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Duan, Wu and Zhou (FOCS 2023) recently obtained the improved upper bound on the exponent of square matrix multiplication $\omega<2.3719$ by introducing a new approach to quantify and compensate the ``combination loss" in prior analyses of powers of the Coppersmith-Winograd tensor. In this paper we show how to use this new approach to improve the exponent of rectangular matrix multiplication as well. Our main technical contribution is showing how to combine this analysis of the combination loss and the analysis of the fourth power of the Coppersmith-Winograd tensor in the context of rectangular matrix multiplication developed by Le Gall and Urrutia (SODA 2018).

We find a succinct expression for computing the sequence $x_t = a_t x_{t-1} + b_t$ in parallel with two prefix sums, given $t = (1, 2, \dots, n)$, $a_t \in \mathbb{R}^n$, $b_t \in \mathbb{R}^n$, and initial value $x_0 \in \mathbb{R}$. On $n$ parallel processors, the computation of $n$ elements incurs $\mathcal{O}(\log n)$ time and $\mathcal{O}(n)$ space. Sequences of this form are ubiquitous in science and engineering, making efficient parallelization useful for a vast number of applications. We implement our expression in software, test it on parallel hardware, and verify that it executes faster than sequential computation by a factor of $\frac{n}{\log n}$.

Join-preserving maps on the discrete time scale $\omega^+$, referred to as time warps, have been proposed as graded modalities that can be used to quantify the growth of information in the course of program execution. The set of time warps forms a simple distributive involutive residuated lattice -- called the time warp algebra -- that is equipped with residual operations relevant to potential applications. In this paper, we show that although the time warp algebra generates a variety that lacks the finite model property, it nevertheless has a decidable equational theory. We also describe an implementation of a procedure for deciding equations in this algebra, written in the OCaml programming language, that makes use of the Z3 theorem prover.

Secure multiparty computation (MPC) schemes allow two or more parties to conjointly compute a function on their private input sets while revealing nothing but the output. Existing state-of-the-art number-theoretic-based designs face the threat of attacks through quantum algorithms. In this context, we present secure MPC protocols that can withstand quantum attacks. We first present the design and analysis of an information-theoretic secure oblivious linear evaluation (OLE), namely ${\sf qOLE}$ in the quantum domain, and show that our ${\sf qOLE}$ is safe from external attacks. In addition, our scheme satisfies all the security requirements of a secure OLE. We further utilize ${\sf qOLE}$ as a building block to construct a quantum-safe multiparty private set intersection (MPSI) protocol.

Given a finite point set $P$ in ${\mathbb R}^d$, and $\epsilon>0$ we say that $N\subseteq{ \mathbb R}^d$ is a weak $\epsilon$-net if it pierces every convex set $K$ with $|K\cap P|\geq \epsilon |P|$. We show that for any finite point set in dimension $d\geq 3$, and any $\epsilon>0$, one can construct a weak $\epsilon$-net whose cardinality is $\displaystyle O^*\left(\frac{1}{\epsilon^{2.558}}\right)$ in dimension $d=3$, and $\displaystyle o\left(\frac{1}{\epsilon^{d-1/2}}\right)$ in all dimensions $d\geq 4$. To be precise, our weak $\epsilon$-net has cardinality $\displaystyle O\left(\frac{1}{\epsilon^{\alpha_d+\gamma}}\right)$ for any $\gamma>0$, with $$ \alpha_d= \left\{ \begin{array}{l} 2.558 & \text{if} \ d=3 \\3.48 & \text{if} \ d=4 \\\left(d+\sqrt{d^2-2d}\right)/2 & \text{if} \ d\geq 5. \end{array}\right\} $$ This is the first significant improvement of the bound of $\displaystyle \tilde{O}\left(\frac{1}{\epsilon^d}\right)$ that was obtained in 1993 by Chazelle, Edelsbrunner, Grigni, Guibas, Sharir, and Welzl for general point sets in dimension $d\geq 3$.

In this paper, we prove the first \emph{super-polynomial} and, in fact, \emph{exponential} lower bound for the model of \emph{sum of read-once oblivious algebraic branching programs} (ROABPs). In particular, we give an explicit polynomial such that any sum of ROABPs (equivalently, sum of \emph{ordered} set-multilinear branching programs, each with a possibly different ordering) computing it must have exponential size. This result generalizes the seminal work of Nisan (STOC 1991), which proved an exponential lower bound for a single ROABP. It also strengthens the work of Arvind and Raja (Chic. J. Theor. Comput. Sci., 2016), as well as the work of Bhargav, Dwivedi, and Saxena (2023), both of which established lower bounds against certain restricted versions of this model, and strongly answers an open question from both papers that asked to prove super-polynomial lower bounds for the corresponding \emph{unrestricted} model. The significance of our lower bounds is underscored by the recent work of Bhargav, Dwivedi, and Saxena (2023), which showed that super-polynomial lower bounds against a sum of ordered set-multilinear branching programs -- for a polynomial of sufficiently low degree -- would imply super-polynomial lower bounds against general ABPs, thereby resolving Valiant's longstanding conjecture that the permanent polynomial can not be computed efficiently by ABPs. More precisely, their work shows that if one could obtain such lower bounds when the degree is bounded by $O(\log n/ \log \log n)$, then it would imply super-polynomial lower bounds against general ABPs. In this paper, we show super-polynomial lower bounds against this model for a polynomial whose degree is as small as $\omega(\log n)$. Prior to our work, showing such lower bounds was open \emph{irrespective} of the assumption on the degree.

