arXiv subjects

Mohsen Ghaffari

Publications and source records attributed to Mohsen Ghaffari.

At least 55 records · Page 3Linked to original sources

arXiv2022

Near-Optimal Distributed Dominating Set in Bounded Arboricity Graphs

We describe a simple deterministic $O( \varepsilon^{-1} \log Δ)$ round distributed algorithm for $(2α+1)(1 + \varepsilon)$ approximation of minimum weighted dominating set on graphs with arboricity at most $α$. Here $Δ$ denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation [Kuhn, Moscibroda, and Wattenhofer JACM'16]. Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized $O(α^2)$ approximation in $O(\log n)$ rounds [Lenzen and Wattenhofer DISC'10], a deterministic $O(α\log Δ)$ approximation in $O(\log Δ)$ rounds [Lenzen and Wattenhofer DISC'10], a deterministic $O(α)$ approximation in $O(\log^2 Δ)$ rounds [implicit in Bansal and Umboh IPL'17 and Kuhn, Moscibroda, and Wattenhofer SODA'06], and a randomized $O(α)$ approximation in $O(α\log n)$ rounds [Morgan, Solomon and Wein DISC'21]. We also provide a randomized $O(α\logΔ)$ round distributed algorithm that sharpens the approximation factor to $α(1+o(1))$. If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve $α- 1 - \varepsilon$ approximation [Bansal and Umboh IPL'17].

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arXiv2022

Universally-Optimal Distributed Exact Min-Cut

We present a universally-optimal distributed algorithm for the exact weighted min-cut. The algorithm is guaranteed to complete in $\widetilde{O}(D + \sqrt{n})$ rounds on every graph, recovering the recent result of Dory, Efron, Mukhopadhyay, and Nanongkai~[STOC'21], but runs much faster on structured graphs. Specifically, the algorithm completes in $\widetilde{O}(D)$ rounds on (weighted) planar graphs or, more generally, any (weighted) excluded-minor family. We obtain this result by designing an aggregation-based algorithm: each node receives only an aggregate of the messages sent to it. While somewhat restrictive, recent work shows any such black-box algorithm can be simulated on any minor of the communication network. Furthermore, we observe this also allows for the addition of (a small number of) arbitrarily-connected virtual nodes to the network. We leverage these capabilities to design a min-cut algorithm that is significantly simpler compared to prior distributed work. We hope this paper showcases how working within this paradigm yields simple-to-design and ultra-efficient distributed algorithms for global problems. Our main technical contribution is a distributed algorithm that, given any tree $T$, computes the minimum cut that $2$-respects $T$ (i.e., cuts at most $2$ edges of $T$) in universally near-optimal time. Moreover, our algorithm gives a \emph{deterministic} $\widetilde{O}(D)$-round 2-respecting cut solution for excluded-minor families and a \emph{deterministic} $\widetilde{O}(D + \sqrt{n})$-round solution for general graphs, the latter resolving a question of Dory, et al.~[STOC'21]

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arXiv2022

Improved Massively Parallel Computation Algorithms for MIS, Matching, and Vertex Cover

We present $O(\log\log n)$-round algorithms in the Massively Parallel Computation (MPC) model, with $\tilde{O}(n)$ memory per machine, that compute a maximal independent set, a $1+ε$ approximation of maximum matching, and a $2+ε$ approximation of minimum vertex cover, for any $n$-vertex graph and any constant $ε>0$. These improve the state of the art as follows: - Our MIS algorithm leads to a simple $O(\log\log Δ)$-round MIS algorithm in the Congested Clique model of distributed computing, which improves on the $\tilde{O}(\sqrt{\log Δ})$-round algorithm of Ghaffari [PODC'17]. - Our $O(\log\log n)$-round $(1+ε)$-approximate maximum matching algorithm simplifies or improves on the following prior work: $O(\log^2\log n)$-round $(1+ε)$-approximation algorithm of Czumaj et al. [STOC'18] and $O(\log\log n)$-round $(1+ε)$-approximation algorithm of Assadi et al. [SODA'19]. - Our $O(\log\log n)$-round $(2+ε)$-approximate minimum vertex cover algorithm improves on an $O(\log\log n)$-round $O(1)$-approximation of Assadi et al. [arXiv'17].

