IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v192y2025ics0960077925000025.html
   My bibliography  Save this article

Fractal information density

Author

Listed:
  • Ratsaby, Joel

Abstract

Fractal sets are generated by simple generating formulas (iterated functions) and therefore have an almost zero algorithmic (Kolmogorov) complexity. Yet when observed as data with no knowledge of the iterated function, for instance, when observing pixel values of any region of a fractal image, the fractal set is very complex. It has rich and complicated patterns that appear at any arbitrary level of magnification. This suggests that fractal sets have a rich information content despite their essentially zero algorithmic complexity. This highlights a significant gap between algorithmic complexity of sets and their information richness. To explain this, we propose an information-based complexity measure of fractal sets. We extend a well-known notion of compression ratio of general binary sequences to two-dimensional sets and apply it to fractal sets. We introduce a notion of set information density and boundary information density, and as an application, we estimate them for two well-known fractal sets.

Suggested Citation

  • Ratsaby, Joel, 2025. "Fractal information density," Chaos, Solitons & Fractals, Elsevier, vol. 192(C).
    • Handle: RePEc:eee:chsofr:v:192:y:2025:i:c:s0960077925000025
    DOI: 10.1016/j.chaos.2025.115989
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077925000025
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2025.115989?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. V. Chvatal, 1979. "A Greedy Heuristic for the Set-Covering Problem," Mathematics of Operations Research, INFORMS, vol. 4(3), pages 233-235, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Davidov, Sreten & Pantoš, Miloš, 2017. "Planning of electric vehicle infrastructure based on charging reliability and quality of service," Energy, Elsevier, vol. 118(C), pages 1156-1167.
    2. Song, Zhe & Kusiak, Andrew, 2010. "Mining Pareto-optimal modules for delayed product differentiation," European Journal of Operational Research, Elsevier, vol. 201(1), pages 123-128, February.
    3. Seona Lee & Sang-Ho Lee & HyungJune Lee, 2020. "Timely directional data delivery to multiple destinations through relay population control in vehicular ad hoc network," International Journal of Distributed Sensor Networks, , vol. 16(5), pages 15501477209, May.
    4. Zhuang, Yanling & Zhou, Yun & Yuan, Yufei & Hu, Xiangpei & Hassini, Elkafi, 2022. "Order picking optimization with rack-moving mobile robots and multiple workstations," European Journal of Operational Research, Elsevier, vol. 300(2), pages 527-544.
    5. Menghong Li & Yingli Ran & Zhao Zhang, 2022. "A primal-dual algorithm for the minimum power partial cover problem," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1913-1923, October.
    6. Wang, Yiyuan & Pan, Shiwei & Al-Shihabi, Sameh & Zhou, Junping & Yang, Nan & Yin, Minghao, 2021. "An improved configuration checking-based algorithm for the unicost set covering problem," European Journal of Operational Research, Elsevier, vol. 294(2), pages 476-491.
    7. C Guéret & N Jussien & O Lhomme & C Pavageau & C Prins, 2003. "Loading aircraft for military operations," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 54(5), pages 458-465, May.
    8. Keisuke Murakami, 2018. "Iterative Column Generation Algorithm for Generalized Multi-Vehicle Covering Tour Problem," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 35(04), pages 1-22, August.
    9. Dongyue Liang & Zhao Zhang & Xianliang Liu & Wei Wang & Yaolin Jiang, 2016. "Approximation algorithms for minimum weight partial connected set cover problem," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 696-712, February.
    10. Abdullah Alshehri & Mahmoud Owais & Jayadev Gyani & Mishal H. Aljarbou & Saleh Alsulamy, 2023. "Residual Neural Networks for Origin–Destination Trip Matrix Estimation from Traffic Sensor Information," Sustainability, MDPI, vol. 15(13), pages 1-21, June.
    11. Wedelin, Dag, 1995. "The design of a 0-1 integer optimizer and its application in the Carmen system," European Journal of Operational Research, Elsevier, vol. 87(3), pages 722-730, December.
    12. Victor Reyes & Ignacio Araya, 2021. "A GRASP-based scheme for the set covering problem," Operational Research, Springer, vol. 21(4), pages 2391-2408, December.
    13. Owais, Mahmoud & Moussa, Ghada S. & Hussain, Khaled F., 2019. "Sensor location model for O/D estimation: Multi-criteria meta-heuristics approach," Operations Research Perspectives, Elsevier, vol. 6(C).
    14. Dan Garber, 2021. "Efficient Online Linear Optimization with Approximation Algorithms," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 204-220, February.
    15. Manki Min & Oleg Prokopyev & Panos M. Pardalos, 2006. "Optimal solutions to minimum total energy broadcasting problem in wireless ad hoc networks," Journal of Combinatorial Optimization, Springer, vol. 11(1), pages 59-69, February.
    16. Manki Min & Panos M. Pardalos, 2007. "Total energy optimal multicasting in wireless ad hoc networks," Journal of Combinatorial Optimization, Springer, vol. 13(4), pages 365-378, May.
    17. Qi Wang & Bo Hou & Gengsheng Zhang & Yisheng Zhou & Wen Liu, 2025. "Approximation algorithms for the partition set cover problem with penalties," Journal of Combinatorial Optimization, Springer, vol. 49(5), pages 1-15, July.
    18. Davidov, Sreten & Pantoš, Miloš, 2017. "Stochastic expansion planning of the electric-drive vehicle charging infrastructure," Energy, Elsevier, vol. 141(C), pages 189-201.
    19. Vazifeh, Mohammad M. & Zhang, Hongmou & Santi, Paolo & Ratti, Carlo, 2019. "Optimizing the deployment of electric vehicle charging stations using pervasive mobility data," Transportation Research Part A: Policy and Practice, Elsevier, vol. 121(C), pages 75-91.
    20. Dimitri Watel & Marc-Antoine Weisser, 2016. "A practical greedy approximation for the directed Steiner tree problem," Journal of Combinatorial Optimization, Springer, vol. 32(4), pages 1327-1370, November.

    More about this item

    Keywords

    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:192:y:2025:i:c:s0960077925000025. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.