arXiv subjects

Manuela Fischer

Publications and source records attributed to Manuela Fischer.

14 recordsLinked to original sources

arXiv2023

Deterministic Massively Parallel Symmetry Breaking for Sparse Graphs

We consider the problem of designing deterministic graph algorithms for the model of Massively Parallel Computation (MPC) that improve with the sparsity of the input graph, as measured by the notion of arboricity. For the problems of maximal independent set (MIS), maximal matching (MM), and vertex coloring, we improve the state of the art as follows. Let $\lambda$ denote the arboricity of the $n$-node input graph with maximum degree $\Delta$. MIS and MM: We develop a deterministic low-space MPC algorithm that reduces the maximum degree to $poly(\lambda)$ in $O(\log \log n)$ rounds, improving and simplifying the randomized $O(\log \log n)$-round $poly(\max(\lambda, \log n))$-degree reduction of Ghaffari, Grunau, Jin [DISC'20]. Our approach when combined with the state-of-the-art $O(\log \Delta + \log \log n)$-round algorithm by Czumaj, Davies, Parter [SPAA'20, TALG'21] leads to an improved deterministic round complexity of $O(\log \lambda + \log \log n)$ for MIS and MM in low-space MPC. We also extend above MIS and MM algorithms to work with linear global memory. Specifically, we show that both problems can be solved in deterministic time $O(\min(\log n, \log \lambda \cdot \log \log n))$, and even in $O(\log \log n)$ time for graphs with arboricity at most $\log^{O(1)} \log n$. In this setting, only a $O(\log^2 \log n)$-running time bound for trees was known due to Latypov and Uitto [ArXiv'21]. Vertex Coloring: We present a $O(1)$-round deterministic algorithm for the problem of $O(\lambda)$-coloring in linear-memory MPC with relaxed global memory of $n \cdot poly(\lambda)$ that solves the problem after just one single graph partitioning step. This matches the state-of-the-art randomized round complexity by Ghaffari and Sayyadi [ICALP'19] and improves upon the deterministic $O(\lambda^{\epsilon})$-round algorithm by Barenboim and Khazanov [CSR'18].

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arXiv2022

Fast Distributed Brooks' Theorem

We give a randomized $\Delta$-coloring algorithm in the LOCAL model that runs in $\text{poly} \log \log n$ rounds, where $n$ is the number of nodes of the input graph and $\Delta$ is its maximum degree. This means that randomized $\Delta$-coloring is a rare distributed coloring problem with an upper and lower bound in the same ballpark, $\text{poly}\log\log n$, given the known $\Omega(\log_\Delta\log n)$ lower bound [Brandt et al., STOC '16]. Our main technical contribution is a constant time reduction to a constant number of $(\text{deg}+1)$-list coloring instances, for $\Delta = \omega(\log^4 n)$, resulting in a $\text{poly} \log\log n$-round CONGEST algorithm for such graphs. This reduction is of independent interest for other settings, including providing a new proof of Brooks' theorem for high degree graphs, and leading to a constant-round Congested Clique algorithm in such graphs. When $\Delta=\omega(\log^{21} n)$, our algorithm even runs in $O(\log^* n)$ rounds, showing that the base in the $\Omega(\log_\Delta\log n)$ lower bound is unavoidable. Previously, the best LOCAL algorithm for all considered settings used a logarithmic number of rounds. Our result is the first CONGEST algorithm for $\Delta$-coloring non-constant degree graphs.

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arXiv2022

Exponential Speedup Over Locality in MPC with Optimal Memory

Locally Checkable Labeling (LCL) problems are graph problems in which a solution is correct if it satisfies some given constraints in the local neighborhood of each node. Example problems in this class include maximal matching, maximal independent set, and coloring problems. A successful line of research has been studying the complexities of LCL problems on paths/cycles, trees, and general graphs, providing many interesting results for the LOCAL model of distributed computing. In this work, we initiate the study of LCL problems in the low-space Massively Parallel Computation (MPC) model. In particular, on forests, we provide a method that, given the complexity of an LCL problem in the LOCAL model, automatically provides an exponentially faster algorithm for the low-space MPC setting that uses optimal global memory, that is, truly linear. While restricting to forests may seem to weaken the result, we emphasize that all known (conditional) lower bounds for the MPC setting are obtained by lifting lower bounds obtained in the distributed setting in tree-like networks (either forests or high girth graphs), and hence the problems that we study are challenging already on forests. Moreover, the most important technical feature of our algorithms is that they use optimal global memory, that is, memory linear in the number of edges of the graph. In contrast, most of the state-of-the-art algorithms use more than linear global memory. Further, they typically start with a dense graph, sparsify it, and then solve the problem on the residual graph, exploiting the relative increase in global memory. On forests, this is not possible, because the given graph is already as sparse as it can be, and using optimal memory requires new solutions.

