Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [CW18] improved the state-of-the-art for light spanners by constructing a $(2k-1)(1+\epsilon)$-spanner with $O(n^{1+1/k})$ edges and $O_\epsilon(n^{1/k})$ lightness. Soon after, Filtser and Solomon [FS19] showed that the classic greedy spanner construction achieves the same bounds The major drawback of the greedy spanner is its running time of $O(mn^{1+1/k})$ (which is faster than [CW16]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness $\Omega_\epsilon(kn^{1/k})$, even when randomization is used. The contribution of this paper is deterministic spanner constructions that are fast, and achieve similar bounds as the state-of-the-art slower constructions. Our first result is an $O_\epsilon(n^{2+1/k+\epsilon'})$ time spanner construction which achieves the state-of-the-art bounds. Our second result is an $O_\epsilon(m + n\log n)$ time construction of a spanner with $(2k-1)(1+\epsilon)$ stretch, $O(\log k\cdot n^{1+1/k})$ edges and $O_\epsilon(\log k\cdot n^{1/k})$ lightness. This is an exponential improvement in the dependence on $k$ compared to the previous result with such running time. Finally, for the important special case where $k=\log n$, for every constant $\epsilon>0$, we provide an $O(m+n^{1+\epsilon})$ time construction that produces an $O(\log n)$-spanner with $O(n)$ edges and $O(1)$ lightness which is asymptotically optimal. This is the first known sub-quadratic construction of such a spanner for any $k = \omega(1)$. To achieve our constructions, we show a novel deterministic incremental approximate distance oracle, which may be of independent interest.
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Graph Convolutional Networks (GCNs) are one of the most popular architectures that are used to solve classification problems accompanied by graphical information. We present a rigorous theoretical understanding of the effects of graph convolutions in multi-layer networks. We study these effects through the node classification problem of a non-linearly separable Gaussian mixture model coupled with a stochastic block model. First, we show that a single graph convolution expands the regime of the distance between the means where multi-layer networks can classify the data by a factor of at least $1/\sqrt[4]{\mathbb{E}{\rm deg}}$, where $\mathbb{E}{\rm deg}$ denotes the expected degree of a node. Second, we show that with a slightly stronger graph density, two graph convolutions improve this factor to at least $1/\sqrt[4]{n}$, where $n$ is the number of nodes in the graph. Finally, we provide both theoretical and empirical insights into the performance of graph convolutions placed in different combinations among the layers of a network, concluding that the performance is mutually similar for all combinations of the placement. We present extensive experiments on both synthetic and real-world data that illustrate our results.
While algorithms for planar graphs have received a lot of attention, few papers have focused on the additional power that one gets from assuming an embedding of the graph is available. While in the classic sequential setting, this assumption gives no additional power (as a planar graph can be embedded in linear time), we show that this is far from being the case in other settings. We assume that the embedding is straight-line, but our methods also generalize to non-straight-line embeddings. Specifically, we focus on sublinear-time computation and massively parallel computation (MPC). Our main technical contribution is a sublinear-time algorithm for computing a relaxed version of an $r$-division. We then show how this can be used to estimate Lipschitz additive graph parameters. This includes, for example, the maximum matching, maximum independent set, or the minimum dominating set. We also show how this can be used to solve some property testing problems with respect to the vertex edit distance. In the second part of our paper, we show an MPC algorithm that computes an $r$-division of the input graph. We show how this can be used to solve various classical graph problems with space per machine of $O(n^{2/3+\epsilon})$ for some $\epsilon>0$, and while performing $O(1)$ rounds. This includes for example approximate shortest paths or the minimum spanning tree. Our results also imply an improved MPC algorithm for Euclidean minimum spanning tree.
We present a sheaf-theoretic construction of shape space -- the space of all shapes. We do this by describing a homotopy sheaf on the poset category of constructible sets, where each set is mapped to its Persistent Homology Transform (PHT). Recent results that build on fundamental work of Schapira have shown that this transform is injective, thus making the PHT a good summary object for each shape. Our homotopy sheaf result allows us to "glue" PHTs of different shapes together to build up the PHT of a larger shape. In the case where our shape is a polyhedron we prove a generalized nerve lemma for the PHT. Finally, by re-examining the sampling result of Smale-Niyogi-Weinberger, we show that we can reliably approximate the PHT of a manifold by a polyhedron up to arbitrary precision.
