eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-26
29:1
29:14
10.4230/LIPIcs.ESA.2020.29
article
More on Change-Making and Related Problems
Chan, Timothy M.
1
https://orcid.org/0000-0002-8093-0675
He, Qizheng
1
Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, USA
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).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol173-esa2020/LIPIcs.ESA.2020.29/LIPIcs.ESA.2020.29.pdf
Coin changing
knapsack
dynamic programming
Frobenius problem
fine-grained complexity