Related papers: New Lower Bounds for Trace Reconstruction
We consider an \emph{approximate} version of the trace reconstruction problem, where the goal is to recover an unknown string $s\in\{0,1\}^n$ from $m$ traces (each trace is generated independently by passing $s$ through a probabilistic…
We study the space complexity of sketching cuts and Laplacian quadratic forms of graphs. We show that any data structure which approximately stores the sizes of all cuts in an undirected graph on $n$ vertices up to a $1+\epsilon$ error must…
We prove the following asymptotically tight lower bound for $k$-color discrepancy: For any $k \geq 2$, there exists a hypergraph with $n$ hyperedges such that its $k$-color discrepancy is at least $\Omega(\sqrt{n})$. This improves on the…
We prove a lower bound of $\Omega(n^2/\log^2 n)$ on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial $f(x_1, \ldots, x_n)$. Our approach expands and improves upon a result of Raz,…
We show optimal lower bounds for spanning forest computation in two different models: * One wants a data structure for fully dynamic spanning forest in which updates can insert or delete edges amongst a base set of $n$ vertices. The sole…
Locally repairable codes(LRCs) play important roles in distributed storage systems(DSS). LRCs with small locality have their own advantages since fewer available symbols are needed in the recovery of erased symbols. In this paper, we prove…
In the Steiner point removal (SPR) problem, we are given a (weighted) graph $G$ and a subset $T$ of its vertices called terminals, and the goal is to compute a (weighted) graph $H$ on $T$ that is a minor of $G$, such that the distance…
Fault-tolerant spanners are fundamental objects that preserve distances in graphs even under edge failures. A long line of work culminating in Bodwin, Dinitz, Robelle (SODA 2022) gives $(2k-1)$-stretch, $f$-fault-tolerant spanners with…
The problem of constructing dense subsets S of {1,2,..,n} that contain no arithmetic triple was introduced by Erdos and Turan in 1936. They have presented a construction with |S| = \Omega(n^{\log_3 2}) elements. Their construction was…
Let $M_n$ be the minimal position at generation $n$, of a real-valued branching random walk in the boundary case. As $n \to \infty$, $M_n- {3 \over 2} \log n$ is tight (see [1][9][2]). We establish here a law of iterated logarithm for the…
A new risk bound is presented for the problem of convex/concave function estimation, using the least squares estimator. The best known risk bound, as had appeared in \citet{GSvex}, scaled like $\log(en) n^{-4/5}$ under the mean squared…
We show in this note that the average number of terms in the optimal double-base number system is in Omega(n / log n). The lower bound matches the upper bound shown earlier by Dimitrov, Imbert, and Mishra (Math. of Comp. 2008).
The graph reconstruction problem has been extensively studied under various query models. In this paper, we propose a new query model regarding the number of connected components, which is one of the most basic and fundamental graph…
Let $S(t) = \frac{1}{\pi}\Im \log\zeta\left(\frac{1}{2}+it\right)$. We prove an unconditional lower bound on the measure of the sets $\{t\in [T,2T] \colon S(t) \geq V\}$ for $\sqrt{\log\log T} \leq V \ll \left(\frac{\log T}{\log \log…
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…
This paper studies the lower bound complexity for the optimization problem whose objective function is the average of $n$ individual smooth convex functions. We consider the algorithm which gets access to gradient and proximal oracle for…
Mean-based reconstruction is a fundamental, natural approach to worst-case trace reconstruction over channels with synchronization errors. It is known that $\exp(O(n^{1/3}))$ traces are necessary and sufficient for mean-based worst-case…
It has been established that when the gradient coding problem is distributed among $n$ servers, the computation load (number of stored data partitions) of each worker is at least $s+1$ in order to resists $s$ stragglers. This scheme incurs…
The obstacle number is a new graph parameter introduced by Alpert, Koch, and Laison (2010). Mukkamala etal (2012) show that there exist graphs with n vertices having obstacle number in Omega(n/\log n). In this note, we up this lower bound…
We show the validity of Nachman's procedure (Ann. Math. 128(3):531-576, 1988) for reconstructing a conductivity function $\gamma$ in a smooth bounded domain $\Omega \subset \mathbb{R}^n$ ($n\geq 3$) from its Dirichlet-to-Neumann map…