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arXiv2021

Deterministic Distributed Vertex Coloring: Simpler, Faster, and without Network Decomposition

We present a simple deterministic distributed algorithm that computes a $(Δ+1)$-vertex coloring in $O(\log^2 Δ\cdot \log n)$ rounds. The algorithm can be implemented with $O(\log n)$-bit messages. The algorithm can also be extended to the more general $(degree+1)$-list coloring problem. Obtaining a polylogarithmic-time deterministic algorithm for $(Δ+1)$-vertex coloring had remained a central open question in the area of distributed graph algorithms since the 1980s, until a recent network decomposition algorithm of Rozhoň and Ghaffari [STOC'20]. The current state of the art is based on an improved variant of their decomposition, which leads to an $O(\log^5 n)$-round algorithm for $(Δ+1)$-vertex coloring. Our coloring algorithm is completely different and considerably simpler and faster. It solves the coloring problem in a direct way, without using network decomposition, by gradually rounding a certain fractional color assignment until reaching an integral color assignments. Moreover, via the approach of Chang, Li, and Pettie [STOC'18], this improved deterministic algorithm also leads to an improvement in the complexity of randomized algorithms for $(Δ+1)$-coloring, now reaching the bound of $O(\log^3\log n)$ rounds. As a further application, we also provide faster deterministic distributed algorithms for the following variants of the vertex coloring problem. In graphs of arboricity $a$, we show that a $(2+ε)a$-vertex coloring can be computed in $O(\log^3 a\cdot\log n)$ rounds. We also show that for $Δ\geq 3$, a $Δ$-coloring of a $Δ$-colorable graph $G$ can be computed in $O(\log^2 Δ\cdot\log^2 n)$ rounds.

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arXiv2021

Massively Parallel Algorithms for Distance Approximation and Spanners

Over the past decade, there has been increasing interest in distributed/parallel algorithms for processing large-scale graphs. By now, we have quite fast algorithms -- usually sublogarithmic-time and often $poly(\log\log n)$-time, or even faster -- for a number of fundamental graph problems in the massively parallel computation (MPC) model. This model is a widely-adopted theoretical abstraction of MapReduce style settings, where a number of machines communicate in an all-to-all manner to process large-scale data. Contributing to this line of work on MPC graph algorithms, we present $poly(\log k) \in poly(\log\log n)$ round MPC algorithms for computing $O(k^{1+{o(1)}})$-spanners in the strongly sublinear regime of local memory. To the best of our knowledge, these are the first sublogarithmic-time MPC algorithms for spanner construction. As primary applications of our spanners, we get two important implications, as follows: -For the MPC setting, we get an $O(\log^2\log n)$-round algorithm for $O(\log^{1+o(1)} n)$ approximation of all pairs shortest paths (APSP) in the near-linear regime of local memory. To the best of our knowledge, this is the first sublogarithmic-time MPC algorithm for distance approximations. -Our result above also extends to the Congested Clique model of distributed computing, with the same round complexity and approximation guarantee. This gives the first sub-logarithmic algorithm for approximating APSP in weighted graphs in the Congested Clique model.

