Let P be a set of n points in ℝ^d where each point p ∈ P carries a weight drawn from a commutative monoid (ℳ, +, 0). Given a d-rectangle r_upd (i.e., an orthogonal rectangle in ℝ^d) and a value Δ ∈ ℳ, a range update adds Δ to the weight of every point p ∈ P∩ r_upd; given a d-rectangle r_qry, a range sum query returns the total weight of the points in P ∩ r_qry. The goal is to store P in a structure to support updates and queries with attractive performance guarantees. We describe a structure of Õ(n) space that handles an update in Õ(T_upd) time and a query in Õ(T_qry) time for arbitrary functions T_upd(n) and T_qry(n) satisfying T_upd ⋅ T_qry = n. The result holds for any fixed dimensionality d ≥ 2. Our query-update tradeoff is tight up to a polylog factor subject to the OMv-conjecture.
@InProceedings{lu_et_al:LIPIcs.ISAAC.2022.57, author = {Lu, Shangqi and Tao, Yufei}, title = {{Range Updates and Range Sum Queries on Multidimensional Points with Monoid Weights}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {57:1--57:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.57}, URN = {urn:nbn:de:0030-drops-173427}, doi = {10.4230/LIPIcs.ISAAC.2022.57}, annote = {Keywords: Range Updates, Range Sum Queries, Data Structures, Lower Bounds} }
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