We give a fast algorithm for sampling uniform solutions of general constraint satisfaction problems (CSPs) in a local lemma regime. The expected running time of our algorithm is near-linear in $n$ and a fixed polynomial in $\Delta$, where $n$ is the number of variables and $\Delta$ is the max degree of constraints. Previously, up to similar conditions, sampling algorithms with running time polynomial in both $n$ and $\Delta$, only existed for the almost atomic case, where each constraint is violated by a small number of forbidden local configurations. Our sampling approach departs from all previous fast algorithms for sampling LLL, which were based on Markov chains. A crucial step of our algorithm is a recursive marginal sampler that is of independent interests. Within a local lemma regime, this marginal sampler can draw a random value for a variable according to its marginal distribution, at a local cost independent of the size of the CSP.
We study the distributed minimum spanning tree (MST) problem, a fundamental problem in distributed computing. It is well-known that distributed MST can be solved in $\tilde{O}(D+\sqrt{n})$ rounds in the standard CONGEST model (where $n$ is the network size and $D$ is the network diameter) and this is essentially the best possible round complexity (up to logarithmic factors). However, in resource-constrained networks such as ad hoc wireless and sensor networks, nodes spending so much time can lead to significant spending of resources such as energy. Motivated by the above consideration, we study distributed algorithms for MST under the \emph{sleeping model} [Chatterjee et al., PODC 2020], a model for design and analysis of resource-efficient distributed algorithms. In the sleeping model, a node can be in one of two modes in any round -- \emph{sleeping} or \emph{awake} (unlike the traditional model where nodes are always awake). Only the rounds in which a node is \emph{awake} are counted, while \emph{sleeping} rounds are ignored. A node spends resources only in the awake rounds and hence the main goal is to minimize the \emph{awake complexity} of a distributed algorithm, the worst-case number of rounds any node is awake. We present deterministic and randomized distributed MST algorithms that have an \emph{optimal} awake complexity of $O(\log n)$ time with a matching lower bound. We also show that our randomized awake-optimal algorithm has essentially the best possible round complexity by presenting a lower bound of $\tilde{\Omega}(n)$ on the product of the awake and round complexity of any distributed algorithm (including randomized) that outputs an MST, where $\tilde{\Omega}$ hides a $1/(\text{polylog } n)$ factor.
Many existing algorithms for streaming geometric data analysis have been plagued by exponential dependencies in the space complexity, which are undesirable for processing high-dimensional data sets. In particular, once $d\geq\log n$, there are no known non-trivial streaming algorithms for problems such as maintaining convex hulls and L\"owner-John ellipsoids of $n$ points, despite a long line of work in streaming computational geometry since [AHV04]. We simultaneously improve these results to $\mathrm{poly}(d,\log n)$ bits of space by trading off with a $\mathrm{poly}(d,\log n)$ factor distortion. We achieve these results in a unified manner, by designing the first streaming algorithm for maintaining a coreset for $\ell_\infty$ subspace embeddings with $\mathrm{poly}(d,\log n)$ space and $\mathrm{poly}(d,\log n)$ distortion. Our algorithm also gives similar guarantees in the \emph{online coreset} model. Along the way, we sharpen results for online numerical linear algebra by replacing a log condition number dependence with a $\log n$ dependence, answering a question of [BDM+20]. Our techniques provide a novel connection between leverage scores, a fundamental object in numerical linear algebra, and computational geometry. For $\ell_p$ subspace embeddings, we give nearly optimal trade-offs between space and distortion for one-pass streaming algorithms. For instance, we give a deterministic coreset using $O(d^2\log n)$ space and $O((d\log n)^{1/2-1/p})$ distortion for $p>2$, whereas previous deterministic algorithms incurred a $\mathrm{poly}(n)$ factor in the space or the distortion [CDW18]. Our techniques have implications in the offline setting, where we give optimal trade-offs between the space complexity and distortion of subspace sketch data structures. To do this, we give an elementary proof of a "change of density" theorem of [LT80] and make it algorithmic.
