IA Scholar Query: Polylogarithmic-time deterministic network decomposition and distributed derandomization.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgFri, 23 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Deterministic Distributed Sparse and Ultra-Sparse Spanners and Connectivity Certificates
https://scholar.archive.org/work/finy2wwnkrhdpag6u6gmmpy46m
This paper presents efficient distributed algorithms for a number of fundamental problems in the area of graph sparsification: We provide the first deterministic distributed algorithm that computes an ultra-sparse spanner in polylog(n) rounds in weighted graphs. Concretely, our algorithm outputs a spanning subgraph with only n+o(n) edges in which the pairwise distances are stretched by a factor of at most O(log n · 2^O(log^* n)). We provide a polylog(n)-round deterministic distributed algorithm that computes a spanner with stretch (2k-1) and O(nk + n^1 + 1/klog k) edges in unweighted graphs and with O(n^1 + 1/k k) edges in weighted graphs. We present the first polylog(n)-round randomized distributed algorithm that computes a sparse connectivity certificate. For an n-node graph G, a certificate for connectivity k is a spanning subgraph H that is k-edge-connected if and only if G is k-edge-connected, and this subgraph H is called sparse if it has O(nk) edges. Our algorithm achieves a sparsity of (1 + o(1))nk edges, which is within a 2(1 + o(1)) factor of the best possible.Marcel Bezdrighin, Michael Elkin, Mohsen Ghaffari, Christoph Grunau, Bernhard Haeupler, Saeed Ilchi, Václav Rozhoňwork_finy2wwnkrhdpag6u6gmmpy46mFri, 23 Sep 2022 00:00:00 GMTUndirected (1+ε)-Shortest Paths via Minor-Aggregates: Near-Optimal Deterministic Parallel Distributed Algorithms
https://scholar.archive.org/work/avgfthkhgzgltjmalqqxpw37rq
This paper presents near-optimal deterministic parallel and distributed algorithms for computing (1+ε)-approximate single-source shortest paths in any undirected weighted graph. On a high level, we deterministically reduce this and other shortest-path problems to Õ(1) Minor-Aggregations. A Minor-Aggregation computes an aggregate (e.g., max or sum) of node-values for every connected component of some subgraph. Our reduction immediately implies: Optimal deterministic parallel (PRAM) algorithms with Õ(1) depth and near-linear work. Universally-optimal deterministic distributed (CONGEST) algorithms, whenever deterministic Minor-Aggregate algorithms exist. For example, an optimal Õ(HopDiameter(G))-round deterministic CONGEST algorithm for excluded-minor networks. Several novel tools developed for the above results are interesting in their own right: A local iterative approach for reducing shortest path computations "up to distance D" to computing low-diameter decompositions "up to distance D/2". Compared to the recursive vertex-reduction approach of [Li20], our approach is simpler, suitable for distributed algorithms, and eliminates many derandomization barriers. A simple graph-based Õ(1)-competitive ℓ_1-oblivious routing based on low-diameter decompositions that can be evaluated in near-linear work. The previous such routing [ZGY+20] was n^o(1)-competitive and required n^o(1) more work. A deterministic algorithm to round any fractional single-source transshipment flow into an integral tree solution. The first distributed algorithms for computing Eulerian orientations.Václav Rozhoň and Christoph Grunau and Bernhard Haeupler and Goran Zuzic and Jason Liwork_avgfthkhgzgltjmalqqxpw37rqFri, 23 Sep 2022 00:00:00 GMTImproved Distributed Network Decomposition, Hitting Sets, and Spanners, via Derandomization
https://scholar.archive.org/work/lnohfvsjgbdirlx37d5cq6dbcu
