Results 2411 - 2420 of 23934

We introduce a novel technique for ``lifting'' dimension lower bounds for linear sketches in the continuous setting to dimension lower bounds for linear sketches with polynomially-bounded integer entries when the input is a polynomially-bounded integer vector. Using this technique, we obtain the first optimal sketching lower bounds for discrete inputs in a data stream, for classical problems such as approximating the frequency moments, estimating the operator norm, and compressed sensing. Additionally, we lift the adaptive attack of Hardt and Woodruff (STOC, 2013) for breaking any real-valued linear sketch via a sequence of real-valued queries, and show how to obtain an attack on any integer-valued linear sketch using integer-valued queries. This shows that there is no linear sketch in a data stream with insertions and deletions that is adversarially robust for approximating any $L_p$ norm of the input. This resolves a central open question for adversarially robust streaming algorithms. To do so, we introduce a new pre-processing technique of independent interest which, given an integer-valued linear sketch, increases the dimension of the sketch by only a constant factor in order to make the orthogonal lattice to its row span smooth, enabling us to leverage results in lattice theory on discrete Gaussian distributions and reason that efficient discrete sketches imply efficient continuous sketches. Our work resolves open questions from the Banff '14 and '17 workshops on Communication Complexity and Applications, as well as the STOC '21 and FOCS '23 workshops on adaptivity and robustness.
 

Based on joint work with Elena Gribelyuk, Honghao Lin, Huacheng Yu, and Samson Zhou.

Cardinality sketches are compact data structures used to represent sets or vectors, and they are widely deployed in practice. These sketches are space-efficient, typically requiring only logarithmic storage in the input size. Importantly, they are composable, meaning that the sketch of a union of sets can be computed from the sketches of the individual sets. They enable approximate computation of cardinality (i.e., the number of nonzero entries). Statistical guarantees ensure accurate answers to a number of queries that is exponential in the sketch size k when the queries are non-adaptive. However, these guarantees degrade significantly, to only quadratic in k, when the queries are adaptive.

In this talk, we demonstrate that this quadratic vulnerability to adaptive queries is inherent for broad classes of cardinality sketches, based on the combinatorial properties of composable mappings.

Joint work with: Sara Ahmadian, Jelani Nelson, Tamás Sarlós, Mihor Singhal, and Uri Stemmer

Computing matchings in general graphs plays a central role in graph algorithms. However, despite the recent interest in differentially private graph algorithms, there has been limited work on private matchings. Almost all existing work focuses on estimating the size of the maximum matching, whereas in many applications, the matching itself is the object of interest. There is currently only a single work on private algorithms for computing matching solutions by [HHRRW STOC'14]. Moreover, their work focuses on allocation problems and hence is limited to bipartite graphs. In this talk, we give a number of new results in the domain of differentially private matchings with algorithms drawing inspiration from fast algorithms in distributed computing. 

Joint work with Michael Dinitz, George Z. Li, and Felix Zhou.

A partially dynamic graph is a graph that undergoes edge insertions or deletions, but not both. In this talk, I present a unifying framework that yields the first almost-optimal, almost-linear time algorithms for many well-studied problems on partially dynamic graphs [Chen-Kyng-Liu-Meierhans-Probst-Gutenberg, STOC’24; Brand-Chen-Kyng-Liu-Meierhans-Probst Gutenberg-Sachdevea, FOCS’24]. These problems include cycle detection, strongly connected components, s-t distances, transshipment, bipartite matching, maximum flow, and minimum-cost flow. We achieve this unification by solving the partially dynamic threshold minimum-cost flow problem. We solve these problems by combining a partially dynamic L1 interior point method (Brand-Liu-Sidford STOC'23) with powerful new data structures that solve fully-dynamic APSP and min-cut with sub-polynomial approximation quality and sub-polynomial update and query time.