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arXiv2021

Strong-Diameter Network Decomposition

Network decomposition is a central concept in the study of distributed graph algorithms. We present the first polylogarithmic-round deterministic distributed algorithm with small messages that constructs a strong-diameter network decomposition with polylogarithmic parameters. Concretely, a ($C$, $D$) strong-diameter network decomposition is a partitioning of the nodes of the graph into disjoint clusters, colored with $C$ colors, such that neighboring clusters have different colors and the subgraph induced by each cluster has a diameter at most $D$. In the weak-diameter variant, the requirement is relaxed by measuring the diameter of each cluster in the original graph, instead of the subgraph induced by the cluster. A recent breakthrough of Rozhoň and Ghaffari [STOC 2020] presented the first $\text{poly}(\log n)$-round deterministic algorithm for constructing a weak-diameter network decomposition where $C$ and $D$ are both in $\text{poly}(\log n)$. Their algorithm uses small $O(\log n)$-bit messages. One can transform their algorithm to a strong-diameter network decomposition algorithm with similar parameters. However, that comes at the expense of requiring unbounded messages. The key remaining qualitative question in the study of network decompositions was whether one can achieve a similar result for strong-diameter network decompositions using small messages. We resolve this question by presenting a novel technique that can transform any black-box weak-diameter network decomposition algorithm to a strong-diameter one, using small messages and with only moderate loss in the parameters.

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arXiv2020

Improved MPC Algorithms for MIS, Matching, and Coloring on Trees and Beyond

We present $O(\log\log n)$ round scalable Massively Parallel Computation algorithms for maximal independent set and maximal matching, in trees and more generally graphs of bounded arboricity, as well as for constant coloring trees. Following the standards, by a scalable MPC algorithm, we mean that these algorithms can work on machines that have capacity/memory as small as $n^δ$ for any positive constant $δ<1$. Our results improve over the $O(\log^2\log n)$ round algorithms of Behnezhad et al. [PODC'19]. Moreover, our matching algorithm is presumably optimal as its bound matches an $Ω(\log\log n)$ conditional lower bound of Ghaffari, Kuhn, and Uitto [FOCS'19].

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arXiv2020

Low-Congestion Shortcuts for Graphs Excluding Dense Minors

We prove that any $n$-node graph $G$ with diameter $D$ admits shortcuts with congestion $O(δD \log n)$ and dilation $O(δD)$, where $δ$ is the maximum edge-density of any minor of $G$. Our proof is simple, elementary, and constructive - featuring a $\tildeΘ(δD)$-round distributed construction algorithm. Our results are tight up to $\tilde{O}(1)$ factors and generalize, simplify, unify, and strengthen several prior results. For example, for graphs excluding a fixed minor, i.e., graphs with constant $δ$, only a $\tilde{O}(D^2)$ bound was known based on a very technical proof that relies on the Robertson-Seymour Graph Structure Theorem. A direct consequence of our result is that many graph families, including any minor-excluded ones, have near-optimal $\tildeΘ(D)$-round distributed algorithms for many fundamental communication primitives and optimization problems including minimum spanning tree, minimum cut, and shortest-path approximations.

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arXiv2020

Improved Distributed $Δ$-Coloring

We present a randomized distributed algorithm that computes a $Δ$-coloring in any non-complete graph with maximum degree $Δ\geq 4$ in $O(\log Δ) + 2^{O(\sqrt{\log\log n})}$ rounds, as well as a randomized algorithm that computes a $Δ$-coloring in $O((\log \log n)^2)$ rounds when $Δ\in [3, O(1)]$. Both these algorithms improve on an $O(\log^3 n/\log Δ)$-round algorithm of Panconesi and Srinivasan~[STOC'1993], which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an $Ω(\log\log n)$ round lower bound of Brandt et al.~[STOC'16].

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arXiv2020

Improved Deterministic Network Decomposition

Network decomposition is a central tool in distributed graph algorithms. We present two improvements on the state of the art for network decomposition, which thus lead to improvements in the (deterministic and randomized) complexity of several well-studied graph problems. - We provide a deterministic distributed network decomposition algorithm with $O(\log^5 n)$ round complexity, using $O(\log n)$-bit messages. This improves on the $O(\log^7 n)$-round algorithm of Rozhoň and Ghaffari [STOC'20], which used large messages, and their $O(\log^8 n)$-round algorithm with $O(\log n)$-bit messages. This directly leads to similar improvements for a wide range of deterministic and randomized distributed algorithms, whose solution relies on network decomposition, including the general distributed derandomization of Ghaffari, Kuhn, and Harris [FOCS'18]. - One drawback of the algorithm of Rozhoň and Ghaffari, in the $\mathsf{CONGEST}$ model, was its dependence on the length of the identifiers. Because of this, for instance, the algorithm could not be used in the shattering framework in the $\mathsf{CONGEST}$ model. Thus, the state of the art randomized complexity of several problems in this model remained with an additive $2^{O(\sqrt{\log\log n})}$ term, which was a clear leftover of the older network decomposition complexity [Panconesi and Srinivasan STOC'92]. We present a modified version that remedies this, constructing a decomposition whose quality does not depend on the identifiers, and thus improves the randomized round complexity for various problems.

