Related papers: Faster Min-Plus Product for Monotone Instances
We consider a problem first proposed by Mahler and Popken in 1953 and later developed by Coppersmith, Erd\H{o}s, Guy, Isbell, Selfridge, and others. Let $f(n)$ be the complexity of $n \in \mathbb{Z^{+}}$, where $f(n)$ is defined as the…
Given a $k\times n$ integer primitive matrix $\bf{A}$ (i.e., a matrix can be extended to an $n\times n$ unimodular matrix over the integers) with the maximal absolute value of entries $\|\bf{A}\|$ bounded by {an integer} $\lambda$ from…
We study the central All-Pairs Shortest Paths (APSP) problem under the restriction that there are at most $d$ distinct weights on the outgoing edges from every node. For $d=n$ this is the classical (unrestricted) APSP problem that is…
Matrix $M$ is {\em $k$-concise} if the finite entries of each column of $M$ consist of $k$ or less intervals of identical numbers. We give an $O(n+m)$-time algorithm to compute the row minima of any $O(1)$-concise $n\times m$ matrix. Our…
We consider the problem of multiplying sparse matrices (over a semiring) where the number of non-zero entries is larger than main memory. In the classical paper of Hong and Kung (STOC '81) it was shown that to compute a product of dense $U…
We prove lower bounds on the number of product gates in bilinear and quadratic circuits that compute the product of two $n \cross n$ matrices over finite fields. In particular we obtain the following results: 1. We show that the number of…
Almost asymptotically tight lower bounds are derived for the Input/Output (I/O) complexity $IO_\mathcal{A}\left(n,M\right)$ of a general class of hybrid algorithms computing the product of two integers, each represented with $n$ digits in a…
A zero-one matrix is a matrix with entries from $\{0, 1\}$. We study monoids containing only such matrices. A finite set of zero-one matrices generating such a monoid can be seen as the matrix representation of an unambiguous finite…
Recent advances in randomized incremental methods for minimizing $L$-smooth $\mu$-strongly convex finite sums have culminated in tight complexity of $\tilde{O}((n+\sqrt{n L/\mu})\log(1/\epsilon))$ and $O(n+\sqrt{nL/\epsilon})$, where…
In moldable job scheduling, we are provided $m$ identical machines and $n$ jobs that can be executed on a variable number of machines. The execution time of each job depends on the number of machines assigned to execute that job. For the…
In this paper, we present algorithms to solve matrix multiplication problems in the MPC model. In particular, we consider the problem under various processor/memory constraints in the MPC model and prove the following results. 1.…
We disprove a conjecture by Ye and Lim, by showing that there are $3 \times 3$ complex matrices which can't be expressed as the product of two Toeplitz matrices of the same size. We also improve previous estimates by Ye and Lim on the…
We propose a non-commutative algorithm for multiplying 2x2 matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast algorithms, and a time complexity…
For a matrix $A\in \mathbb{R}^{n\times d}$ with $n\geq d$, we consider the dual problems of $\min \|Ax-b\|_p^p, \, b\in \mathbb{R}^n$ and $\min_{A^\top x=b} \|x\|_p^p,\, b\in \mathbb{R}^d$. We improve the runtimes for solving these problems…
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…
Makespan scheduling on identical machines is one of the most basic and fundamental packing problems studied in the discrete optimization literature. It asks for an assignment of $n$ jobs to a set of $m$ identical machines that minimizes the…
We study fundamental problems in linear algebra, such as finding a maximal linearly independent subset of rows or columns (a basis), solving linear regression, or computing a subspace embedding. For these problems, we consider input…
Computing the edit distance of two strings is one of the most basic problems in computer science and combinatorial optimization. Tree edit distance is a natural generalization of edit distance in which the task is to compute a measure of…
Given a set $S$ of $n$ points in $\mathbb{R}^d$, the Closest Pair problem is to find a pair of distinct points in $S$ at minimum distance. When $d$ is constant, there are efficient algorithms that solve this problem, and fast approximate…
Consider an undirected weighted graph $G = (V,E,w)$. We study the problem of computing $(1+\epsilon)$-approximate shortest paths for $S \times V$, for a subset $S \subseteq V$ of $|S| = n^r$ sources, for some $0 < r \le 1$. We devise a…