1,721,031 research outputs found
Randomized (Delta+1)-Coloring in O(log* Delta) Congested Clique Rounds
No full text(Delta+1)-vertex coloring is one of the most fundamental symmetry breaking graph problems, receiving tremendous amount of attention over the last decades. We consider the congested clique model where in each round, every pair of vertices can exchange O(log n) bits of information.
In a recent breakthrough, Yi-Jun Chang, Wenzheng Li, and Seth Pettie [CLP-STOC'18] presented a randomized (Delta+1)-list coloring algorithm in the LOCAL model that works in O(log^*n+Det_{deg}(log log n)) rounds, where Det_{deg}(n') is the deterministic LOCAL complexity of (deg+1)-list coloring algorithm on n'-vertex graphs. Unfortunately, the CLP algorithm uses large messages and hence cannot be efficiently implemented in the congested clique model when the maximum degree Delta is large (in particular, when Delta=omega(sqrt{n})).
Merav Parter [P-ICALP'18] recently provided a randomized (Delta+1)-coloring algorithm in O(log log Delta * log^* Delta) congested clique rounds based on a careful partitioning of the input graph into almost-independent subgraphs with maximum degree sqrt{n}. In this work, we significantly improve upon this result and present a randomized (Delta+1)-coloring algorithm with O(log^* Delta) rounds, with high probability. At the heart of our algorithm is an adaptation of the CLP algorithm for coloring a subgraph with o(n) vertices and maximum degree Omega(n^{5/8}) in O(log^* Delta) rounds. The approach is built upon a combination of techniques, this includes: the graph sparsification of [Parter-ICALP'18], and a palette sampling technique adopted to the CLP framework
Distributed Constructions of Dual-Failure Fault-Tolerant Distance Preservers
Full text linkFault tolerant distance preservers (spanners) are sparse subgraphs that preserve (approximate) distances between given pairs of vertices under edge or vertex failures. So-far, these structures have been studied thoroughly mainly from a centralized viewpoint. Despite the fact fault tolerant preservers are mainly motivated by the error-prone nature of distributed networks, not much is known on the distributed computational aspects of these structures.
In this paper, we present distributed algorithms for constructing fault tolerant distance preservers and +2 additive spanners that are resilient to at most two edge faults. Prior to our work, the only non-trivial constructions known were for the single fault and single source setting by [Ghaffari and Parter SPAA'16].
Our key technical contribution is a distributed algorithm for computing distance preservers w.r.t. a subset S of source vertices, resilient to two edge faults. The output structure contains a BFS tree BFS(s,G ⧵ {e₁,e₂}) for every s ∈ S and every e₁,e₂ ∈ G. The distributed construction of this structure is based on a delicate balance between the edge congestion (formed by running multiple BFS trees simultaneously) and the sparsity of the output subgraph. No sublinear-round algorithms for constructing these structures have been known before
Distributed Planar Reachability in Nearly Optimal Time
Full text linkWe present nearly optimal distributed algorithms for fundamental reachability problems in planar graphs. In the single-source reachability problem given is an n-vertex directed graph G = (V,E) and a source node s, it is required to determine the subset of nodes that are reachable from s in G. We present the first distributed reachability algorithm for planar graphs that runs in nearly optimal time of Õ(D) rounds, where D is the undirected diameter of the graph. This improves the complexity of Õ(D²) rounds implied by the recent work of [Li and Parter, STOC'19].
We also consider the more general reachability problem of identifying the strongly connected components (SCCs) of the graph. We present an Õ(D)-round algorithm that computes for each node in the graph an identifier of its strongly connected component in G. No non-trivial upper bound for this problem (even in general graphs) has been known before.
Our algorithms are based on characterizing the structural interactions between balanced cycle separators. We show that the reachability relations between separator nodes can be compressed due to a Monge-like property of their directed shortest paths. The algorithmic results are obtained by combining this structural characterization with the recursive graph partitioning machinery of [Li and Parter, STOC'19]
Small Cuts and Connectivity Certificates: A Fault Tolerant Approach
Full text linkWe revisit classical connectivity problems in the {CONGEST} model of distributed computing. By using techniques from fault tolerant network design, we show improved constructions, some of which are even "local" (i.e., with O~(1) rounds) for problems that are closely related to hard global problems (i.e., with a lower bound of Omega(Diam+sqrt{n}) rounds).
Distributed Minimum Cut: Nanongkai and Su presented a randomized algorithm for computing a (1+epsilon)-approximation of the minimum cut using O~(D +sqrt{n}) rounds where D is the diameter of the graph. For a sufficiently large minimum cut lambda=Omega(sqrt{n}), this is tight due to Das Sarma et al. [FOCS '11], Ghaffari and Kuhn [DISC '13].
- Small Cuts: A special setting that remains open is where the graph connectivity lambda is small (i.e., constant). The only lower bound for this case is Omega(D), with a matching bound known only for lambda <= 2 due to Pritchard and Thurimella [TALG '11]. Recently, Daga, Henzinger, Nanongkai and Saranurak [STOC '19] raised the open problem of computing the minimum cut in poly(D) rounds for any lambda=O(1). In this paper, we resolve this problem by presenting a surprisingly simple algorithm, that takes a completely different approach than the existing algorithms. Our algorithm has also the benefit that it computes all minimum cuts in the graph, and naturally extends to vertex cuts as well. At the heart of the algorithm is a graph sampling approach usually used in the context of fault tolerant (FT) design.