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arXiv2020

A Massively Parallel Algorithm for Minimum Weight Vertex Cover

We present a massively parallel algorithm, with near-linear memory per machine, that computes a $(2+\varepsilon)$-approximation of minimum-weight vertex cover in $O(\log\log d)$ rounds, where $d$ is the average degree of the input graph. Our result fills the key remaining gap in the state-of-the-art MPC algorithms for vertex cover and matching problems; two classic optimization problems, which are duals of each other. Concretely, a recent line of work---by Czumaj et al. [STOC'18], Ghaffari et al. [PODC'18], Assadi et al. [SODA'19], and Gamlath et al. [PODC'19]---provides $O(\log\log n)$ time algorithms for $(1+\varepsilon)$-approximate maximum weight matching as well as for $(2+\varepsilon)$-approximate minimum cardinality vertex cover. However, the latter algorithm does not work for the general weighted case of vertex cover, for which the best known algorithm remained at $O(\log n)$ time complexity.

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arXiv2020

Polylogarithmic-Time Deterministic Network Decomposition and Distributed Derandomization

We present a simple polylogarithmic-time deterministic distributed algorithm for network decomposition. This improves on a celebrated $2^{O(\sqrt{\log n})}$-time algorithm of Panconesi and Srinivasan [STOC'92] and settles a central and long-standing question in distributed graph algorithms. It also leads to the first polylogarithmic-time deterministic distributed algorithms for numerous other problems, hence resolving several well-known and decades-old open problems, including Linial's question about the deterministic complexity of maximal independent set [FOCS'87; SICOMP'92]---which had been called the most outstanding problem in the area. The main implication is a more general distributed derandomization theorem: Put together with the results of Ghaffari, Kuhn, and Maus [STOC'17] and Ghaffari, Harris, and Kuhn [FOCS'18], our network decomposition implies that $$\mathsf{P}\textit{-}\mathsf{RLOCAL} = \mathsf{P}\textit{-}\mathsf{LOCAL}.$$ That is, for any problem whose solution can be checked deterministically in polylogarithmic-time, any polylogarithmic-time randomized algorithm can be derandomized to a polylogarithmic-time deterministic algorithm. Informally, for the standard first-order interpretation of efficiency as polylogarithmic-time, distributed algorithms do not need randomness for efficiency. By known connections, our result leads also to substantially faster randomized distributed algorithms for a number of well-studied problems including $(Δ+1)$-coloring, maximal independent set, and Lovász Local Lemma, as well as massively parallel algorithms for $(Δ+1)$-coloring.

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arXiv2019

On Derandomizing Local Distributed Algorithms

The gap between the known randomized and deterministic local distributed algorithms underlies arguably the most fundamental and central open question in distributed graph algorithms. In this paper, we develop a generic and clean recipe for derandomizing LOCAL algorithms. We also exhibit how this simple recipe leads to significant improvements on a number of problem. Two main results are: - An improved distributed hypergraph maximal matching algorithm, improving on Fischer, Ghaffari, and Kuhn [FOCS'17], and giving improved algorithms for edge-coloring, maximum matching approximation, and low out-degree edge orientation. The first gives an improved algorithm for Open Problem 11.4 of the book of Barenboim and Elkin, and the last gives the first positive resolution of their Open Problem 11.10. - An improved distributed algorithm for the Lovász Local Lemma, which gets closer to a conjecture of Chang and Pettie [FOCS'17], and moreover leads to improved distributed algorithms for problems such as defective coloring and $k$-SAT.

