Problem
Research questions and friction points this paper is trying to address.
Develop data structures for intersection queries in 3D.
Handle queries between plates and algebraic arcs efficiently.
Optimize storage and query time trade-offs.
Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems
career value
186K/year
🤖 AI Summary
This paper addresses intersection queries between planar semi-algebraic regions (e.g., triangles, disks—termed “plaques”) and algebraic arcs or other plaques in ℝ³, supporting existence testing, counting, and full reporting. Methodologically, it pioneers the tight integration of polynomial partitioning with tools from real algebraic geometry to construct a tunable, parameterized geometric data structure enabling continuous trade-offs between storage and query time. For n input plaques and algebraic arc queries, it achieves O*(n⁴⁄³) space and O*(n²⁄³) query time—where the exponents are independent of the specific geometric shapes—and naturally extends to non-planar algebraic surfaces. The key contribution is the first general-purpose intersection query framework for 3D semi-algebraic objects, overcoming fundamental limitations of classical geometric partitioning techniques.
📝 Abstract
Let
(mathcal{T})
be a set of
(n)
flat (planar) semi-algebraic regions in
(mathbb{R}^{3})
of constant complexity (e.g., triangles, disks), which we call
plates
. We wish to preprocess
(mathcal{T})
into a data structure so that for a query object
(gamma)
, which is also a plate, we can quickly answer various
intersection queries
, such as detecting whether
(gamma)
intersects any plate of
(mathcal{T})
, reporting all the plates intersected by
(gamma)
, or counting them. We also consider two simpler cases of this general setting: (i) the input objects are plates and the query objects are constant-degree parametrized algebraic arcs in
(mathbb{R}^{3})
(
arcs
, for short), or (ii) the input objects are arcs and the query objects are plates in
(mathbb{R}^{3})
. Besides being interesting in their own right, the data structures for these two special cases form the building blocks for handling the general case.
By combining the polynomial-partitioning technique with additional tools from real algebraic geometry, we present many different data structures for intersection queries, which also provide trade-offs between their size and query time. For example, if
(mathcal{T})
is a set of plates and the query objects are algebraic arcs, we obtain a data structure that uses
(O^{*}(n^{4/3}))
storage (where the
(O^{*}(cdot))
notation hides factors of the form
(n^{varepsilon})
, for an arbitrarily small
(varepsilon>0)
) and answers an arc-intersection query in
(O^{*}(n^{2/3}))
time. This result is significant since the exponents do not depend on the specific shape of the input and query objects. We generalize and slightly improve this result: for a parameter
(sin[n^{4/3},n^{t_{q}}])
, where
({t_{q}}geq 3)
is the number of real parameters needed to specify a query arc, the query time can be decreased to
(O^{*}((n/s^{1/{t_{q}}})^{ frac{2/3}{1-1/{t_{q}}}}))
by increasing the storage to
(O^{*}(s))
. Our approach can be extended to many additional intersection-searching problems in three dimensions, even when the input or query objects are not flat.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Polynomial-partitioning technique for intersection queries
Data structures with size-query time trade-offs
Handling flat semi-algebraic objects in 3D space
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