This paper presents significantly improved deterministic algorithms for some of the key problems in the area of distributed graph algorithms, including network decomposition, hitting sets, and spanners. As the main ingredient in these results, we develop novel randomized distributed algorithms that we can analyze using only pairwise independence, and we can thus derandomize efficiently. As our most prominent end-result, we obtain a deterministic construction for O(log n)-color O(log n ·logloglog n)-strong diameter network decomposition in Õ(log^3 n) rounds. This is the first construction that achieves almost log n in both parameters, and it improves on a recent line of exciting progress on deterministic distributed network decompositions [Rozhoň, Ghaffari STOC'20; Ghaffari, Grunau, Rozhoň SODA'21; Chang, Ghaffari PODC'21; Elkin, Haeupler, Rozhoň, Grunau FOCS'22].Mohsen Ghaffari, Christoph Grunau, Bernhard Haeupler, Saeed Ilchi, Václav Rozhoňwork_lnohfvsjgbdirlx37d5cq6dbcuFri, 23 Sep 2022 00:00:00 GMTLocal Distributed Rounding: Generalized to MIS, Matching, Set Cover, and Beyond
https://scholar.archive.org/work/rkpzdjoeg5cydcveey4imjqgxu
We develop a general deterministic distributed method for locally rounding fractional solutions of graph problems for which the analysis can be broken down into analyzing pairs of vertices. Roughly speaking, the method can transform fractional/probabilistic label assignments of the vertices into integral/deterministic label assignments for the vertices, while approximately preserving a potential function that is a linear combination of functions, each of which depends on at most two vertices (subject to some conditions usually satisfied in pairwise analyses). The method unifies and significantly generalizes prior work on deterministic local rounding techniques [Ghaffari, Kuhn FOCS'21; Harris FOCS'19; Fischer, Ghaffari, Kuhn FOCS'17; Fischer DISC'17] to obtain polylogarithmic-time deterministic distributed solutions for combinatorial graph problems. Our general rounding result enables us to locally and efficiently derandomize a range of distributed algorithms for local graph problems, including maximal independent set (MIS), maximum-weight independent set approximation, and minimum-cost set cover approximation. As a highlight, we in particular obtain a deterministic O(log^2Δ·log n)-round algorithm for computing an MIS in the LOCAL model and an almost as efficient O(log^2Δ·loglogΔ·log n)-round deterministic MIS algorithm in the CONGEST model. As a result, the best known deterministic distributed time complexity of the four most widely studied distributed symmetry breaking problems (MIS, maximal matching, (Δ+1)-vertex coloring, and (2Δ-1)-edge coloring) is now O(log^2Δ·log n). Our new MIS algorithm is also the first direct polylogarithmic-time deterministic distributed MIS algorithm, which is not based on network decomposition.Salwa Faour, Mohsen Ghaffari, Christoph Grunau, Fabian Kuhn, Václav Rozhoňwork_rkpzdjoeg5cydcveey4imjqgxuFri, 23 Sep 2022 00:00:00 GMTÕptimal Vertex Fault-Tolerant Spanners in Õptimal Time: Sequential, Distributed and Parallel
https://scholar.archive.org/work/4435vcjedze67euljnqyfudpr4
We (nearly) settle the time complexity for computing vertex fault-tolerant (VFT) spanners with optimal sparsity (up to polylogarithmic factors). VFT spanners are sparse subgraphs that preserve distance information, up to a small multiplicative stretch, in the presence of vertex failures. These structures were introduced by [Chechik et al., STOC 2009] and have received a lot of attention since then. We provide algorithms for computing nearly optimal f-VFT spanners for any n-vertex m-edge graph, with near optimal running time in several computational models: - A randomized sequential algorithm with a runtime of O(m) (i.e., independent in the number of faults f). The state-of-the-art time bound is O(f^1-1/k· n^2+1/k+f^2 m) by [Bodwin, Dinitz and Robelle, SODA 2021]. - A distributed congest algorithm of O(1) rounds. Improving upon [Dinitz and Robelle, PODC 2020] that obtained FT spanners with near-optimal sparsity in O(f^2) rounds. - A PRAM (CRCW) algorithm with O(m) work and O(1) depth. Prior bounds implied by [Dinitz and Krauthgamer, PODC 2011] obtained sub-optimal FT spanners using O(f^3m) work and O(f^3) depth. An immediate corollary provides the first nearly-optimal PRAM algorithm for computing nearly optimal λ-vertex connectivity certificates using polylogarithmic depth and near-linear work. This improves the state-of-the-art parallel bounds of O(1) depth and O(λ m) work, by [Karger and Motwani, STOC'93].Merav Parterwork_4435vcjedze67euljnqyfudpr4Wed, 07 Sep 2022 00:00:00 GMTDeterministic Low-Diameter Decompositions for Weighted Graphs and Distributed and Parallel Applications
