2026-07-23T22:34:22Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:15500 2024-03-06T10:55:27Z ddc:004 open_access

Computing Shapley Values for Mean Width in 3-D Tan, Shuhao Shapley value mean width dynamic convolution 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Shuhao Tan 2021 Is Part Of LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ISAAC.2021.67 urn:nbn:de:0030-drops-155008 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.67 eng https://creativecommons.org/licenses/by/4.0/legalcode