Multi-Resolution Hashing for Fast Pairwise Summations

A basic computational primitive in the analysis of massive datasets is summing simple functions over a large number of objects. Modern applications pose an additional challenge in that such functions often depend on a parameter vector $y$ (query) that is unknown a priori. Given a set of points $X\subset \mathbb{R}^{d}$ and a pairwise function $w:\mathbb{R}^{d}\times \mathbb{R}^{d}\to [0,1]$, we study the problem of designing a data-structure that enables sublinear-time approximation of the summation $Z_{w}(y)=\frac{1}{|X|}\sum_{x\in X}w(x,y)$ for any query $y\in \mathbb{R}^{d}$. By combining ideas from Harmonic Analysis (partitions of unity and approximation theory) with Hashing-Based-Estimators [Charikar, Siminelakis FOCS'17], we provide a general framework for designing such data structures through hashing that reaches far beyond what previous techniques allowed. A key design principle is a collection of $T\geq 1$ hashing schemes with collision probabilities $p_{1},\ldots, p_{T}$ such that $\sup_{t\in [T]}\{p_{t}(x,y)\} = \Theta(\sqrt{w(x,y)})$. This leads to a data-structure that approximates $Z_{w}(y)$ using a sub-linear number of samples from each hash family. Using this new framework along with Distance Sensitive Hashing [Aumuller, Christiani, Pagh, Silvestri PODS'18], we show that such a collection can be constructed and evaluated efficiently for any log-convex function $w(x,y)=e^{\phi(\langle x,y\rangle)}$ of the inner product on the unit sphere $x,y\in \mathcal{S}^{d-1}$. Our method leads to data structures with sub-linear query time that significantly improve upon random sampling and can be used for Kernel Density or Partition Function Estimation. We provide extensions of our result from the sphere to $\mathbb{R}^{d}$ and from scalar functions to vector functions.Comment: 39 pages, 3 figure

Fully Polynomial Time Approximation Schemes for Stochastic Dynamic Programs

We present a framework for obtaining fully polynomial time approximation schemes (FPTASs) for stochastic univariate dynamic programs with either convex or monotone single-period cost functions. This framework is developed through the establishment of two sets of computational rules, namely, the calculus of K-approximation functions and the calculus of K-approximation sets. Using our framework, we provide the first FPTASs for several NP-hard problems in various fields of research such as knapsack models, logistics, operations management, economics, and mathematical finance. Extensions of our framework via the use of the newly established computational rules are also discussed

04091 Abstracts Collection -- Data Structures

From 22.02. to 27.02.2004, Dagstuhl Seminar "Data Structures" was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar are put together in this paper. The first section describes the seminar topics and goals in general