This paper presents new deterministic and distributed low-diameter decomposition algorithms for weighted graphs. In particular, we show that if one can efficiently compute approximate distances in a parallel or a distributed setting, one can also efficiently compute low-diameter decompositions. This consequently implies solutions to many fundamental distance based problems using a polylogarithmic number of approximate distance computations. Our low-diameter decomposition generalizes and extends the line of work starting from [Rozho\v{n}, Ghaffari STOC 2020] to weighted graphs in a very model-independent manner. Moreover, our clustering results have additional useful properties, including strong-diameter guarantees, separation properties, restricting cluster centers to specified terminals, and more. Applications include: -- The first near-linear work and polylogarithmic depth randomized and deterministic parallel algorithm for low-stretch spanning trees (LSST) with polylogarithmic stretch. Previously, the best parallel LSST algorithm required $m \cdot n^{o(1)}$ work and $n^{o(1)}$ depth and was inherently randomized. No deterministic LSST algorithm with truly sub-quadratic work and sub-linear depth was known. -- The first near-linear work and polylogarithmic depth deterministic algorithm for computing an $\ell_1$-embedding into polylogarithmic dimensional space with polylogarithmic distortion. The best prior deterministic algorithms for $\ell_1$-embeddings either require large polynomial work or are inherently sequential. Even when we apply our techniques to the classical problem of computing a ball-carving with strong-diameter $O(\log^2 n)$ in an unweighted graph, our new clustering algorithm still leads to an improvement in round complexity from $O(\log^{10} n)$ rounds [Chang, Ghaffari PODC 21] to $O(\log^{4} n)$.
Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is better modelled as a weighted hypergraph that keeps evolving with time. This motivates the problem of maintaining the densest subhypergraph of a weighted hypergraph in a {\em dynamic setting}, where the input keeps changing via a sequence of updates (hyperedge insertions/deletions). Previously, the only known algorithm for this problem was due to Hu et al. [HWC17]. This algorithm worked only on unweighted hypergraphs, and had an approximation ratio of $(1+\epsilon)r^2$ and an update time of $O(\text{poly} (r, \log n))$, where $r$ denotes the maximum rank of the input across all the updates. We obtain a new algorithm for this problem, which works even when the input hypergraph is weighted. Our algorithm has a significantly improved (near-optimal) approximation ratio of $(1+\epsilon)$ that is independent of $r$, and a similar update time of $O(\text{poly} (r, \log n))$. It is the first $(1+\epsilon)$-approximation algorithm even for the special case of weighted simple graphs. To complement our theoretical analysis, we perform experiments with our dynamic algorithm on large-scale, real-world data-sets. Our algorithm significantly outperforms the state of the art [HWC17] both in terms of accuracy and efficiency.
Many mathematical objects can be represented as functors from finitely-presented categories $\mathsf{C}$ to $\mathsf{Set}$. For instance, graphs are functors to $\mathsf{Set}$ from the category with two parallel arrows. Such functors are known informally as $\mathsf{C}$-sets. In this paper, we describe and implement an extension of $\mathsf{C}$-sets having data attributes with fixed types, such as graphs with labeled vertices or real-valued edge weights. We call such structures "acsets," short for "attributed $\mathsf{C}$-sets." Derived from previous work on algebraic databases, acsets are a joint generalization of graphs and data frames. They also encompass more elaborate graph-like objects such as wiring diagrams and Petri nets with rate constants. We develop the mathematical theory of acsets and then describe a generic implementation in the Julia programming language, which uses advanced language features to achieve performance comparable with specialized data structures.
Most existing works of polar codes focus on the analysis of block error probability. However, in many scenarios, bit error probability is also important for evaluating the performance of channel codes. In this paper, we establish a new framework to analyze the bit error probability of polar codes. Specifically, by revisiting the error event of bit-channel, we first introduce the conditional bit error probability as a metric to evaluate the reliability of bit-channel for both systematic and non-systematic polar codes. Guided by the concept of polar subcode, we then derive an upper bound on the conditional bit error probability of each bit-channel, and accordingly, an upper bound on the bit error probability of polar codes. Based on these, two types of construction metrics aiming at minimizing the bit error probability of polar codes are proposed, which are of linear computational complexity and explicit forms. Simulation results show that the polar codes constructed by the proposed methods can outperform those constructed by the conventional methods.
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