Fully Retroactive Approximate Range and Nearest Neighbor Searching

We describe fully retroactive dynamic data structures for approximate range reporting and approximate nearest neighbor reporting. We show how to maintain, for any positive constant $d$, a set of $n$ points in $\R^d$ indexed by time such that we can perform insertions or deletions at any point in the timeline in $O(\log n)$ amortized time. We support, for any small constant $\epsilon>0$, $(1+\epsilon)$-approximate range reporting queries at any point in the timeline in $O(\log n + k)$ time, where $k$ is the output size. We also show how to answer $(1+\epsilon)$-approximate nearest neighbor queries for any point in the past or present in $O(\log n)$ time.Comment: 24 pages, 4 figures. To appear at the 22nd International Symposium on Algorithms and Computation (ISAAC 2011

Exploring Privacy Preservation in Outsourced K-Nearest Neighbors with Multiple Data Owners

The k-nearest neighbors (k-NN) algorithm is a popular and effective classification algorithm. Due to its large storage and computational requirements, it is suitable for cloud outsourcing. However, k-NN is often run on sensitive data such as medical records, user images, or personal information. It is important to protect the privacy of data in an outsourced k-NN system. Prior works have all assumed the data owners (who submit data to the outsourced k-NN system) are a single trusted party. However, we observe that in many practical scenarios, there may be multiple mutually distrusting data owners. In this work, we present the first framing and exploration of privacy preservation in an outsourced k-NN system with multiple data owners. We consider the various threat models introduced by this modification. We discover that under a particularly practical threat model that covers numerous scenarios, there exists a set of adaptive attacks that breach the data privacy of any exact k-NN system. The vulnerability is a result of the mathematical properties of k-NN and its output. Thus, we propose a privacy-preserving alternative system supporting kernel density estimation using a Gaussian kernel, a classification algorithm from the same family as k-NN. In many applications, this similar algorithm serves as a good substitute for k-NN. We additionally investigate solutions for other threat models, often through extensions on prior single data owner systems

Three Approaches to Building Time-Windowed Geometric Data Structures

Given a set of geometric objects (points or line segments) each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems. We present algorithms to preprocess for the time-windowed closest pair decision problem in O(n) expected time, for the time-windowed 2D diameter decision problem in O(n log n) time, the time-windowed 2D convex hull area decision problem in O(n α(n) log n) time (where α is the inverse Ackermann function), and the time-windowed 3D diameter decision and orthogonal segment intersection detection problems in O(n polylog n) time. Our first approach is to reduce the closest pair decision problem to 2D dominance range emptiness using grids to compute candidate satisfying pairs. We extend this approach to find the closest pair of points by reducing the problem to 2D dominance range minimum, which we further reduce to 2D point location. Our second approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our third approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions

Conditional Lower Bounds for Dynamic Geometric Measure Problems

We give new polynomial lower bounds for a number of dynamic measure problems in computational geometry. These lower bounds hold in the Word-RAM model, conditioned on the hardness of either 3SUM, APSP, or the Online Matrix-Vector Multiplication problem [Henzinger et al., STOC 2015]. In particular we get lower bounds in the incremental and fully-dynamic settings for counting maximal or extremal points in R^3, different variants of Klee's Measure Problem, problems related to finding the largest empty disk in a set of points, and querying the size of the i'th convex layer in a planar set of points. We also answer a question of Chan et al. [SODA 2022] by giving a conditional lower bound for dynamic approximate square set cover. While many conditional lower bounds for dynamic data structures have been proven since the seminal work of Patrascu [STOC 2010], few of them relate to computational geometry problems. This is the first paper focusing on this topic. Most problems we consider can be solved in O(n log n) time in the static case and their dynamic versions have only been approached from the perspective of improving known upper bounds. One exception to this is Klee's measure problem in R^2, for which Chan [CGTA 2010] gave an unconditional ${\Omega}(\sqrt{n})$ lower bound on the worst-case update time. By a similar approach, we show that such a lower bound also holds for an important special case of Klee's measure problem in R^3 known as the Hypervolume Indicator problem, even for amortized runtime in the incremental setting.Comment: Improved presentation, improved the reduction for the Hypervolume Indicator problem and added a reduction for dynamic approximate square set cove

