O-notation in algorithm analysis

TL;DR

This paper rigorously characterizes the O-notation used in algorithm analysis by establishing its properties, equivalence to linear dominance, and its behavior under various manipulations and theorems.

Contribution

It introduces a formal framework for O-notation based on primitive properties, unifies existing definitions, and extends its applicability through O-mappings and master theorems.

Findings

O-notation is equivalent to linear dominance

Characterization of O-notation via limits over filters

Master theorems hold under linear dominance

Abstract

We provide an extensive list of desirable properties for an O-notation --- as used in algorithm analysis --- and reduce them to 8 primitive properties. We prove that the primitive properties are equivalent to the definition of the O-notation as linear dominance. We abstract the existing definitions of the O-notation under local linear dominance, and show that it has a characterization by limits over filters for positive functions. We define the O-mappings as a general tool for manipulating the O-notation, and show that Master theorems hold under linear dominance.