Totally Greedy Sequences Defined by Second-Order Linear Recurrences With Constant Coefficients

TL;DR

This paper investigates totally greedy sequences generated by second-order linear recurrences, establishing conditions under which these sequences ensure the greedy algorithm yields optimal coin representations for all prefixes.

Contribution

It introduces the concept of totally greedy sequences derived from second-order recurrences and provides sufficient conditions for their total greediness.

Findings

Identifies conditions for sequences to be totally greedy

Analyzes subsequences of second-order linear recurrence sequences

Provides proofs for the total greediness of specific sequences

Abstract

The change-making problem consists of representing a certain amount of money with the least possible number of coins, from a given, pre-established set of denominations. The greedy algorithm works by choosing the coins of largest possible denomination first. This greedy strategy does not always produce the least number of coins, except when the set of denominations obeys certain properties. We call a set of denominations with these properties a greedy set. If the set of denominations is an infinite sequence, we call it totally greedy if every prefix subset is greedy. In this paper we investigate some totally greedy sequences arising from second-order linear recurrences with constant coefficients, as well as their subsequences, and we prove sufficient conditions under which these sequences are totally greedy.