The Total Acquisition Number Of The Randomly Weighted Path

Anant Godbole, et al. "The Total Acquisition Number Of The Randomly Weighted Path." Discussiones Mathematicae Graph Theory 37.4 (2017): 919-934. <http://eudml.org/doc/288291>.

@article{AnantGodbole2017,
abstract = {There exists a significant body of work on determining the acquisition number at(G) of various graphs when the vertices of those graphs are each initially assigned a unit weight. We determine properties of the acquisition number of the path, star, complete, complete bipartite, cycle, and wheel graphs for variations on this initial weighting scheme, with the majority of our work focusing on the expected acquisition number of randomly weighted graphs. In particular, we bound the expected acquisition number E(at(Pn)) of the n-path when n distinguishable “units” of integral weight, or chips, are randomly distributed across its vertices between 0.242n and 0.375n. With computer support, we improve it by showing that E(at(Pn)) lies between 0.29523n and 0.29576n. We then use subadditivity to show that the limiting ratio lim E(at(Pn))/n exists, and simulations reveal more exactly what the limiting value equals. The Hoeffding-Azuma inequality is used to prove that the acquisition number is tightly concentrated around its expected value. Additionally, in a different context, we offer a non-optimal acquisition protocol algorithm for the randomly weighted path and exactly compute the expected size of the resultant residual set.},
author = {Anant Godbole, Elizabeth Kelley, Emily Kurtz, Paweł Prałat, Yiguang Zhang},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {total acquisition number; Poissonization; dePoissonization; Maxwell-Boltzman and Bose-Einstein allocation.},
language = {eng},
number = {4},
pages = {919-934},
title = {The Total Acquisition Number Of The Randomly Weighted Path},
url = {http://eudml.org/doc/288291},
volume = {37},
year = {2017},
}

TY - JOUR
AU - Anant Godbole
AU - Elizabeth Kelley
AU - Emily Kurtz
AU - Paweł Prałat
AU - Yiguang Zhang
TI - The Total Acquisition Number Of The Randomly Weighted Path
JO - Discussiones Mathematicae Graph Theory
PY - 2017
VL - 37
IS - 4
SP - 919
EP - 934
AB - There exists a significant body of work on determining the acquisition number at(G) of various graphs when the vertices of those graphs are each initially assigned a unit weight. We determine properties of the acquisition number of the path, star, complete, complete bipartite, cycle, and wheel graphs for variations on this initial weighting scheme, with the majority of our work focusing on the expected acquisition number of randomly weighted graphs. In particular, we bound the expected acquisition number E(at(Pn)) of the n-path when n distinguishable “units” of integral weight, or chips, are randomly distributed across its vertices between 0.242n and 0.375n. With computer support, we improve it by showing that E(at(Pn)) lies between 0.29523n and 0.29576n. We then use subadditivity to show that the limiting ratio lim E(at(Pn))/n exists, and simulations reveal more exactly what the limiting value equals. The Hoeffding-Azuma inequality is used to prove that the acquisition number is tightly concentrated around its expected value. Additionally, in a different context, we offer a non-optimal acquisition protocol algorithm for the randomly weighted path and exactly compute the expected size of the resultant residual set.
LA - eng
KW - total acquisition number; Poissonization; dePoissonization; Maxwell-Boltzman and Bose-Einstein allocation.
UR - http://eudml.org/doc/288291
ER -