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arXiv2022

Improved Deterministic Connectivity in Massively Parallel Computation

A long line of research about connectivity in the Massively Parallel Computation model has culminated in the seminal works of Andoni et al. [FOCS'18] and Behnezhad et al. [FOCS'19]. They provide a randomized algorithm for low-space MPC with conjectured to be optimal round complexity $O(\log D + \log \log_{\frac m n} n)$ and $O(m)$ space, for graphs on $n$ vertices with $m$ edges and diameter $D$. Surprisingly, a recent result of Coy and Czumaj [STOC'22] shows how to achieve the same deterministically. Unfortunately, however, their algorithm suffers from large local computation time. We present a deterministic connectivity algorithm that matches all the parameters of the randomized algorithm and, in addition, significantly reduces the local computation time to nearly linear. Our derandomization method is based on reducing the amount of randomness needed to allow for a simpler efficient search. While similar randomness reduction approaches have been used before, our result is not only strikingly simpler, but it is the first to have efficient local computation. This is why we believe it to serve as a starting point for the systematic development of computation-efficient derandomization approaches in low-memory MPC.

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arXiv2021

Triangle resilience of the square of a Hamilton cycle in random graphs

Since first introduced by Sudakov and Vu in 2008, the study of resilience problems in random graphs received a lot of attention in probabilistic combinatorics. Of particular interest are resilience problems of spanning structures. It is known that for spanning structures which contain many triangles, local resilience cannot prevent an adversary from destroying all copies of the structure by removing a negligible amount of edges incident to every vertex. In this paper we generalise the notion of local resilience to $H$-resilience and demonstrate its usefulness on the containment problem of the square of a Hamilton cycle. In particular, we show that there exists a constant $C > 0$ such that if $p \geq C\log^3 n/\sqrt{n}$ then w.h.p. in every subgraph $G$ of a random graph $G_{n, p}$ there exists the square of a Hamilton cycle, provided that every vertex of $G$ remains on at least a $(4/9 + o(1))$-fraction of its triangles from $G_{n, p}$. The constant $4/9$ is optimal and the value of $p$ slightly improves on the best-known appearance threshold of such a structure and is optimal up to the logarithmic factor.

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arXiv2021

Deterministic $(1+\varepsilon)$-Approximate Maximum Matching with $\mathsf{poly}(1/\varepsilon)$ Passes in the Semi-Streaming Model and Beyond

We present a deterministic $(1+\varepsilon)$-approximate maximum matching algorithm in $\mathsf{poly} 1/\varepsilon$ passes in the semi-streaming model, solving the long-standing open problem of breaking the exponential barrier in the dependence on $1/\varepsilon$. Our algorithm exponentially improves on the well-known randomized $(1/\varepsilon)^{O(1/\varepsilon)}$-pass algorithm from the seminal work by McGregor~[APPROX05], the recent deterministic algorithm by Tirodkar with the same pass complexity~[FSTTCS18]. Up to polynomial factors in $1/\varepsilon$, our work matches the state-of-the-art deterministic $(\log n / \log \log n) \cdot (1/\varepsilon)$-pass algorithm by Ahn and Guha~[TOPC18], that is allowed a dependence on the number of nodes $n$. Our result also makes progress on the Open Problem 60 at sublinear.info. Moreover, we design a general framework that simulates our approach for the streaming setting in other models of computation. This framework requires access to an algorithm computing an $O(1)$-approximate maximum matching and an algorithm for processing disjoint $(\mathsf{poly} 1 / \varepsilon)$-size connected components. Instantiating our framework in $\mathsf{CONGEST}$ yields a $\mathsf{poly}(\log{n}, 1/\varepsilon)$ round algorithm for computing $(1+\varepsilon$)-approximate maximum matching. In terms of the dependence on $1/\varepsilon$, this result improves exponentially state-of-the-art result by Lotker, Patt-Shamir, and Pettie~[LPSP15]. Our framework leads to the same quality of improvement in the context of the Massively Parallel Computation model as well.