https://scholar.archive.org/work/tjrbck7p2ffx3olx4nwdvnooim
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 [Rozhoň, 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 · 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 ℓ_1-embedding into polylogarithmic dimensional space with polylogarithmic distortion. The best prior deterministic algorithms for ℓ_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).Václav Rozhoň, Michael Elkin, Christoph Grunau, Bernhard Haeuplerwork_tjrbck7p2ffx3olx4nwdvnooimSat, 03 Sep 2022 00:00:00 GMTImproved Distributed Algorithms for the Lovász Local Lemma and Edge Coloring
https://scholar.archive.org/work/df5cn72adrgzhp2hrmj34rdm4q
The Lovász Local Lemma is a classic result in probability theory that is often used to prove the existence of combinatorial objects via the probabilistic method. In its simplest form, it states that if we have n 'bad events', each of which occurs with probability at most p and is independent of all but d other events, then under certain criteria on p and d, all of the bad events can be avoided with positive probability. While the original proof was existential, there has been much study on the algorithmic Lovász Local Lemma: that is, designing an algorithm which finds an assignment of the underlying random variables such that all the bad events are indeed avoided. Notably, the celebrated result of Moser and Tardos [JACM '10] also implied an efficient distributed algorithm for the problem, running in O(log^2 n) rounds. For instances with low d, this was improved to O(d^2+log^O(1)log n) by Fischer and Ghaffari [DISC '17], a result that has proven highly important in distributed complexity theory (Chang and Pettie [SICOMP '19]). We give an improved algorithm for the Lovász Local Lemma, providing a trade-off between the strength of the criterion relating p and d, and the distributed round complexity. In particular, in the same regime as Fischer and Ghaffari's algorithm, we improve the round complexity to O(d/log d+log^O(1)log n). At the other end of the trade-off, we obtain a log^O(1)log n round complexity for a substantially wider regime than previously known. As our main application, we also give the first log^O(1)log n-round distributed algorithm for the problem of Δ+o(Δ)-edge coloring a graph of maximum degree Δ. This is an almost exponential improvement over previous results: no prior log^o(1) n-round algorithm was known even for 2Δ-2-edge coloring.Peter Davieswork_df5cn72adrgzhp2hrmj34rdm4qThu, 18 Aug 2022 00:00:00 GMTFast Distributed Vertex Splitting with Applications
https://scholar.archive.org/work/mbfj7vwwy5gm5bnhzz22soxshi
We present polyloglog n-round randomized distributed algorithms to compute vertex splittings, a partition of the vertices of a graph into k parts such that a node of degree d(u) has ≈ d(u)/k neighbors in each part. Our techniques can be seen as the first progress towards general polyloglog n-round algorithms for the Lovász Local Lemma. As the main application of our result, we obtain a randomized polyloglog n-round CONGEST algorithm for (1+ϵ)Δ-edge coloring n-node graphs of sufficiently large constant maximum degree Δ, for any ϵ>0. Further, our results improve the computation of defective colorings and certain tight list coloring problems. All the results improve the state-of-the-art round complexity exponentially, even in the LOCAL model.Magnús M. Halldórsson, Yannic Maus, Alexandre Nolinwork_mbfj7vwwy5gm5bnhzz22soxshiWed, 17 Aug 2022 00:00:00 GMTImproved Throughput for All-or-Nothing Multicommodity Flows with Arbitrary Demands
https://scholar.archive.org/work/nmd2np4gsnbjjavehov76c5v2q