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arXiv2019

Faster Algorithms for Edge Connectivity via Random $2$-Out Contractions

We provide a simple new randomized contraction approach to the global minimum cut problem for simple undirected graphs. The contractions exploit 2-out edge sampling from each vertex rather than the standard uniform edge sampling. We demonstrate the power of our new approach by obtaining better algorithms for sequential, distributed, and parallel models of computation. Our end results include the following randomized algorithms for computing edge connectivity with high probability: -- Two sequential algorithms with complexities $O(m \log n)$ and $O(m+n \log^3 n)$. These improve on a long line of developments including a celebrated $O(m \log^3 n)$ algorithm of Karger [STOC'96] and the state of the art $O(m \log^2 n (\log\log n)^2)$ algorithm of Henzinger et al. [SODA'17]. Moreover, our $O(m+n \log^3 n)$ algorithm is optimal whenever $m = Ω(n \log^3 n)$. Within our new time bounds, whp, we can also construct the cactus representation of all minimal cuts. -- An $Õ(n^{0.8} D^{0.2} + n^{0.9})$ round distributed algorithm, where D denotes the graph diameter. This improves substantially on a recent breakthrough of Daga et al. [STOC'19], which achieved a round complexity of $Õ(n^{1-1/353}D^{1/353} + n^{1-1/706})$, hence providing the first sublinear distributed algorithm for exactly computing the edge connectivity. -- The first $O(1)$ round algorithm for the massively parallel computation setting with linear memory per machine.

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arXiv2019

Improved Network Decompositions using Small Messages with Applications on MIS, Neighborhood Covers, and Beyond

Network decompositions, as introduced by Awerbuch, Luby, Goldberg, and Plotkin [FOCS'89], are one of the key algorithmic tools in distributed graph algorithms. We present an improved deterministic distributed algorithm for constructing network decompositions of power graphs using small messages, which improves upon the algorithm of Ghaffari and Kuhn [DISC'18]. In addition, we provide a randomized distributed network decomposition algorithm, based on our deterministic algorithm, with failure probability exponentially small in the input size that works with small messages as well. Compared to the previous algorithm of Elkin and Neiman [PODC'16], our algorithm achieves a better success probability at the expense of its round complexity, while giving a network decomposition of the same quality. As a consequence of the randomized algorithm for network decomposition, we get a faster randomized algorithm for computing a Maximal Independent Set, improving on a result of Ghaffari [SODA'19]. Other implications of our improved deterministic network decomposition algorithm are: a faster deterministic distributed algorithms for constructing spanners and approximations of distributed set cover, improving results of Ghaffari, and Kuhn [DISC'18] and Deurer, Kuhn, and Maus [PODC'19]; and faster a deterministic distributed algorithm for constructing neighborhood covers, resolving an open question of Elkin [SODA'04].

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arXiv2019

Improved Distributed Approximations for Minimum-Weight Two-Edge-Connected Spanning Subgraph

The minimum-weight $2$-edge-connected spanning subgraph (2-ECSS) problem is a natural generalization of the well-studied minimum-weight spanning tree (MST) problem, and it has received considerable attention in the area of network design. The latter problem asks for a minimum-weight subgraph with an edge connectivity of $1$ between each pair of vertices while the former strengthens this edge-connectivity requirement to $2$. Despite this resemblance, the 2-ECSS problem is considerably more complex than MST. While MST admits a linear-time centralized exact algorithm, 2-ECSS is NP-hard and the best known centralized approximation algorithm for it (that runs in polynomial time) gives a $2$-approximation. In this paper, we give a deterministic distributed algorithm with round complexity of $\widetilde{O}(D+\sqrt{n})$ that computes a $(5+ε)$-approximation of 2-ECSS, for any constant $ε>0$. Up to logarithmic factors, this complexity matches the $\widetildeΩ(D+\sqrt{n})$ lower bound that can be derived from Das Sarma et al. [STOC'11], as shown by Censor-Hillel and Dory [OPODIS'17]. Our result is the first distributed constant approximation for 2-ECSS in the nearly optimal time and it improves on a recent randomized algorithm of Dory [PODC'18], which achieved an $O(\log n)$-approximation in $\widetilde{O}(D+\sqrt{n})$ rounds. We also present an alternative algorithm for $O(\log n)$-approximation, whose round complexity is linear in the low-congestion shortcut parameter of the network, following a framework introduced by Ghaffari and Haeupler [SODA'16]. This algorithm has round complexity $\widetilde{O}(D+\sqrt{n})$ in worst-case networks but it provably runs much faster in many well-behaved graph families of interest. For instance, it runs in $\widetilde{O}(D)$ time in planar networks and those with bounded genus, bounded path-width or bounded tree-width.

