Let $A, B \in \mathbb{Z}_{\ge 0}^n$ be nonnegative vectors and let $t = |\operatorname{supp}(A \star B)|$. We give a Las Vegas algorithm that computes $A \star B$ in $O(t \log t)$ expected time. More generally, for every $0<\delta \le \frac{1}{2}$, the algorithm terminates within $O(t \log t \log \frac{1}{\delta})$ time with probability at least $1 - \delta$. The algorithm uses dense convolution, linear hashing, and the length reduction of \cite{BFN22}. Its main ingredient is a carry-free representation of the indices as vectors of constant dimension $d$ whose coordinates have size $O(t / \log t)$. We can then take our hash function to be the inner product with a random element of $\mathbb{F}_p^d$ for a prime $p$ of size $\Omega(t / \log t)$: this preserves addition and gives collision probability exactly $1/p$, while identities regarding the moments of the vectors identify and recover the isolated terms as in \cite{BFN22}. Our expected running time matches that of Jin and Xu~\cite{JX24} while using substantially different tools and yielding a simpler algorithm. Note that their algorithm also terminates within $O(t \log t)$ time with probability at least $1 - \frac{1}{t}$, while our tail bound is weaker.
We give an efficient algorithm with improved algorithmic guarantees for the (offline) Beck--Fiala problem when the sets have bounded size. Let $A$ be an arbitrary matrix $A\in\{0,1\}^{m\times n}$ with at most $d$ ones per column and at most $s$ ones per row. Let $\log^*$ denote the iterated logarithm and $\ell_j$ denot...
Let $A\in\mathbb{R}^{m\times n}$ have columns of Euclidean norm at most one. We prove that $\operatorname{disc}(A)\le2395\left(1+\log_+\frac n9\right)^{1/4}+2\sqrt2$. Here $\log_+t=\max\{0,\log t\}$. Building on Bansal and Jiang's affine spectral independence framework, we remove the $(\log\log n)^{7/4}$ factor from th...
Let $P_n$ be the matrix of a random permutation of $n$ symbols and let $M_n=\log\max_{|z|=1}|\det(I-zP_n)|$. Cook and Zeitouni proved that $M_n/\log n$ converges in probability to a constant $x_0$ for a uniform permutation. We show that the $\sqrt{\log n}$ fluctuations of $M_n$ are carried entirely by the number of cyc...
The Koml\'os conjecture, now a theorem, asserts that whenever the columns of a matrix $A\in\mathbb{R}^{m\times n}$ have Euclidean norm at most one, some signs $\varepsilon\in\{-1,1\}^n$ make every coordinate of $A\varepsilon$ bounded by an absolute constant. Guo, Fang, and Lu gave the first polynomial-time algorithm fo...
Let $\nu$ be a compactly supported Borel probability measure on $\mathbb R^{d}$ with $\nu(B(x,r))\leq Cr^{a}$ for some $a>0$. Refine a dyadic cube exactly when its mass is at least $t$, and let $\mathcal{L}_{\nu}(t)$ be the mean depth at which this refinement stops. We show that the lower and upper geometric-mean quant...
Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1...
Egor D. Kosov, A. Zhukova· 1 citation· ⚡1
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