ADVANCED INTRAVASCULAR MAGNETIC RESONANCE IMAGING WITH INTERACTION

Intravascular (IV) Magnetic Resonance Imaging (MRI) is a specialized class of interventional MRI (iMRI) techniques that acquire MRI images through blood vessels to guide, identify and/or treat pathologies inside the human body which are otherwise difficult to locate and treat precisely. Here, interactions based on real-time computations and feedback are explored to improve the accuracy and efficiency of IVMRI procedures. First, an IV MRI-guided high-intensity focused ultrasound (HIFU) ablation method is developed for targeting perivascular pathology with minimal injury to the vessel wall. To take advantage of real-time feedback, a software interface is developed for monitoring thermal dose with real-time MRI thermometry, and an MRI-guided ablation protocol developed and tested on muscle and liver tissue ex vivo. It is shown that, with cumulative thermal dose monitored with MRI thermometry, lesion location and dimensions can be estimated consistently, and desirable thermal lesions can be achieved in animals in vivo. Second, to achieve fully interactive IV MRI, high-resolution real-time 10 frames-per-second (fps) MRI endoscopy is developed as an advance over prior methods of MRI endoscopy. Intravascular transmit-receive MRI endoscopes are fabricated for highly under-sampled radial-projection MRI in a clinical 3Tesla MRI scanner. Iterative nonlinear reconstruction is accelerated using graphics processor units (GPU) to achieve true real-time endoscopy visualization at the scanner. The results of high-speed MRI endoscopy at 6-10 fps are consistent with fully-sampled MRI endoscopy and histology, with feasibility demonstrated in vivo in a large animal model. Last, a general framework for automatic imaging contrast tuning over MRI protocol parameters is explored. The framework reveals typical signal patterns over different protocol parameters from calibration imaging data and applies this knowledge to design efficient acquisition strategies and predicts contrasts under unacquired protocols. An external computer in real-time communication with the MRI console is utilized for online processing and controlling MRI acquisitions. This workflow enables machine learning for optimizing acquisition strategies in general, and provides a foundation for efficiently tuning MRI protocol parameters to perform interventional MRI in the highly varying and interactive environments commonly in play. This work is loosely inspired by prior research on extremely accelerated MRI relaxometry using the minimal-acquisition linear algebraic modeling (SLAM) method

How Fast Can We Play Tetris Greedily With Rectangular Pieces?

Consider a variant of Tetris played on a board of width $w$ and infinite height, where the pieces are axis-aligned rectangles of arbitrary integer dimensions, the pieces can only be moved before letting them drop, and a row does not disappear once it is full. Suppose we want to follow a greedy strategy: let each rectangle fall where it will end up the lowest given the current state of the board. To do so, we want a data structure which can always suggest a greedy move. In other words, we want a data structure which maintains a set of $O(n)$ rectangles, supports queries which return where to drop the rectangle, and updates which insert a rectangle dropped at a certain position and return the height of the highest point in the updated set of rectangles. We show via a reduction to the Multiphase problem [P\u{a}tra\c{s}cu, 2010] that on a board of width $w=\Theta(n)$, if the OMv conjecture [Henzinger et al., 2015] is true, then both operations cannot be supported in time $O(n^{1/2-\epsilon})$ simultaneously. The reduction also implies polynomial bounds from the 3-SUM conjecture and the APSP conjecture. On the other hand, we show that there is a data structure supporting both operations in $O(n^{1/2}\log^{3/2}n)$ time on boards of width $n^{O(1)}$, matching the lower bound up to a $n^{o(1)}$ factor.Comment: Correction of typos and other minor correction