Throughput is a main performance objective in communication networks. This paper considers a fundamental maximum throughput routing problem -- the all-or-nothing multicommodity flow (ANF) problem -- in arbitrary directed graphs and in the practically relevant but challenging setting where demands can be (much) larger than the edge capacities. Hence, in addition to assigning requests to valid flows for each routed commodity, an admission control mechanism is required which prevents overloading the network when routing commodities. We make several contributions. On the theoretical side we obtain substantially improved bi-criteria approximation algorithms for this NP-hard problem. We present two non-trivial linear programming relaxations and show how to convert their fractional solutions into integer solutions via randomized rounding. One is an exponential-size formulation (solvable in polynomial time using a separation oracle) that considers a "packing" view and allows a more flexible approach, while the other is a compact (polynomial-size) edge-flow formulation that allows for easy solving via standard LP solvers. We obtain a polynomial-time randomized algorithm that yields an arbitrarily good approximation on the weighted throughput, while violating the edge capacity constraints by only a small multiplicative factor. We also describe a deterministic rounding algorithm by derandomization, using the method of pessimistic estimators. We complement our theoretical results with a proof of concept empirical evaluation.Anya Chaturvedi, Chandra Chekuri, Mengxue Liu, Andréa W. Richa, Mattias Rost, Stefan Schmid, Jamison Weberwork_nmd2np4gsnbjjavehov76c5v2qTue, 26 Jul 2022 00:00:00 GMTDistributed Edge Coloring in Time Polylogarithmic in Δ
https://scholar.archive.org/work/3phqv2vtefbwldn7tvzjxjn5d4
We provide new deterministic algorithms for the edge coloring problem, which is one of the classic and highly studied distributed local symmetry breaking problems. As our main result, we show that a (2Δ − 1)-edge coloring can be computed in time poly log Δ + 𝑂 (log * 𝑛) in the LOCAL model. This improves a result of Balliu, Kuhn, and Olivetti [PODC '20], who gave an algorithm with a quasi-polylogarithmic dependency on Δ. We further show that in the CONGEST model, an (8 + 𝜀)Δ-edge coloring can be computed in poly log Δ + 𝑂 (log * 𝑛) rounds. The best previous 𝑂 (Δ)-edge coloring algorithm that can be implemented in the CONGEST model is by Barenboim and Elkin [PODC '11] and it computes a 2 𝑂 (1/𝜀) Δedge coloring in time 𝑂 (Δ 𝜀 + log * 𝑛) for any 𝜀 ∈ (0, 1]. CCS CONCEPTS • Theory of computation → Distributed algorithms.Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivettiwork_3phqv2vtefbwldn7tvzjxjn5d4Wed, 20 Jul 2022 00:00:00 GMTNear Optimal Linear Algebra in the Online and Sliding Window Models
https://scholar.archive.org/work/65xfc5nxxzbmfaahx26vhcecsa
We initiate the study of numerical linear algebra in the sliding window model, where only the most recent W updates in a stream form the underlying data set. We first introduce a unified row-sampling based framework that gives randomized algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and ℓ_1-subspace embeddings in the sliding window model, which often use nearly optimal space and achieve nearly input sparsity runtime. Our algorithms are based on "reverse online" versions of offline sampling distributions such as (ridge) leverage scores, ℓ_1 sensitivities, and Lewis weights to quantify both the importance and the recency of a row. Our row-sampling framework rather surprisingly implies connections to the well-studied online model; our structural results also give the first sample optimal (up to lower order terms) online algorithm for low-rank approximation/projection-cost preservation. Using this powerful primitive, we give online algorithms for column/row subset selection and principal component analysis that resolves the main open question of Bhaskara et. al.,(FOCS 2019). We also give the first online algorithm for ℓ_1-subspace embeddings. We further formalize the connection between the online model and the sliding window model by introducing an additional unified framework for deterministic algorithms using a merge and reduce paradigm and the concept of online coresets. Our sampling based algorithms in the row-arrival online model yield online coresets, giving deterministic algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and ℓ_1-subspace embeddings in the sliding window model that use nearly optimal space.Vladimir Braverman, Petros Drineas, Cameron Musco, Christopher Musco, Jalaj Upadhyay, David P. Woodruff, Samson Zhouwork_65xfc5nxxzbmfaahx26vhcecsaTue, 19 Jul 2022 00:00:00 GMTDeterministic Distributed Sparse and Ultra-Sparse Spanners and Connectivity Certificates
