License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2021.67
URN: urn:nbn:de:0030-drops-155008
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/15500/
Tan, Shuhao
Computing Shapley Values for Mean Width in 3-D
Abstract
The Shapley value is a classical concept from game theory, which is used to evaluate the importance of a player in a cooperative setting. Assuming that players are inserted in a uniformly random order, the Shapley value of a player p is the expected increase in the value of the characteristic function when p is inserted. Cabello and Chan (SoCG 2019) recently showed how to adapt this to a geometric context on planar point sets. For example, when the characteristic function is the area of the convex hull, the Shapley value of a point is the average amount by which the convex-hull area increases when this point is added to the set. Shapley values can be viewed as an indication of the relative importance/impact of a point on the function of interest.
In this paper, we present an efficient algorithm for computing Shapley values in 3-dimensional space, where the function of interest is the mean width of the point set. Our algorithm runs in O(n³log²n) time and O(n) space. This result can be generalized to any point set in d-dimensional space (d ≥ 3) to compute the Shapley values for the mean volume of the convex hull projected onto a uniformly random (d - 2)-subspace in O(n^d log²n) time and O(n) space. These results are based on a new data structure for a dynamic variant of the convolution problem, which is of independent interest. Our data structure supports incremental modifications to n-element vectors (including cyclical rotation by one position). We show that n operations can be executed in O(n log²n) time and O(n) space.
BibTeX - Entry
@InProceedings{tan:LIPIcs.ISAAC.2021.67,
author = {Tan, Shuhao},
title = {{Computing Shapley Values for Mean Width in 3-D}},
booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
pages = {67:1--67:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-214-3},
ISSN = {1868-8969},
year = {2021},
volume = {212},
editor = {Ahn, Hee-Kap and Sadakane, Kunihiko},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/15500},
URN = {urn:nbn:de:0030-drops-155008},
doi = {10.4230/LIPIcs.ISAAC.2021.67},
annote = {Keywords: Shapley value, mean width, dynamic convolution}
}
|
Keywords: |
|
Shapley value, mean width, dynamic convolution |
|
Collection: |
|
32nd International Symposium on Algorithms and Computation (ISAAC 2021) |
|
Issue Date: |
|
2021 |
|
Date of publication: |
|
30.11.2021 |
