Topic
Cartesian product
About: Cartesian product is a research topic. Over the lifetime, 3154 publications have been published within this topic receiving 42585 citations. The topic is also known as: direct product of sets & Cartesian product of sets.
Papers published on a yearly basis
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Papers
TL;DR: This paper introduces a product quantization-based approach for approximate nearest neighbor search to decompose the space into a Cartesian product of low-dimensional subspaces and to quantize each subspace separately.
Abstract: This paper introduces a product quantization-based approach for approximate nearest neighbor search. The idea is to decompose the space into a Cartesian product of low-dimensional subspaces and to quantize each subspace separately. A vector is represented by a short code composed of its subspace quantization indices. The euclidean distance between two vectors can be efficiently estimated from their codes. An asymmetric version increases precision, as it computes the approximate distance between a vector and a code. Experimental results show that our approach searches for nearest neighbors efficiently, in particular in combination with an inverted file system. Results for SIFT and GIST image descriptors show excellent search accuracy, outperforming three state-of-the-art approaches. The scalability of our approach is validated on a data set of two billion vectors.
3,320 citations
Book•
1 Jan 1998
TL;DR: A survey of domination-related parameters topics on directed graphs graphs can be found in this article with respect to the domination number bondage, insensitivity, and reinforcement of graph dominating functions.
Abstract: LP-duality, complementarity and generality of graphical subset parameters dominating functions in graphs fractional domination and related parameters majority domination and its generalizations convexity of external domination-related functions of graphs combinatorial problems on chessboards - II domination in cartesian products - Vizing's conjecture algorithms complexity results domination parameters of a graph global domination distance domination in graphs domatic numbers of graphs and their variants - a survey domination-related parameters topics on domination in directed graphs graphs critical with respect to the domination number bondage, insensitivity and reinforcement.
1,661 citations
805 citations
TL;DR: Inductively defined FM-sets involving the name-abstraction set former can correctly encode syntax modulo renaming of bound variables, and the standard theory of algebraic data types can be extended to encompass signatures involving binding operators.
Abstract: The permutation model of set theory with atoms (FM-sets), devised by Fraenkel and Mostowski in the 1930s, supports notions of `name-abstraction' and `fresh name' that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variable-binding operations. Inductively defined FM-sets involving the name-abstraction set former (together with Cartesian product and disjoint union) can correctly encode syntax modulo renaming of bound variables. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated notion of structural recursion for defining syntax-manipulating functions (such as capture avoiding substitution, set of free variables, etc.) and a notion of proof by structural induction, both of which remain pleasingly close to informal practice in computer science.
642 citations
Book•
19 Jul 1990
TL;DR: Polymorphic sets: the semantics of the judgement forms general rules enumeration sets Cartesian product of a family of sets equality sets natural numbers lists cartesian product two sets disjoint union of two sets Disjoint Union of small sets (the first universe) well-orderings general trees.
Abstract: Part 1 Polymorphic sets: the semantics of the judgement forms general rules enumeration sets Cartesian product of a family of sets equality sets natural numbers lists cartesian product of two sets disjoint union of two sets disjoint union of a family of sets the set of small sets (the first universe) well-orderings general trees. Part 2 Subsets: subsets in the basic set theory the subset theory. Part 3 Monomorphic sets: types defining sets in terms of types. Part 4 Examples: some small examples program derivation specification of abstract data types.
573 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 58 |
| 2024 | 117 |
| 2023 | 226 |
| 2022 | 347 |
| 2021 | 206 |
| 2020 | 222 |