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arXiv2019

Breaking the Linear-Memory Barrier in MPC: Fast MIS on Trees with Strongly Sublinear Memory

Recently, studying fundamental graph problems in the \emph{Massively Parallel Computation (MPC) framework, inspired by the MapReduce paradigm, has gained a lot of attention. An assumption common to a vast majority of approaches is to allow $\widetildeΩ(n)$ memory per machine, where $n$ is the number of nodes in the graph and $\widetildeΩ$ hides polylogarithmic factors. However, as pointed out by Karloff et al. [SODA'10] and Czumaj et al. [STOC'18], it might be unrealistic for a single machine to have linear or only slightly sublinear memory. In this paper, we thus study a more practical variant of the MPC model which only requires substantially sublinear or even subpolynomial memory per machine. In contrast to the linear-memory MPC model and also to streaming algorithms, in this low-memory MPC setting, a single machine will only see a small number of nodes in the graph. We introduce a new and strikingly simple technique to cope with this imposed locality. In particular, we show that the Maximal Independent Set (MIS) problem can be solved efficiently, that is, in $O(\log^3 \log n)$ rounds, when the input graph is a tree. This constitutes an almost exponential speed-up over the low-memory MPC algorithm in $O(\sqrt{\log n})$-algorithm in a concurrent work by Ghaffari and Uitto [SODA'19] and substantially reduces the local memory from $\widetildeΩ(n)$ required by the recent $O(\log \log n)$-round MIS algorithm of Ghaffari et al. [PODC'18] to $n^α$ for any $α>0$, without incurring a significant loss in the round complexity. Moreover, it demonstrates how to make use of the all-to-all communication in the MPC model to almost exponentially improve on the corresponding bound in the $\mathsf{LOCAL}$ and $\mathsf{PRAM}$ models by Lenzen and Wattenhofer [PODC'11].

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arXiv2019

Tight Analysis of Randomized Greedy MIS

We provide a tight analysis which settles the round complexity of the well-studied parallel randomized greedy MIS algorithm, thus answering the main open question of Blelloch, Fineman, and Shun [SPAA'12]. The parallel/distributed randomized greedy Maximal Independent Set (MIS) algorithm works as follows. An order of the vertices is chosen uniformly at random. Then, in each round, all vertices that appear before their neighbors in the order are added to the independent set and removed from the graph along with their neighbors. The main question of interest is the number of rounds it takes until the graph is empty. This algorithm has been studied since 1987, initiated by Coppersmith, Raghavan, and Tompa [FOCS'87], and the previously best known bounds were $O(\log n)$ rounds in expectation for Erdős-Rényi random graphs by Calkin and Frieze [Random Struc. \& Alg. '90] and $O(\log^2 n)$ rounds with high probability for general graphs by Blelloch, Fineman, and Shun [SPAA'12]. We prove a high probability upper bound of $O(\log n)$ on the round complexity of this algorithm in general graphs, and that this bound is tight. This also shows that parallel randomized greedy MIS is as fast as the celebrated algorithm of Luby [STOC'85, JALG'86].

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arXiv2018

The Complexity of $(Δ+ 1)$Coloring inCongested Clique, Massively Parallel Computation,and Centralized Local Computation

We present new randomized algorithms that improve the complexity of the classic $(Δ+1)$-coloring problem, and its generalization $(Δ+1)$-list-coloring, in three well-studied models of distributed, parallel, and centralized computation: Distributed Congested Clique: We present an $O(1)$-round randomized algorithm for $(Δ+1)$-list coloring in the congested clique model of distributed computing. This settles the asymptotic complexity of this problem. It moreover improves upon the $O(\log^\ast Δ)$-round randomized algorithms of Parter and Su [DISC'18] and $O((\log\log Δ)\cdot \log^\ast Δ)$-round randomized algorithm of Parter [ICALP'18]. Massively Parallel Computation: We present a $(Δ+1)$-list coloring algorithm with round complexity $O(\sqrt{\log\log n})$ in the Massively Parallel Computation (MPC) model with strongly sublinear memory per machine. This algorithm uses a memory of $O(n^α)$ per machine, for any desirable constant $α>0$, and a total memory of $\widetilde{O}(m)$, where $m$ is the size of the graph. Notably, this is the first coloring algorithm with sublogarithmic round complexity, in the sublinear memory regime of MPC. For the quasilinear memory regime of MPC, an $O(1)$-round algorithm was given very recently by Assadi et al. [SODA'19]. Centralized Local Computation: We show that $(Δ+1)$-list coloring can be solved with $Δ^{O(1)} \cdot O(\log n)$ query complexity, in the centralized local computation model. The previous state-of-the-art for $(Δ+1)$-list coloring in the centralized local computation model are based on simulation of known LOCAL algorithms.