https://scholar.archive.org/work/v7gisf3wrzdefbqiszp3pwdpyy
This paper presents efficient distributed algorithms for a number of fundamental problems in the area of graph sparsification: • We provide the first deterministic distributed algorithm that computes an ultra-sparse spanner in polylog(𝑛) rounds in weighted graphs. Concretely, our algorithm outputs a spanning subgraph with only 𝑛 + 𝑜 (𝑛) edges in which the pairwise distances are stretched by a factor of at most 𝑂 (log 𝑛 • 2 𝑂 (log * 𝑛) ). • We provide a polylog(𝑛)-round deterministic distributed algorithm that computes a spanner with stretch (2𝑘 − 1) and 𝑂 (𝑛𝑘 + 𝑛 1+ 1 𝑘 log 𝑘) edges in unweighted graphs and with 𝑂 (𝑛 1+ 1 𝑘 𝑘) edges in weighted graphs. • We present the first polylog(𝑛) round randomized distributed algorithm that computes a sparse connectivity certificate. For an 𝑛-node graph 𝐺, a certificate for connectivity 𝑘 is a spanning subgraph 𝐻 that is 𝑘-edge-connected if and only if 𝐺 is 𝑘-edge-connected, and this subgraph 𝐻 is called sparse if it has 𝑂 (𝑛𝑘) edges. Our algorithm achieves a sparsity of (1 + 𝑜 (1))𝑛𝑘 edges, which is within a 2(1 + 𝑜 (1)) factor of the best possible. CCS CONCEPTS • Theory of computation → Distributed algorithms.Marcel Bezdrighin, Michael Elkin, Mohsen Ghaffari, Christoph Grunau, Bernhard Haeupler, Saeed Ilchi, Václav Rozhoňwork_v7gisf3wrzdefbqiszp3pwdpyyMon, 11 Jul 2022 00:00:00 GMTLIPIcs, Volume 230, ITC 2022, Complete Volume
https://scholar.archive.org/work/x5cobg6anzbgjazexwg7mkanie
LIPIcs, Volume 230, ITC 2022, Complete VolumeDana Dachman-Soledwork_x5cobg6anzbgjazexwg7mkanieThu, 30 Jun 2022 00:00:00 GMTDistributed Edge Coloring in Time Polylogarithmic in Δ
https://scholar.archive.org/work/oos4tohnmbez5a6lvegqtzy2ju
We provide new deterministic algorithms for the edge coloring problem, which is one of the classic and highly studied distributed local symmetry breaking problems. As our main result, we show that a (2Δ-1)-edge coloring can be computed in time polylogΔ + O(log^* n) in the LOCAL model. This improves a result of Balliu, Kuhn, and Olivetti [PODC '20], who gave an algorithm with a quasi-polylogarithmic dependency on Δ. We further show that in the CONGEST model, an (8+ε)Δ-edge coloring can be computed in polylogΔ + O(log^* n) rounds. The best previous O(Δ)-edge coloring algorithm that can be implemented in the CONGEST model is by Barenboim and Elkin [PODC '11] and it computes a 2^O(1/ε)Δ-edge coloring in time O(Δ^ε + log^* n) for any ε∈(0,1].Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivettiwork_oos4tohnmbez5a6lvegqtzy2juThu, 02 Jun 2022 00:00:00 GMTUniversally-Optimal Distributed Exact Min-Cut
https://scholar.archive.org/work/l6qdoqmw6fexnm7dqsn4vecike
We present a universally-optimal distributed algorithm for the exact weighted min-cut. The algorithm is guaranteed to complete in O(D + √(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 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 deterministic O(D)-round 2-respecting cut solution for excluded-minor families and a deterministic O(D + √(n))-round solution for general graphs, the latter resolving a question of Dory, et al. [STOC'21]Mohsen Ghaffari, Goran Zuzicwork_l6qdoqmw6fexnm7dqsn4vecikeMon, 30 May 2022 00:00:00 GMTData-heterogeneity-aware Mixing for Decentralized Learning
https://scholar.archive.org/work/prhdjmu3sfb2vl7whmcsifdixa
Decentralized learning provides an effective framework to train machine learning models with data distributed over arbitrary communication graphs. However, most existing approaches toward decentralized learning disregard the interaction between data heterogeneity and graph topology. In this paper, we characterize the dependence of convergence on the relationship between the mixing weights of the graph and the data heterogeneity across nodes. We propose a metric that quantifies the ability of a graph to mix the current gradients. We further prove that the metric controls the convergence rate, particularly in settings where the heterogeneity across nodes dominates the stochasticity between updates for a given node. Motivated by our analysis, we propose an approach that periodically and efficiently optimizes the metric using standard convex constrained optimization and sketching techniques. Through comprehensive experiments on standard computer vision and NLP benchmarks, we show that our approach leads to improvement in test performance for a wide range of tasks.Yatin Dandi, Anastasia Koloskova, Martin Jaggi, Sebastian U. Stichwork_prhdjmu3sfb2vl7whmcsifdixaWed, 13 Apr 2022 00:00:00 GMTImproved Distributed Fractional Coloring Algorithms
