Related papers: Subquadratic Encodings for Point Configurations
A simple method to produce a random order type is to take the order type of a random point set. We conjecture that many probability distributions on order types defined in this way are heavily concentrated and therefore sample inefficiently…
We consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering \topk{} queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be…
Encoding data structures store enough information to answer the queries they are meant to support but not enough to recover their underlying datasets. In this paper we give the first encoding data structure for the challenging problem of…
An arrangement of $n$ curves in the plane is given. The query is a point $q$ and the goal is to find the face of the arrangement that contains $q$. A data-structure for point-location, preprocesses the curves into a data structure of…
We consider the following problem: given three sets of real numbers, output a word-RAM data structure from which we can efficiently recover the sign of the sum of any triple of numbers, one in each set. This is similar to a previous work by…
Most of the attention in statistical compression is given to the space used by the compressed sequence, a problem completely solved with optimal prefix codes. However, in many applications, the storage space used to represent the prefix…
Hilbert order is widely applied in many areas. However, most of the algorithms are confined to low dimensional cases. In this paper, algorithms for encoding and decoding arbitrary dimensional Hilbert order are presented. Eight algorithms…
One of the crucial generic techniques for quantum computation is amplitude encoding. Although several approaches have been proposed, each of them often requires exponential classical-computational cost or an oracle whose explicit…
We study first-order optimization algorithms under the constraint that the descent direction is quantized using a pre-specified budget of $R$-bits per dimension, where $R \in (0 ,\infty)$. We propose computationally efficient optimization…
Neural language models process sequences of words, but the mathematical operations inside them are insensitive to the order in which words appear. Positional encodings are the component added to remedy this. Despite their importance,…
One of the most basic computational problems is the task of finding a desired item in an ordered list of N items. While the best classical algorithm for this problem uses log_2 N queries to the list, a quantum computer can solve the problem…
Fast encoding and decoding of codes have been always an important topic in code theory as well as complexity theory. Although encoding is easier than decoding in general, designing an encoding algorithm of codes of length $N$ with…
Let $X$ be a set of points in $\mathbb{R}^2$ and $\mathcal{O}$ be a set of geometric objects in $\mathbb{R}^2$, where $|X| + |\mathcal{O}| = n$. We study the problem of computing a minimum subset $\mathcal{O}^* \subseteq \mathcal{O}$ that…
Product quantization-based approaches are effective to encode high-dimensional data points for approximate nearest neighbor search. The space is decomposed into a Cartesian product of low-dimensional subspaces, each of which generates a sub…
We present an exact $n$-qubit computational-basis amplitude encoder of real- or complex-valued data vectors of $d=\binom{n}{k}$ components into a subspace of fixed Hamming weight $k$. This represents a polynomial space compression of degree…
The order type of a point set in $R^d$ maps each $(d{+}1)$-tuple of points to its orientation (e.g., clockwise or counterclockwise in $R^2$). Two point sets $X$ and $Y$ have the same order type if there exists a mapping $f$ from $X$ to $Y$…
Positional encodings enable Transformers to incorporate sequential information, yet their theoretical understanding remains limited to two properties: distance attenuation and translation invariance. Because natural language lacks purely…
An archetypal problem discussed in computer science is the problem of searching for a given number in a given set of numbers. Other than sequential search, the classic solution is to sort the list of numbers and then apply binary search.…
This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right…
A constant-workspace algorithm has read-only access to an input array and may use only O(1) additional words of $O(\log n)$ bits, where $n$ is the size of the input. We assume that a simple $n$-gon is given by the ordered sequence of its…