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arXiv2018

Matching and MIS for Uniformly Sparse Graphs in the Low-Memory MPC Model

The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale parallel computation frameworks and has recently gained a lot of importance, especially in the context of classic graph problems. Unsatisfactorily, all current $\text{poly} (\log \log n)$-round MPC algorithms seem to get fundamentally stuck at the linear-memory barrier: their efficiency crucially relies on each machine having space at least linear in the number $n$ of nodes. As this might not only be prohibitively large, but also allows for easy if not trivial solutions for sparse graphs, we are interested in the low-memory MPC model, where the space per machine is restricted to be strongly sublinear, that is, $n^δ$ for any $0<δ<1$. We devise a degree reduction technique that reduces maximal matching and maximal independent set in graphs with arboricity $λ$ to the corresponding problems in graphs with maximum degree $\text{poly}(λ)$ in $O(\log^2 \log n)$ rounds. This gives rise to $O\left(\log^2\log n + T(\text{poly} λ)\right)$-round algorithms, where $T(Δ)$ is the $Δ$-dependency in the round complexity of maximal matching and maximal independent set in graphs with maximum degree $Δ$. A concurrent work by Ghaffari and Uitto shows that $T(Δ)=O(\sqrt{\log Δ})$. For graphs with arboricity $λ=\text{poly}(\log n)$, this almost exponentially improves over Luby's $O(\log n)$-round PRAM algorithm [STOC'85, JALG'86], and constitutes the first $\text{poly} (\log \log n)$-round maximal matching algorithm in the low-memory MPC model, thus breaking the linear-memory barrier. Previously, the only known subpolylogarithmic algorithm, due to Lattanzi et al. [SPAA'11], required strongly superlinear, that is, $n^{1+Ω(1)}$, memory per machine.

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arXiv2018

A Simple Parallel and Distributed Sampling Technique: Local Glauber Dynamics

\emph{Sampling} constitutes an important tool in a variety of areas: from machine learning and combinatorial optimization to computational physics and biology. A central class of sampling algorithms is the \emph{Markov Chain Monte Carlo} method, based on the construction of a Markov chain with the desired sampling distribution as its stationary distribution. Many of the traditional Markov chains, such as the \emph{Glauber dynamics}, do not scale well with increasing dimension. To address this shortcoming, we propose a simple local update rule based on the Glauber dynamics that leads to efficient parallel and distributed algorithms for sampling from Gibbs distributions. Concretely, we present a Markov chain that mixes in $O(\log n)$ rounds when Dobrushin's condition for the Gibbs distribution is satisfied. This improves over the \emph{LubyGlauber} algorithm by Feng, Sun, and Yin [PODC'17], which needs $O(Δ\log n)$ rounds, and their \emph{LocalMetropolis} algorithm, which converges in $O(\log n)$ rounds but requires a considerably stronger mixing condition. Here, $n$ denotes the number of nodes in the graphical model inducing the Gibbs distribution, and $Δ$ its maximum degree. In particular, our method can sample a uniform proper coloring with $αΔ$ colors in $O(\log n)$ rounds for any $α>2$, which almost matches the threshold of the sequential Glauber dynamics and improves on the $α>2 +\sqrt{2}$ threshold of Feng et al.

