Theoretical formulation and analysis of the deterministic dendritic cell algorithm

TL;DR

This paper provides a formal mathematical definition and runtime analysis of the deterministic Dendritic Cell Algorithm (dDCA), addressing previous ambiguities and exploring its computational complexity and behavior.

Contribution

It introduces a formal set-theoretic definition of the dDCA and analyzes its runtime complexity, including effects of segmentation, which were lacking in prior empirical studies.

Findings

Standard dDCA has worst-case runtime O(n^2)

Segmentation alters runtime to O(max(nN, nz))

Two runtime variables are formulated for further analysis

Abstract

As one of the emerging algorithms in the field of Artificial Immune Systems (AIS), the Dendritic Cell Algorithm (DCA) has been successfully applied to a number of challenging real-world problems. However, one criticism is the lack of a formal definition, which could result in ambiguity for understanding the algorithm. Moreover, previous investigations have mainly focused on its empirical aspects. Therefore, it is necessary to provide a formal definition of the algorithm, as well as to perform runtime analyses to revealits theoretical aspects. In this paper, we define the deterministic version of the DCA, named the dDCA, using set theory and mathematical functions. Runtime analyses of the standard algorithm and the one with additional segmentation are performed. Our analysis suggests that the standard dDCA has a runtime complexity of O(n2) for the worst-case scenario, where n is the number of input data instances. The introduction of segmentation changes the algorithm's worst case runtime complexity to O(max(nN; nz)), for DC population size N with size of each segment z. Finally, two runtime variables of the algorithm are formulated based on the input data, to understand its runtime behaviour as guidelines for further development.