https://scholar.archive.org/work/p2ftpgu2jzf7naxcq3q5ufmuxa
We prove new bounds on the distributed fractional coloring problem in the LOCAL model. A fractional c-coloring of a graph G = (V,E) is a fractional covering of the nodes of G with independent sets such that each independent set I of G is assigned a fractional value λ_I ∈ [0,1]. The total value of all independent sets of G is at most c, and for each node v ∈ V, the total value of all independent sets containing v is at least 1. Equivalently, fractional c-colorings can also be understood as multicolorings as follows. For some natural numbers p and q such that p/q ≤ c, each node v is assigned a set of at least q colors from {1,...,p} such that adjacent nodes are assigned disjoint sets of colors. The minimum c for which a fractional c-coloring of a graph G exists is called the fractional chromatic number χ_f(G) of G. Recently, [Bousquet, Esperet, and Pirot; SIROCCO '21] showed that for any constant ε > 0, a fractional (Δ+ε)-coloring can be computed in Δ^{O(Δ)} + O(Δ⋅log^* n) rounds. We show that such a coloring can be computed in only O(log² Δ) rounds, without any dependency on n. We further show that in O((log n)/ε) rounds, it is possible to compute a fractional (1+ε)χ_f(G)-coloring, even if the fractional chromatic number χ_f(G) is not known. That is, the fractional coloring problem can be approximated arbitrarily well by an efficient algorithm in the LOCAL model. For the standard coloring problem, it is only known that an O((log n)/(log log n))-approximation can be computed in polylogarithmic time in the LOCAL model. We also show that our distributed fractional coloring approximation algorithm is best possible. We show that in trees, which have fractional chromatic number 2, computing a fractional (2+ε)-coloring requires at least Ω((log n)/ε) rounds. We finally study fractional colorings of regular grids. In [Bousquet, Esperet, and Pirot; SIROCCO '21], it is shown that in regular grids of bounded dimension, a fractional (2+ε)-coloring can be computed in time O(log^* n). We show that such a coloring can even be co [...]Alkida Balliu, Fabian Kuhn, Dennis Olivetti, Quentin Bramas, Vincent Gramoli, Alessia Milaniwork_p2ftpgu2jzf7naxcq3q5ufmuxaMon, 28 Feb 2022 00:00:00 GMTDeterministic Distributed Sparse and Ultra-Sparse Spanners and Connectivity Certificates
https://scholar.archive.org/work/dgpjqgelzrae3i2zlrglmgwrwu
In this thesis we study the problem of computing graph spanners in the CONGEST model of distributed computation. Given a graph G, a graph spanner is a subgraph of G that preserves the shortest distance between nodes up to some distortion error -typically multiplicative and/or additive. Of particular interest are spanners with O(n) and n + o(n) edges which are called sparse and ultra-sparse respectively. Graph spanners have many applications ranging from shortest path approximation, message routing, distance oracles, sparse connectivity certificates and many others. In this thesis we present the following distributed algorithms in the CONGEST model: • We provide the first deterministic distributed algorithm that computes an ultra-sparse spanner in polylog(n) rounds in weighted and unweighted graphs. Concretely, our algorithm outputs a spanning subgraph with only n + o(n) edges in which the pairwise distances are stretched by a factor of at most Ō(log n). • We provide a polylog(n)-round deterministic distributed algorithm that computes a spanner with stretch (2k − 1) and O(nk + n 1+ 1 k log k) edges in unweighted graphs and with O(n 1+ 1 k k) edges in weighted graphs. I would like first and foremost to thank Saeed Iilchi for his help and amazing guidance over the course of my thesis. Without his help this thesis would not be possible. Also, I would like to thank Prof. Mohsen Ghaffari for presenting me this interesting topic and providing valuable insights. Finally, I would like to thank my family for their continuous support and love.Marcel Bezdrighin, Mohsen Ghaffari, Saeed Ilchi Ghazaanwork_dgpjqgelzrae3i2zlrglmgwrwuExpander Decomposition and Pruning: Faster, Stronger, and Simpler
