Related papers: Compressed Representations of Permutations, and Ap…

We survey permutation-based methods for approximate k-nearest neighbor search. In these methods, every data point is represented by a ranked list of pivots sorted by the distance to this point. Such ranked lists are called permutations. The…

Let $\pi=(\pi_1,\pi_2,\hdots,\pi_n)$ be permutation of the elements $1,2,\hdots,n. $ Positive integer $k\leq2^{n-1}$ we call index of $\pi,$ if in its binary notation as $n$-digital binary number, the 1's correspond to the ascent points. We…

The data structure at the core of large-scale search engines is the inverted index, which is essentially a collection of sorted integer sequences called inverted lists. Because of the many documents indexed by such engines and stringent…

For a permutation $\pi$, and the corresponding permutation matrix, we introduce the notion of {\em discrete derivative}, obtained by taking differences of successive entries in $\pi$. We characterize the possible derivatives of…

We derive a generating function for the number of integer compositions of $n$ into $k$ parts (i.e., $k$-compositions of $n$) with a given number of inversions, and obtain similar results for $k$-compositions of $n$ with a given number of…

Pasting and Reversing operations have been used successfully over the set of integer numbers, simple permutations, rings and recently over a generalized vector product. In this paper, these operations are defined from a natural way to be…

This paper extends the sample complexity theory for ill-posed inverse problems developed in a recent work by the authors [`Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform', J. Eur. Math. Soc.,…

Representing sorted integer sequences in small space is a central problem for large-scale retrieval systems such as Web search engines. Efficient query resolution, e.g., intersection or random access, is achieved by carefully partitioning…

We investigate lossy compression (source coding) of data in the form of permutations. This problem has direct applications in the storage of ordinal data or rankings, and in the analysis of sorting algorithms. We analyze the rate-distortion…

Storage systems often rely on multiple copies of the same compressed data, enabling recovery in case of binary data errors, of course, at the expense of a higher storage cost. In this paper we show that a wiser method of duplication entails…

Sorted data is usually easier to compress than unsorted permutations of the same data. This motivates a simple compression scheme: specify the sorted permutation of the data along with a representation of the sorted data compressed…

Recently Implicit Neural Representations (INRs) gained attention as a novel and effective representation for various data types. Thus far, prior work mostly focused on optimizing their reconstruction performance. This work investigates INRs…

A permutation on an alphabet $ \Sigma $, is a sequence where every element in $ \Sigma $ occurs precisely once. Given a permutation $ \pi $= ($\pi_{1} $, $ \pi_{2} $, $ \pi_{3} $,....., $ \pi_{n} $) over the alphabet $ \Sigma $ =$\{ $0, 1,…

A permutation array $A$ is a set of permutations on a finite set $\Omega$, say of size $n$. Given distinct permutations $\pi, \sigma\in \Omega$, we let $hd(\pi, \sigma) = |\{ x\in \Omega: \pi(x) \ne \sigma(x) \}|$, called the Hamming…

Generalizing stack sorting and $c$-sorting for permutations, we define the permutree sorting algorithm. Given two disjoint subsets $U$ and $D$ of $\{2, \dots, n-1\}$, the $(U,D)$-permutree sorting tries to sort the permutation $\pi \in…

Compressed file formats are the corner stone of efficient data storage and transmission, yet their potential for representation learning remains largely underexplored. We introduce TEMPEST (TransformErs froM comPressed rEpreSenTations), a…

Graph reordering is a powerful technique to increase the locality of the representations of graphs, which can be helpful in several applications. We study how the technique can be used to improve compression of graphs and inverted indexes.…

Given a countable set X (usually taken to be the natural numbers or the integers), an infinite permutation \pi of X is a linear ordering of X. This paper investigates the combinatorial complexity of the infinite permutation on the natural…

Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion…

In [4] we describe a variation of the classical permutation decoding algorithm that can be applied to any binary affine-invariant code; in particular, it can be applied to first-order Reed-Muller codes successfully. In this paper we study…