We study three levels in a hierarchy of nondeterminism: A nondeterministic automaton $\mathcal{A}$ is determinizable by pruning (DBP) if we can obtain a deterministic automaton equivalent to $\mathcal{A}$ by removing some of its transitions. Then, $\mathcal{A}$ is history deterministic (HD) if its nondeterministic choices can be resolved in a way that only depends on the past. Finally, $\mathcal{A}$ is semantically deterministic (SD) if different nondeterministic choices in $\mathcal{A}$ lead to equivalent states. Some applications of automata in formal methods require deterministic automata, yet in fact can use automata with some level of nondeterminism. For example, DBP automata are useful in the analysis of online algorithms, and HD automata are useful in synthesis and control. For automata on finite words, the three levels in the hierarchy coincide. We study the hierarchy for B\"uchi, co-B\"uchi, and weak automata on infinite words. We show that the hierarchy is strict, study the expressive power of the different levels in it, as well as the complexity of deciding the membership of a language in a given level. Finally, we describe a probability-based analysis of the hierarchy, which relates the level of nondeterminism with the probability that a random run on a word in the language is accepting. We relate the latter to nondeterministic automata that can be used when reasoning about probabilistic systems.

We study the following combinatorial problem. Given a set of $n$ y-monotone \emph{wires}, a \emph{tangle} determines the order of the wires on a number of horizontal \emph{layers} such that the orders of the wires on any two consecutive layers differ only in swaps of neighboring wires. Given a multiset~$L$ of \emph{swaps} (that is, unordered pairs of wires) and an initial order of the wires, a tangle \emph{realizes}~$L$ if each pair of wires changes its order exactly as many times as specified by~$L$. \textsc{List-Feasibility} is the problem of finding a tangle that realizes a given list~$L$ if such a tangle exists. \textsc{Tangle-Height Minimization} is the problem of finding a tangle that realizes a given list and additionally uses the minimum number of layers. \textsc{List-Feasibility} (and therefore \textsc{Tangle-Height Minimization}) is NP-hard [Yamanaka, Horiyama, Uno, Wasa; CCCG 2018]. We prove that \textsc{List-Feasibility} remains NP-hard if every pair of wires swaps only a constant number of times. On the positive side, we present an algorithm for \textsc{Tangle-Height Minimization} that computes an optimal tangle for $n$ wires and a given list~$L$ of swaps in $O((2|L|/n^2+1)^{n^2/2} \cdot \varphi^n \cdot n)$ time, where $\varphi \approx 1.618$ is the golden ratio and $|L|$ is the total number of swaps in~$L$. From this algorithm, we derive a simpler and faster version to solve \textsc{List-Feasibility}. We also use the algorithm to show that \textsc{List-Feasibility} is in NP and fixed-parameter tractable with respect to the number of wires. For \emph{simple} lists, where every swap occurs at most once, we show how to solve \textsc{Tangle-Height Minimization} in $O(n!\varphi^n)$ time.

We show an $O(n)$-time reduction from the problem of testing whether a multiset of positive integers can be partitioned into two multisets so that the sum of the integers in each multiset is equal to $n/2$ to the problem of testing whether an $n$-vertex biconnected outerplanar DAG admits an upward planar drawing. This constitutes the first barrier to the existence of efficient algorithms for testing the upward planarity of DAGs with no large triconnected minor. We also show a result in the opposite direction. Suppose that partitioning a multiset of positive integers into two multisets so that the sum of the integers in each multiset is $n/2$ can be solved in $f(n)$ time. Let $G$ be an $n$-vertex biconnected outerplanar DAG and $e$ be an edge incident to the outer face of an outerplanar drawing of $G$. Then it can be tested in $O(f(n))$ time whether $G$ admits an upward planar drawing with $e$ on the outer face.

In this article, we present a construction of a spanner on a set of $n$ points in $\mathbf{R}^d$ that we call a heavy path WSPD spanner. The construction is parameterized by a constant $s > 2$ called the separation ratio. The size of the graph is $O(s^dn)$ and the spanning ratio is at most $1 + 2/s + 2/(s - 1)$. We also show that this graph has a hop spanning ratio of at most $2\lg n + 1$. We present a memoryless local routing algorithm for heavy path WSPD spanners. The routing algorithm requires a vertex $v$ of the graph to store $O(\mathrm{deg}(v)\log n)$ bits of information, where $\mathrm{deg}(v)$ is the degree of $v$. The routing ratio is at most $1 + 4/s + 1/(s - 1)$ and at least $1 + 4/s$ in the worst case. The number of edges on the routing path is bounded by $2\lg n + 1$. We then show that the heavy path WSPD spanner can be constructed in metric spaces of bounded doubling dimension. These metric spaces have been studied in computational geometry as a generalization of Euclidean space. We show that, in a metric space with doubling dimension $\lambda$, the heavy path WSPD spanner has size $O(s^\lambda n)$ where $s$ is the separation ratio. The spanning ratio and hop spanning ratio are the same as in the Euclidean case. Finally, we show that the local routing algorithm works in the bounded doubling dimension case. The vertices require the same amount of storage, but the routing ratio becomes at most $1 + (2 + \frac{\tau}{\tau-1})/s + 1/(s - 1)$ in the worst case, where $\tau \ge 11$ is a constant related to the doubling dimension.