Lower Bounds for Oblivious Near-Neighbor Search

We prove an $\Omega(d \lg n/ (\lg\lg n)^2)$ lower bound on the dynamic cell-probe complexity of statistically $\mathit{oblivious}$ approximate-near-neighbor search ($\mathsf{ANN}$) over the $d$-dimensional Hamming cube. For the natural setting of $d = \Theta(\log n)$, our result implies an $\tilde{\Omega}(\lg^2 n)$ lower bound, which is a quadratic improvement over the highest (non-oblivious) cell-probe lower bound for $\mathsf{ANN}$. This is the first super-logarithmic $\mathit{unconditional}$ lower bound for $\mathsf{ANN}$ against general (non black-box) data structures. We also show that any oblivious $\mathit{static}$ data structure for decomposable search problems (like $\mathsf{ANN}$) can be obliviously dynamized with $O(\log n)$ overhead in update and query time, strengthening a classic result of Bentley and Saxe (Algorithmica, 1980).Comment: 28 page

Ramsey-type theorems for metric spaces with applications to online problems

A nearly logarithmic lower bound on the randomized competitive ratio for the metrical task systems problem is presented. This implies a similar lower bound for the extensively studied k-server problem. The proof is based on Ramsey-type theorems for metric spaces, that state that every metric space contains a large subspace which is approximately a hierarchically well-separated tree (and in particular an ultrametric). These Ramsey-type theorems may be of independent interest.Comment: Fix an error in the metadata. 31 pages, 0 figures. Preliminary version in FOCS '01. To be published in J. Comput. System Sc

The achievable region method in the optimal control of queueing systems : formulations, bounds and policies

Cover title.Includes bibliographical references (p. 44-48).Supported in part by a Presidential Young Investigator Award, with matching funds from Draper Laboratory. DDM-9158118Dimitris Bertsimas

The achievable region method in the optimal control of queueing systems : formulations, bounds and policies

Cover title.Includes bibliographical references (p. 44-48).Supported in part by a Presidential Young Investigator Award, with matching funds from Draper Laboratory. DDM-9158118Dimitris Bertsimas

On metric Ramsey-type phenomena

The main question studied in this article may be viewed as a nonlinear analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey theory in combinatorics. Given a finite metric space on n points, we seek its subspace of largest cardinality which can be embedded with a given distortion in Hilbert space. We provide nearly tight upper and lower bounds on the cardinality of this subspace in terms of n and the desired distortion. Our main theorem states that for any epsilon>0, every n point metric space contains a subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the distortion is tight up to the log(1/\epsilon) factor. We further include a comprehensive study of various other aspects of this problem.Comment: 67 pages, published versio

The Predicted-Deletion Dynamic Model: Taking Advantage of ML Predictions, for Free

The main bottleneck in designing efficient dynamic algorithms is the unknown nature of the update sequence. In particular, there are some problems, like 3-vertex connectivity, planar digraph all pairs shortest paths, and others, where the separation in runtime between the best partially dynamic solutions and the best fully dynamic solutions is polynomial, sometimes even exponential. In this paper, we formulate the predicted-deletion dynamic model, motivated by a recent line of empirical work about predicting edge updates in dynamic graphs. In this model, edges are inserted and deleted online, and when an edge is inserted, it is accompanied by a "prediction" of its deletion time. This models real world settings where services may have access to historical data or other information about an input and can subsequently use such information make predictions about user behavior. The model is also of theoretical interest, as it interpolates between the partially dynamic and fully dynamic settings, and provides a natural extension of the algorithms with predictions paradigm to the dynamic setting. We give a novel framework for this model that "lifts" partially dynamic algorithms into the fully dynamic setting with little overhead. We use our framework to obtain improved efficiency bounds over the state-of-the-art dynamic algorithms for a variety of problems. In particular, we design algorithms that have amortized update time that scales with a partially dynamic algorithm, with high probability, when the predictions are of high quality. On the flip side, our algorithms do no worse than existing fully-dynamic algorithms when the predictions are of low quality. Furthermore, our algorithms exhibit a graceful trade-off between the two cases. Thus, we are able to take advantage of ML predictions asymptotically "for free.'