Pachinko
Name
1601.05706.pdf
Description
Submitted version
Size
1.51 MB
Format
Adobe PDF
Checksum (MD5)
5d14b09ca8a84a642ad29ba8fc5d9f0f
Author(s)
Akitaya, Hugo A.
Demaine, Erik D
Demaine, Martin L
Hesterberg, Adam Classen
Hurtado, Ferran
Ku, Jason S
Lynch, Jayson R.
Date Issued
July 2017
Journal
Computational Geometry
Publisher
Elsevier BV
Citation
Akitaya, Hugo A. et al. "Pachinko." Computational Geometry 68 (March 2018): 226-242 © 2017 Elsevier B.V.
Version
Original manuscript
Abstract
Inspired by the Japanese game Pachinko, we study simple (perfectly “inelastic” collisions) dynamics of a unit ball falling amidst point obstacles (pins) in the plane. A classic example is that a checkerboard grid of pins produces the binomial distribution, but what probability distributions result from different pin placements? In the 50–50 model, where the pins form a subset of this grid, not all probability distributions are possible, but surprisingly the uniform distribution is possible for {1,2,4,8,16} possible drop locations. Furthermore, every probability distribution can be approximated arbitrarily closely, and every dyadic probability distribution can be divided by a suitable power of 2 and then constructed exactly (along with extra “junk” outputs). In a more general model, if a ball hits a pin off center, it falls left or right accordingly. Then we prove a universality result: any distribution of n dyadic probabilities, each specified by k bits, can be constructed using O(nk2) pins, which is close to the information-theoretic lower bound of Ω(nk).
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1016/j.comgeo.2017.06.011