https://scholar.archive.org/work/xmynec6eevh6pfqsz6soa4twca
We study the problem of graph clustering where the goal is to partition a graph into clusters, i.e. disjoint subsets of vertices, such that each cluster is well connected internally while sparsely connected to the rest of the graph. In particular, we use a natural bicriteria notion motivated by Kannan, Vempala, and Vetta which we refer to as expander decomposition. Expander decomposition has become one of the building blocks in the design of fast graph algorithms, most notably in the nearly linear time Laplacian solver by Spielman and Teng, and it also has wide applications in practice. We design algorithm for the parametrized version of expander decomposition, where given a graph G of m edges and a parameter ϕ, our algorithm finds a partition of the vertices into clusters such that each cluster induces a subgraph of conductance at least ϕ (i.e. a ϕ expander), and only a O(ϕ) fraction of the edges in G have endpoints across different clusters. Our algorithm runs in O(m/ϕ) time, and is the first nearly linear time algorithm when ϕ is at least 1/log^O(1) m, which is the case in most practical settings and theoretical applications. Previous results either take Ω(m^1+o(1)) time, or attain nearly linear time but with a weaker expansion guarantee where each output cluster is guaranteed to be contained inside some unknown ϕ expander. Our result achieve both nearly linear running time and the strong expander guarantee for clusters. Moreover, a main technique we develop for our result can be applied to obtain a much better expander pruning algorithm, which is the key tool for maintaining an expander decomposition on dynamic graphs. Finally, we note that our algorithm is developed from first principles based on relatively simple and basic techniques, thus making it very likely to be practical.Thatchaphol Saranurak, Di Wangwork_xmynec6eevh6pfqsz6soa4twcaWed, 15 Dec 2021 00:00:00 GMTImproved Distributed Fractional Coloring Algorithms
https://scholar.archive.org/work/adnnorghpzfqhk7qmhikwgxkoq
We prove new bounds on the distributed fractional coloring problem in the LOCAL model. Fractional c-colorings can be understood as multicolorings as follows. For some natural numbers p and q such that p/q≤ c, each node v is assigned a set of at least q colors from {1,...,p} such that adjacent nodes are assigned disjoint sets of colors. The minimum c for which a fractional c-coloring of a graph G exists is called the fractional chromatic number χ_f(G) of G. Recently, [Bousquet, Esperet, and Pirot; SIROCCO '21] showed that for any constant ϵ>0, a fractional (Δ+ϵ)-coloring can be computed in Δ^O(Δ) + O(Δ·log^* n) rounds. We show that such a coloring can be computed in only O(log^2 Δ) rounds, without any dependency on n. We further show that in O(log n/ϵ) rounds, it is possible to compute a fractional (1+ϵ)χ_f(G)-coloring, even if the fractional chromatic number χ_f(G) is not known. That is, this problem can be approximated arbitrarily well by an efficient algorithm in the LOCAL model. For the standard coloring problem, it is only known that an O(log n/loglog n)-approximation can be computed in polylogarithmic time in the LOCAL model. We also show that our distributed fractional coloring approximation algorithm is best possible. We show that in trees, which have fractional chromatic number 2, computing a fractional (2+ϵ)-coloring requires at least Ω(log n/ϵ) rounds. We finally study fractional colorings of regular grids. In [Bousquet, Esperet, and Pirot; SIROCCO '21], it is shown that in regular grids of bounded dimension, a fractional (2+ϵ)-coloring can be computed in time O(log^* n). We show that such a coloring can even be computed in O(1) rounds in the LOCAL model.Alkida Balliu, Fabian Kuhn, Dennis Olivettiwork_adnnorghpzfqhk7qmhikwgxkoqThu, 09 Dec 2021 00:00:00 GMT