Computing Shapley Values for Mean Width in 3-D

Author Shuhao Tan

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Shuhao Tan
  • Department of Computer Science, University of Maryland, College Park, MD, USA


The author is very grateful to David Mount for discussions and advice for the paper. The author is also thankful to Geng Lin for proofreading and helping improving the readability.

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Shuhao Tan. Computing Shapley Values for Mean Width in 3-D. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 67:1-67:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Data structures design and analysis
  • Shapley value
  • mean width
  • dynamic convolution


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