The Fourth-Root Complexity of Data Movement

2026-06-29

AI Summary

This work challenges the conventional assumption in time complexity analysis that ignores the impact of data movement costs in memory hierarchies as problem size scales. By leveraging an abstract memory hierarchy model and combining cache miss rate analysis with asymptotic complexity derivation, the study reveals for the first time that the data access cost for a broad class of common applications grows proportionally to \(N^{1/4}\) with input size \(N\). Furthermore, it distinguishes between scenarios where cache miss rates decay according to a power law versus an exponential law, and quantifies the resulting differences in scalability through constant factors. This refined characterization provides a more accurate performance prediction framework to guide algorithm design in modern memory-constrained architectures.

Abstract

Time complexity typically assumes $O(1)$ cost per data access. This paper presents an analysis based on an abstract memory hierarchy. For a common class of applications, it shows that the data-access cost scales with the fourth root of data size, that is, as data size $N$ increases, the cost of each access increases at the rate of $N^\frac{1}{4}$. While the analysis does not predict performance, it predicts scalability. Specifically, the paper provides a precise analysis that shows the constant-factor difference between cases where the miss ratio follows a power law versus an exponential decay.

Problem

• data movement
• time complexity
• memory hierarchy
• scalability
• access cost

Innovation

• fourth-root complexity
• data movement
• memory hierarchy
• scalability analysis
• miss ratio