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We show that a large fraction of the data-structure lower bounds known today in fact follow by reduction from the communication complexity of lopsided (asymmetric) set disjointness. This includes lower bounds for: * high-dimensional…
In this work, we study the limits of compressed data structures, i.e., structures that support various queries on an input text $T\in\Sigma^n$ using space proportional to the size of $T$ in compressed form. Nearly all fundamental queries…
We consider a fundamental problem in data structures, static predecessor searching: Given a subset S of size n from the universe [m], store S so that queries of the form "What is the predecessor of x in S?" can be answered efficiently. We…
We consider the problem of designing succinct data structures for interval graphs with $n$ vertices while supporting degree, adjacency, neighborhood and shortest path queries in optimal time in the $\Theta(\log n)$-bit word RAM model. The…
The classical branch-and-bound algorithm for the integer feasibility problem has exponential worst case complexity. We prove that it is surprisingly efficient on reformulated problems, in which the columns of the constraint matrix are…
We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length $m$ and a substring of a longer text. We give…
Matrix multiplication is a fundamental task in almost all computational fields, including machine learning and optimization, computer graphics, signal processing, and graph algorithms (static and dynamic). Twin-width is a natural complexity…
A classic data structure problem is to preprocess a string T of length $n$ so that, given a query $q$, we can quickly find all substrings of T with Hamming distance at most $k$ from the query string. Variants of this problem have seen…
One fundamental question in database theory is the following: Given a Boolean conjunctive query Q, what is the best complexity for computing the answer to Q in terms of the input database size N? When restricted to the class of…
The input to the NP-hard Point Line Cover problem (PLC) consists of a set $P$ of $n$ points on the plane and a positive integer $k$, and the question is whether there exists a set of at most $k$ lines which pass through all points in $P$. A…
In this paper, we show that the time complexity of monotone min-plus product of two $n\times n$ matrices is $\tilde{O}(n^{(3+\omega)/2})=\tilde{O}(n^{2.687})$, where $\omega < 2.373$ is the fast matrix multiplication exponent [Alman and…
We resolve the long-standing open problem of Boolean dynamic data structure hardness, proving an unconditional lower bound of $\Omega((\log n / \log\log n)^2)$ for the Multiphase Problem of Patrascu [STOC 2010] (instantiated with Inner…
Given an array of size $n$ from a total order, we consider the problem of constructing a data structure that supports various queries (range minimum/maximum queries with their variants and next/previous larger/smaller queries) efficiently.…
We improve both upper and lower bounds for the distribution-free testing of monotone conjunctions. Given oracle access to an unknown Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ and sampling oracle access to an unknown distribution…
We study the problem of multiplying two bit matrices with entries either over the Boolean algebra $(0,1,\vee,\wedge)$ or over the binary field $(0,1,+,\cdot)$. We engineer high-performance open-source algorithm implementations for…
Communication lower bounds have long been established for matrix multiplication algorithms. However, most methods of asymptotic analysis have either ignored the constant factors or not obtained the tightest possible values. Recent work has…
This paper considers coding for so-called partially stuck (defect) memory cells. Such memory cells can only store partial information as some of their levels cannot be used fully due to, e.g., wearout. First, we present new constructions…
We prove a lower bound $\Omega\left(\frac{k+l}{k^2l^2}N^{2-\frac{k+l+2}{kl}}\right)$ on the maximal possible weight of a $(k,l)$-free (that is, free of all-ones $k\times l$ submatrices) Boolean circulant $N \times N$ matrix. The bound is…
We present a protocol for the Boolean matrix product of two $n\times b$ Boolean matrices on the congested clique designed for the situation when the rows of the first matrix or the columns of the second matrix are highly clustered in the…
A retrieval data structure for a static function $f:S\rightarrow \{0,1\}^r$ supports queries that return $f(x)$ for any $x \in S$. Retrieval data structures can be used to implement a static approximate membership query data structure…