The Money Changing Problem Revisited: Computing the Frobenius Number in Time O(ka ₁)

Abstract

The Money Changing Problem (also known as Equality Constrained Integer Knapsack Problem) is as follows: Let a ₁ < a ₂ < ... < a _k be fixed positive integers with $\gcd(a_1, \dots, a_k) = 1$. Given some integer n, are there non-negative integers x ₁, ..., x _k such that ∑ _i a _i x _i = n? The Frobenius numberg(a ₁, ..., a _k ) is the largest integer n that has no decomposition of the above form.

There exist algorithms that, for fixed k, compute the Frobenius number in time polynomial in log a _k . For variable k, one can compute a residue table of a ₁ words which, in turn, allows to determine the Frobenius number. The best known algorithm for computing the residue table has runtime O(ka ₁ log a ₁) using binary heaps, and O(a ₁ (k+log a ₁)) using Fibonacci heaps. In both cases, O(a ₁) extra memory in addition to the residue table is needed. Here, we present an intriguingly simple algorithm to compute the residue table in time O(ka ₁) and extra memory O(1). In addition to computing the Frobenius number, we can use the residue table to solve the given instance of the Money Changing Problem in constant time, for any n.

Supported by “Deutsche Forschungsgemeinschaft” (BO 1910/1-1 and 1-2) within the Computer Science Action Program