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arXiv2017

Improved Deterministic Distributed Matching via Rounding

We present improved deterministic distributed algorithms for a number of well-studied matching problems, which are simpler, faster, more accurate, and/or more general than their known counterparts. The common denominator of these results is a deterministic distributed rounding method for certain linear programs, which is the first such rounding method, to our knowledge. A sampling of our end results is as follows. -- An $O(\log^2 Δ\cdot \log n)$-round deterministic distributed algorithm for computing a maximal matching, in $n$-node graphs with maximum degree $Δ$. This is the first improvement in about 20 years over the celebrated $O(\log^4 n)$-round algorithm of Hańćkowiak, Karoński, and Panconesi [SODA'98, PODC'99]. -- A deterministic distributed algorithm for computing a $(2+\varepsilon)$-approximation of maximum matching in $O(\log^2 Δ\cdot \log \frac{1}{\varepsilon} + \log^ * n)$ rounds. This is exponentially faster than the classic $O(Δ+\log^* n)$-round $2$-approximation of Panconesi and Rizzi [DIST'01]. With some modifications, the algorithm can also find an $\varepsilon$-maximal matching which leaves only an $\varepsilon$-fraction of the edges on unmatched nodes. -- An $O(\log^2 Δ\cdot \log \frac{1}{\varepsilon} + \log^ * n)$-round deterministic distributed algorithm for computing a $(2+\varepsilon)$-approximation of a maximum weighted matching, and also for the more general problem of maximum weighted $b$-matching. These improve over the $O(\log^4 n \cdot \log_{1+\varepsilon} W)$-round $(6+\varepsilon)$-approximation algorithm of Panconesi and Sozio [DIST'10], where $W$ denotes the maximum normalized weight.

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arXiv2017

Sublogarithmic Distributed Algorithms for Lovász Local lemma, and the Complexity Hierarchy

Locally Checkable Labeling (LCL) problems include essentially all the classic problems of $\mathsf{LOCAL}$ distributed algorithms. In a recent enlightening revelation, Chang and Pettie [arXiv 1704.06297] showed that any LCL (on bounded degree graphs) that has an $o(\log n)$-round randomized algorithm can be solved in $T_{LLL}(n)$ rounds, which is the randomized complexity of solving (a relaxed variant of) the Lovász Local Lemma (LLL) on bounded degree $n$-node graphs. Currently, the best known upper bound on $T_{LLL}(n)$ is $O(\log n)$, by Chung, Pettie, and Su [PODC'14], while the best known lower bound is $Ω(\log\log n)$, by Brandt et al. [STOC'16]. Chang and Pettie conjectured that there should be an $O(\log\log n)$-round algorithm. Making the first step of progress towards this conjecture, and providing a significant improvement on the algorithm of Chung et al. [PODC'14], we prove that $T_{LLL}(n)= 2^{O(\sqrt{\log\log n})}$. Thus, any $o(\log n)$-round randomized distributed algorithm for any LCL problem on bounded degree graphs can be automatically sped up to run in $2^{O(\sqrt{\log\log n})}$ rounds. Using this improvement and a number of other ideas, we also improve the complexity of a number of graph coloring problems (in arbitrary degree graphs) from the $O(\log n)$-round results of Chung, Pettie and Su [PODC'14] to $2^{O(\sqrt{\log\log n})}$. These problems include defective coloring, frugal coloring, and list vertex-coloring.

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arXiv2017

Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching

We present a deterministic distributed algorithm that computes a $(2Δ-1)$-edge-coloring, or even list-edge-coloring, in any $n$-node graph with maximum degree $Δ$, in $O(\log^7 Δ\log n)$ rounds. This answers one of the long-standing open questions of \emph{distributed graph algorithms} from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities were $2^{O(\sqrt{\log n})}$ by Panconesi and Srinivasan [STOC'92] and $\tilde{O}(\sqrtΔ) + O(\log^* n)$ by Fraigniaud, Heinrich, and Kosowski [FOCS'16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of $(2Δ-1)$-edge-coloring to poly$(\log\log n)$ rounds. The key technical ingredient is a deterministic distributed algorithm for \emph{hypergraph maximal matching}, which we believe will be of interest beyond this result. In any hypergraph of rank $r$ --- where each hyperedge has at most $r$ vertices --- with $n$ nodes and maximum degree $Δ$, this algorithm computes a maximal matching in $O(r^5 \log^{6+\log r } Δ\log n)$ rounds. This hypergraph matching algorithm and its extensions lead to a number of other results. In particular, a polylogarithmic-time deterministic distributed maximal independent set algorithm for graphs with bounded neighborhood independence, hence answering Open Problem 5 of Barenboim and Elkin's book, a $((\log Δ/\varepsilon)^{O(\log (1/\varepsilon))})$-round deterministic algorithm for $(1+\varepsilon)$-approximation of maximum matching, and a quasi-polylogarithmic-time deterministic distributed algorithm for orienting $λ$-arboricity graphs with out-degree at most $(1+\varepsilon)λ$, for any constant $\varepsilon>0$, hence partially answering Open Problem 10 of Barenboim and Elkin's book.

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