Range updates and range sum queries on multidimensional points with monoid weights
Let P be a set of n points in Rd where each point p∈P carries a weight drawn from a commutative monoid (M,+,0). Given a d-rectangle rupd (i.e., an orthogonal rectangle in Rd) and a value Δ∈M, a range update adds Δ to the weight of every point p∈P∩rupd; given a d-rectangle rqry, a range sum query ret...
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| Published in | Computational geometry : theory and applications Vol. 115; p. 102030 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
01.12.2023
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0925-7721 |
| DOI | 10.1016/j.comgeo.2023.102030 |
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| Abstract | Let P be a set of n points in Rd where each point p∈P carries a weight drawn from a commutative monoid (M,+,0). Given a d-rectangle rupd (i.e., an orthogonal rectangle in Rd) and a value Δ∈M, a range update adds Δ to the weight of every point p∈P∩rupd; given a d-rectangle rqry, a range sum query returns the total weight of the points in P∩rqry. The goal is to store P in a structure to support updates and queries with attractive performance guarantees. We describe a structure of O˜(n) space that handles an update in O˜(Tupd) time and a query in O˜(Tqry) time for arbitrary functions Tupd(n) and Tqry(n) satisfying Tupd⋅Tqry=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. |
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| AbstractList | Let P be a set of n points in Rd where each point p∈P carries a weight drawn from a commutative monoid (M,+,0). Given a d-rectangle rupd (i.e., an orthogonal rectangle in Rd) and a value Δ∈M, a range update adds Δ to the weight of every point p∈P∩rupd; given a d-rectangle rqry, a range sum query returns the total weight of the points in P∩rqry. The goal is to store P in a structure to support updates and queries with attractive performance guarantees. We describe a structure of O˜(n) space that handles an update in O˜(Tupd) time and a query in O˜(Tqry) time for arbitrary functions Tupd(n) and Tqry(n) satisfying Tupd⋅Tqry=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. |
| ArticleNumber | 102030 |
| Author | Lu, Shangqi Tao, Yufei |
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| Cites_doi | 10.1016/j.tcs.2020.11.034 10.1016/j.jcss.2021.09.004 10.1016/0020-0190(79)90117-0 10.1145/3396375 |
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| Keywords | Lower bounds Data structures Range updates Range sum queries |
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| SubjectTerms | Data structures Lower bounds Range sum queries Range updates |
| Title | Range updates and range sum queries on multidimensional points with monoid weights |
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