A Simple Deterministic Distributed MST Algorithm, with Near-Optimal Time and Message Complexities

Distributed minimum spanning tree (MST) problem is one of the most central and fundamental problems in distributed graph algorithms. Garay et al. \cite{GKP98,KP98} devised an algorithm with running time $O(D + \sqrt{n} \cdot \log^* n)$, where $D$ is the hop-diameter of the input $n$-vertex $m$-edge graph, and with message complexity $O(m + n^{3/2})$. Peleg and Rubinovich \cite{PR99} showed that the running time of the algorithm of \cite{KP98} is essentially tight, and asked if one can achieve near-optimal running time **together with near-optimal message complexity**. In a recent breakthrough, Pandurangan et al. \cite{PRS16} answered this question in the affirmative, and devised a **randomized** algorithm with time $\tilde{O}(D+ \sqrt{n})$ and message complexity $\tilde{O}(m)$. They asked if such a simultaneous time- and message-optimality can be achieved by a **deterministic** algorithm. In this paper, building upon the work of \cite{PRS16}, we answer this question in the affirmative, and devise a **deterministic** algorithm that computes MST in time $O((D + \sqrt{n}) \cdot \log n)$, using $O(m \cdot \log n + n \log n \cdot \log^* n)$ messages. The polylogarithmic factors in the time and message complexities of our algorithm are significantly smaller than the respective factors in the result of \cite{PRS16}. Also, our algorithm and its analysis are very **simple** and self-contained, as opposed to rather complicated previous sublinear-time algorithms \cite{GKP98,KP98,E04b,PRS16}

The Densest k-Subhypergraph Problem

The Densest $k$-Subgraph (D$k$S) problem, and its corresponding minimization problem Smallest $p$-Edge Subgraph (S$p$ES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both D$k$S and S$p$ES from graphs to hypergraphs. We consider the Densest $k$-Subhypergraph problem (given a hypergraph $(V, E)$, find a subset $W\subseteq V$ of $k$ vertices so as to maximize the number of hyperedges contained in $W$) and define the Minimum $p$-Union problem (given a hypergraph, choose $p$ of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an $O(n^{4(4-\sqrt{3})/13 + \epsilon}) \leq O(n^{0.697831+\epsilon})$-approximation (for arbitrary constant $\epsilon > 0$) for Densest $k$-Subhypergraph and an $\tilde O(n^{2/5})$-approximation for Minimum $p$-Union. We also give an $O(\sqrt{m})$-approximation for Minimum $p$-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.Comment: 21 page

Almost Shortest Paths with Near-Additive Error in Weighted Graphs

Let $G=(V,E,w)$ be a weighted undirected graph with $n$ vertices and $m$ edges, and fix a set of $s$ sources $S\subseteq V$. We study the problem of computing {\em almost shortest paths} (ASP) for all pairs in $S \times V$ in both classical centralized and parallel (PRAM) models of computation. Consider the regime of multiplicative approximation of $1+\epsilon$, for an arbitrarily small constant $\epsilon > 0$ . In this regime existing centralized algorithms require $\Omega(\min\{|E|s,n^\omega\})$ time, where $\omega < 2.372$ is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka time) require work $\Omega(\min\{|E|s,n^\omega\})$. Our centralized algorithm has running time $O((m+ ns)n^\rho)$, and its PRAM counterpart has polylogarithmic depth and work $O((m + ns)n^\rho)$, for an arbitrarily small constant $\rho > 0$. For a pair $(s,v) \in S\times V$, it provides a path of length $\hat{d}(s,v)$ that satisfies $\hat{d}(s,v) \le (1+\epsilon)d_G(s,v) + \beta \cdot W(s,v)$, where $W(s,v)$ is the weight of the heaviest edge on some shortest $s-v$ path. Hence our additive term depends linearly on a {\em local} maximum edge weight, as opposed to the global maximum edge weight in previous works. Finally, our $\beta = (1/\rho)^{O(1/\rho)}$. We also extend a centralized algorithm of Dor et al. \cite{DHZ00}. For a parameter $\kappa = 1,2,\ldots$, this algorithm provides for {\em unweighted} graphs a purely additive approximation of $2(\kappa -1)$ for {\em all pairs shortest paths} (APASP) in time $\tilde{O}(n^{2+1/\kappa})$. Within the same running time, our algorithm for {\em weighted} graphs provides a purely additive error of $2(\kappa - 1) W(u,v)$, for every vertex pair $(u,v) \in {V \choose 2}$, with $W(u,v)$ defined as above. On the way to these results we devise a suit of novel constructions of spanners, emulators and hopsets

Bias vs structure of polynomials in large fields, and applications in effective algebraic geometry and coding theory

Let $f$ be a polynomial of degree $d$ in $n$ variables over a finite field $\mathbb{F}$. The polynomial is said to be unbiased if the distribution of $f(x)$ for a uniform input $x \in \mathbb{F}^n$ is close to the uniform distribution over $\mathbb{F}$, and is called biased otherwise. The polynomial is said to have low rank if it can be expressed as a composition of a few lower degree polynomials. Green and Tao [Contrib. Discrete Math 2009] and Kaufman and Lovett [FOCS 2008] showed that bias implies low rank for fixed degree polynomials over fixed prime fields. This lies at the heart of many tools in higher order Fourier analysis. In this work, we extend this result to all prime fields (of size possibly growing with $n$). We also provide a generalization to nonprime fields in the large characteristic case. However, we state all our applications in the prime field setting for the sake of simplicity of presentation. As an immediate application, we obtain improved bounds for a suite of problems in effective algebraic geometry, including Hilbert nullstellensatz, radical membership and counting rational points in low degree varieties. Using the above generalization to large fields as a starting point, we are also able to settle the list decoding radius of fixed degree Reed-Muller codes over growing fields. The case of fixed size fields was solved by Bhowmick and Lovett [STOC 2015], which resolved a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008]. Here, we show that the list decoding radius is equal the minimum distance of the code for all fixed degrees, even when the field size is possibly growing with $n$