Related papers: A Near-Optimal Depth-Hierarchy Theorem for Small-D…

The concept of data depth in non-parametric multivariate descriptive statistics is the generalization of the univariate rank method to multivariate data. Halfspace depth is a measure of data depth. Given a set S of points and a point p, the…

Thorup [FOCS'01, JACM'04] and Klein [SODA'01] independently showed that there exists a $(1+\epsilon)$-approximate distance oracle for planar graphs with $O(n (\log n)\epsilon^{-1})$ space and $O(\epsilon^{-1})$ query time. While the…

We contribute a 2D nearest-neighbor quantum architecture for Shor's algorithm to factor an $n$-bit number in $O(\log^2(n))$ depth. Our implementation uses parallel phase estimation, constant-depth fanout and teleportation, and…

Boolean circuits of McCulloch-Pitts threshold gates are a classic model of neural computation studied heavily in the late 20th century as a model of general computation. Recent advances in large-scale neural computing hardware has made…

It is shown that the minimal depth of an optimal prefix circuit (i.e., a zero-deficiency circuit) on $N$ inputs with fanout bounded by $k$ is ${\log_{\alpha_k} N \pm O(1)}$, where $\alpha_k$ is the unique positive root of the polynomial…

Good approximations have been attained for the sparsest cut problem by rounding solutions to convex relaxations via low-distortion metric embeddings. Recently, Bryant and Tupper showed that this approach extends to the hypergraph setting by…

An open problem in complexity theory is to find the minimal degree of a polynomial representing the $n$-bit OR function modulo composite $m$. This problem is related to understanding the power of circuits with $\text{MOD}_m$ gates where $m$…

Given a graph $G = (V, E)$, we wish to compute a spanning tree whose maximum vertex degree, i.e. tree degree, is as small as possible. Computing the exact optimal solution is known to be NP-hard, since it generalizes the Hamiltonian path…

This paper considers a linear quasi-cyclic product code of two given quasi-cyclic codes of relatively prime lengths over finite fields. We give the spectral analysis of a quasi-cyclic product code in terms of the spectral analysis of the…

We present a top-down lower-bound method for depth-$4$ boolean circuits. In particular, we give a new proof of the well-known result that the parity function requires depth-$4$ circuits of size exponential in $n^{1/3}$. Our proof is an…

We characterize the monotone bounded depth formula complexity for graph homomorphism and colored isomorphism polynomials using a graph parameter called the cost of bounded product depth baggy elimination tree. Using this characterization,…

We prove that a binary linear code of block length $n$ that is locally correctable with $3$ queries against a fraction $\delta > 0$ of adversarial errors must have dimension at most $O_{\delta}(\log^2 n \cdot \log \log n)$. This is almost…

Since the breakthrough superpolynomial multilinear formula lower bounds of Raz (Theory of Computing 2006), proving such lower bounds against multilinear algebraic branching programs (mABPs) has been a longstanding open problem in algebraic…

We show that the permanent cannot be computed by DLOGTIME-uniform threshold or arithmetic circuits of depth o(log log n) and polynomial size.

We initiate a broad study of classical problems in the streaming model with insertions and deletions in the setting where we allow the approximation factor $\alpha$ to be much larger than $1$. Such algorithms can use significantly less…

Recent breakthroughs in quantum query complexity have shown that any formula of size n can be evaluated with O(sqrt(n)log(n)/log log(n)) many quantum queries in the bounded-error setting [FGG08, ACRSZ07, RS08b, Rei09]. In particular, this…

Set cover, over a universe of size $n$, may be modelled as a data-streaming problem, where the $m$ sets that comprise the instance are to be read one by one. A semi-streaming algorithm is allowed only $O(n\, \mathrm{poly}\{\log n, \log…

We show that on graphs with n vertices, the 2-dimensional Weisfeiler-Leman algorithm requires at most O(n^2/log(n)) iterations to reach stabilization. This in particular shows that the previously best, trivial upper bound of O(n^2) is…

We show that any depth-$d$ circuit for determining whether an $n$-node graph has an $s$-to-$t$ path of length at most $k$ must have size $n^{\Omega(k^{1/d}/d)}$. The previous best circuit size lower bounds for this problem were…

Circuit synthesis is the task of decomposing a given logical functionality into a sequence of elementary gates. It is (depth-)optimal if it is impossible to achieve the desired functionality with even shorter circuits. Optimal synthesis is…