2018/08/09 by Shoichi Kamada, Kamada, Shoichi
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Fractal and DNA sequence analysis #Mathematical Dynamics and Fractals #math.CO
paper · pdf · doi:10.48550/arxiv.1808.03033
There is no worth in this manuscript
openalex publication_date 2018/08/09 · arxiv created 2020/08/21 · arxiv updated 2020/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce two frameworks in order to deal with fractal and multi-fractal analysis for subset sum problems where some embedding into the 1-dimensional Euclidean space plays an important role. As one of these frameworks, the notion of the combinatorial q-fractal dimension for a subset sum function is introduced. Thereby, ``non-classical'' generalized dimensions for a family of subset~sum functions can be defined. These generalized dimensions include the box-counting dimension, the information dimension and the correlation dimension as well as the classical case. The combinatorial q-fractal dimension includes the density of the subset sum problem. As the other framework, we construct a self-similar set for a particular subset sum function in a family of subset sum functions by using a graph theoretical technique. In this paper, we give a lower bound for a combinatorial q-fractal dimension and we show the relations between the three parameters: the number of connected components in a graph, the Hausdorff dimension and a combinatorial q-fractal dimension.