Cabello, Sergio ;
Chan, Timothy M.
Computing Shapley Values in the Plane
Abstract
We consider the problem of computing Shapley values for points in the plane, where each point is interpreted as a player, and the value of a coalition is defined by the area of usual geometric objects, such as the convex hull or the minimum axisparallel bounding box.
For sets of n points in the plane, we show how to compute in roughly O(n^{3/2}) time the Shapley values for the area of the minimum axisparallel bounding box and the area of the union of the rectangles spanned by the origin and the input points. When the points form an increasing or decreasing chain, the running time can be improved to nearlinear. In all these cases, we use linearity of the Shapley values and algebraic methods.
We also show that Shapley values for the area of the convex hull or the minimum enclosing disk can be computed in O(n^2) and O(n^3) time, respectively. These problems are closely related to the model of stochastic point sets considered in computational geometry, but here we have to consider random insertion orders of the points instead of a probabilistic existence of points.
BibTeX  Entry
@InProceedings{cabello_et_al:LIPIcs:2019:10424,
author = {Sergio Cabello and Timothy M. Chan},
title = {{Computing Shapley Values in the Plane}},
booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)},
pages = {20:120:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771047},
ISSN = {18688969},
year = {2019},
volume = {129},
editor = {Gill Barequet and Yusu Wang},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/10424},
URN = {urn:nbn:de:0030drops104244},
doi = {10.4230/LIPIcs.SoCG.2019.20},
annote = {Keywords: Shapley values, stochastic computational geometry, convex hull, minimum enclosing disk, bounding box, arrangements, convolutions, airport problem}
}
11.06.2019
Keywords: 

Shapley values, stochastic computational geometry, convex hull, minimum enclosing disk, bounding box, arrangements, convolutions, airport problem 
Seminar: 

35th International Symposium on Computational Geometry (SoCG 2019)

Issue date: 

2019 
Date of publication: 

11.06.2019 