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arXiv2019

On the Use of Randomness in Local Distributed Graph Algorithms

We attempt to better understand randomization in local distributed graph algorithms by exploring how randomness is used and what we can gain from it: - We first ask the question of how much randomness is needed to obtain efficient randomized algorithms. We show that for all locally checkable problems for which polylog $n$-time randomized algorithms exist, there are such algorithms even if either (I) there is a only a single (private) independent random bit in each polylog $n$-neighborhood of the graph, (II) the (private) bits of randomness of different nodes are only polylog $n$-wise independent, or (III) there are only polylog $n$ bits of global shared randomness (and no private randomness). - Second, we study how much we can improve the error probability of randomized algorithms. For all locally checkable problems for which polylog $n$-time randomized algorithms exist, we show that there are such algorithms that succeed with probability $1-n^{-2^{\varepsilon(\log\log n)^2}}$ and more generally $T$-round algorithms, for $T\geq$ polylog $n$, that succeed with probability $1-n^{-2^{\varepsilon\log^2T}}$. We also show that polylog $n$-time randomized algorithms with success probability $1-2^{-2^{\log^\varepsilon n}}$ for some $\varepsilon>0$ can be derandomized to polylog $n$-time deterministic algorithms. Both of the directions mentioned above, reducing the amount of randomness and improving the success probability, can be seen as partial derandomization of existing randomized algorithms. In all the above cases, we also show that any significant improvement of our results would lead to a major breakthrough, as it would imply significantly more efficient deterministic distributed algorithms for a wide class of problems.

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arXiv2019

On the Complexity of Distributed Splitting Problems

One of the fundamental open problems in the area of distributed graph algorithms is the question of whether randomization is needed for efficient symmetry breaking. While there are fast, $\text{poly}\log n$-time randomized distributed algorithms for all of the classic symmetry breaking problems, for many of them, the best deterministic algorithms are almost exponentially slower. The following basic local splitting problem, which is known as the \emph{weak splitting} problem takes a central role in this context: Each node of a graph $G=(V,E)$ has to be colored red or blue such that each node of sufficiently large degree has at least one node of each color among its neighbors. Ghaffari, Kuhn, and Maus [STOC '17] showed that this seemingly simple problem is complete w.r.t. the above fundamental open question in the following sense: If there is an efficient $\text{poly}\log n$-time determinstic distributed algorithm for weak splitting, then there is such an algorithm for all locally checkable graph problems for which an efficient randomized algorithm exists. In this paper, we investigate the distributed complexity of weak splitting and some closely related problems. E.g., we obtain efficient algorithms for special cases of weak splitting, where the graph is nearly regular. In particular, we show that if $δ$ and $Δ$ are the minimum and maximum degrees of $G$ and if $δ=Ω(\log n)$, weak splitting can be solved deterministically in time $O\big(\fracΔδ\cdot\text{poly}(\log n)\big)$. Further, if $δ= Ω(\log\log n)$ and $Δ\leq 2^{\varepsilonδ}$, there is a randomized algorithm with time complexity $O\big(\fracΔδ\cdot\text{poly}(\log\log n)\big)$.

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