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We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of…
We present a new framework for the fast solution of inhomogeneous elliptic boundary value problems in domains with smooth boundaries. High-order solvers based on adaptive box codes or the fast Fourier transform can efficiently treat the…
We consider the problem of finding high dimensional approximate nearest neighbors. Suppose there are d independent rare features, each having its own independent statistics. A point x will have x_{i}=0 denote the absence of feature i, and…
In this paper, we prove that with high probability, random Reed-Solomon codes approach the half-Singleton bound - the optimal rate versus error tradeoff for linear insdel codes - with linear-sized alphabets. More precisely, we prove that,…
Hashing method maps similar data to binary hashcodes with smaller hamming distance, and it has received a broad attention due to its low storage cost and fast retrieval speed. However, the existing limitations make the present algorithms…
Finding the nearest neighbor to a hyperplane (or Point-to-Hyperplane Nearest Neighbor Search, simply P2HNNS) is a new and challenging problem with applications in many research domains. While existing state-of-the-art hashing schemes (e.g.,…
Finding nearest neighbors in high-dimensional spaces is a fundamental operation in many multimedia retrieval applications. Exact tree-based indexing approaches are known to suffer from the notorious curse of dimensionality for…
We prove that $poly(t) \cdot n^{1/D}$-depth local random quantum circuits with two qudit nearest-neighbor gates on a $D$-dimensional lattice with n qudits are approximate $t$-designs in various measures. These include the "monomial"…
An efficient algorithm to solve the $k$ shortest non-homotopic path planning ($k$-SNPP) problem in a 2D environment is proposed in this paper. Motivated by accelerating the inefficient exploration of the homotopy-augmented space of the 2D…
We show tight lower bounds for the entire trade-off between space and query time for the Approximate Near Neighbor search problem. Our lower bounds hold in a restricted model of computation, which captures all hashing-based approaches. In…
Finding nearest neighbors in high-dimensional spaces is a fundamental operation in many diverse application domains. Locality Sensitive Hashing (LSH) is one of the most popular techniques for finding approximate nearest neighbor searches in…
We study the k nearest neighbors problem in the plane for general, convex, pairwise disjoint sites of constant description complexity such as line segments, disks, and quadrilaterals and with respect to a general family of distance…
Nearest neighbor search on high-dimensional vectors is fundamental in modern AI and database systems. In many real-world applications, queries involve constraints on multiple numeric attributes, giving rise to range-filtering approximate…
We give a deterministic nearly-linear time algorithm for approximating any point inside a convex polytope with a sparse convex combination of the polytope's vertices. Our result provides a constructive proof for the Approximate…
We introduce a new class of algorithms for finding a short vector in lattices defined by codes of co-dimension $k$ over $\mathbb{Z}_P^d$, where $P$ is prime. The co-dimension $1$ case is solved by exploiting the packing properties of the…
We give new data-dependent locality sensitive hashing schemes (LSH) for the Earth Mover's Distance ($\mathsf{EMD}$), and as a result, improve the best approximation for nearest neighbor search under $\mathsf{EMD}$ by a quadratic factor.…
This paper describes a novel approach to neighbour-finding in Smoothed Particle Hydrodynamics (SPH) simulations with large dynamic range in smoothing length. This approach is based on hierarchical cell decompositions, sorted interactions,…
We introduce space-efficient plane-sweep algorithms for basic planar geometric problems. It is assumed that the input is in a read-only array of $n$ items and that the available workspace is $\Theta(s)$ bits, where $\lg n \leq s \leq n…
The complexity of nearest-neighbor search dominates the asymptotic running time of many sampling-based motion-planning algorithms. However, collision detection is often considered to be the computational bottleneck in practice. Examining…
We study lower bounds for Locality Sensitive Hashing (LSH) in the strongest setting: point sets in {0,1}^d under the Hamming distance. Recall that here H is said to be an (r, cr, p, q)-sensitive hash family if all pairs x, y in {0,1}^d with…