a new algorithm for the all-targets change-making problem with running time Õ(t^{4/3}), improving a previous Õ(t^{3/2})-time algorithm. 
- We present a very simple Õ(u²+t)-time algorithm for the all-targets change-making problem, where u denotes the maximum coin value. The analysis of the algorithm uses a theorem of Erdős and Graham (1972) on the Frobenius problem. This algorithm can be extended to solve the all-capacities version of the unbounded knapsack problem (for integer item weights bounded by u). 
- For the original (single-target) coin changing problem, we describe a simple modification of one of Chan and He’s algorithms that runs in Õ(u) time (instead of Õ(t)). 
- For the original (single-capacity) unbounded knapsack problem, we describe a simple algorithm that runs in Õ(nu) time, improving previous near-u²-time algorithms. 
- We also observe how one of our ideas implies a new result on the minimum word break problem, an optimization version of a string problem studied by Bringmann et al. (FOCS'17), generalizing change-making (which corresponds to the unary special case).

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Coin changing
  • knapsack
  • dynamic programming
  • Frobenius problem
  • fine-grained complexity

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