- Deterministic Algorithms: While the existing distributed minimum cut algorithms are randomized, our algorithm can be made deterministic within the same round complexity. To obtain this, we introduce a novel definition of universal sets along with their efficient computation. This allows us to derandomize the FT graph sampling technique, which might be of independent interest.
- Computation of all Edge Connectivities: We also consider the more general task of computing the edge connectivity of all the edges in the graph. In the output format, it is required that the endpoints u,v of every edge (u,v) learn the cardinality of the u-v cut in the graph. We provide the first sublinear algorithm for this problem for the case of constant connectivity values. Specifically, by using the recent notion of low-congestion cycle cover, combined with the sampling technique, we compute all edge connectivities in poly(D) * 2^{O(sqrt{log n log log n})} rounds.
Sparse Certificates: For an n-vertex graph G and an integer lambda, a lambda-sparse certificate H is a subgraph H subseteq G with O(lambda n) edges which is lambda-connected iff G is lambda-connected. For D-diameter graphs, constructions of sparse certificates for lambda in {2,3} have been provided by Thurimella [J. Alg. '97] and Dori [PODC '18] respectively using O~(D) number of rounds. The problem of devising such certificates with o(D+sqrt{n}) rounds was left open by Dori [PODC '18] for any lambda >= 4. Using connections to fault tolerant spanners, we considerably improve the round complexity for any lambda in [1,n] and epsilon in (0,1), by showing a construction of (1-epsilon)lambda-sparse certificates with O(lambda n) edges using only O(1/epsilon^2 * log^{2+o(1)} n) rounds
(Delta+1) Coloring in the Congested Clique Model
No full textIn this paper, we present improved algorithms for the (Delta+1) (vertex) coloring problem in the Congested Clique model of distributed computing. In this model, the input is a graph on n nodes, initially each node knows only its incident edges, and per round each two nodes can exchange O(log n) bits of information.
Our key result is a randomized (Delta+1) vertex coloring algorithm that works in O(log log Delta * log^* Delta)-rounds. This is achieved by combining the recent breakthrough result of [Chang-Li-Pettie, STOC'18] in the {LOCAL} model and a degree reduction technique. We also get the following results with high probability: (1) (Delta+1)-coloring for Delta=O((n/log n)^{1-epsilon}) for any epsilon in (0,1), within O(log(1/epsilon)log^* Delta) rounds, and (2) (Delta+Delta^{1/2+o(1)})-coloring within O(log^* Delta) rounds. Turning to deterministic algorithms, we show a (Delta+1)-coloring algorithm that works in O(log Delta) rounds. Our new bounds provide exponential improvements over the state of the art
Graphs Shortcuts: New Bounds and Algorithms (Invited Talk)
No full textFor an n-vertex digraph G = (V,E), a shortcut set is a (small) subset of edges H taken from the transitive closure of G that, when added to G guarantees that the diameter of G ∪ H is small. Shortcut sets, introduced by Thorup in 1993, have a wide range of applications in algorithm design, especially in the context of parallel, distributed and dynamic computation on directed graphs. A folklore result in this context shows that every n-vertex digraph admits a shortcut set of linear size (i.e., of O(n) edges) that reduces the diameter to Õ(√n). Despite extensive research over the years, the question of whether one can reduce the diameter to o(√n) with Õ(n) shortcut edges has been left open.
In this talk, I will present the first improved diameter-sparsity tradeoff for this problem, breaking the √n diameter barrier. Specifically, we show an O(n^ω)-time randomized algorithm for computing a linear shortcut set that reduces the diameter of the digraph to Õ(n^{1/3}). I also present time efficient algorithms for computing these shortcuts and explain the limitations of the current approaches. Finally, I will draw some connections between shortcuts and several forms of graph sparsification (e.g., reachability preservers, spanners). Based on a joint work with Shimon Kogan (SODA 2022, ICALP 2022, FOCS 2022, SODA 2023)
New Additive Emulators
Full text linkFor a given (possibly weighted) graph G = (V,E), an additive emulator H is a weighted graph in V × V that preserves the (all pairs) G-distances up to a small additive stretch. In their breakthrough result, [Abboud and Bodwin, STOC 2016] ruled out the possibility of obtaining o(n^{4/3})-size emulator with n^{o(1)} additive stretch. The focus of our paper is in the following question that has been explicitly stated in many of the prior work on this topic:
What is the minimal additive stretch attainable with linear size emulators?
The only known upper bound for this problem is given by an implicit construction of [Pettie, ICALP 2007] that provides a linear-size emulator with +Õ(n^{1/4}) stretch. No improvement on this problem has been shown since then.
In this work we improve upon the long standing additive stretch of Õ(n^{1/4}), by presenting constructions of linear-size emulators with Õ(n^{0.222}) additive stretch. Our constructions improve the state-of-the-art size vs. stretch tradeoff in the entire regime. For example, for every ε > 1/7, we provide +n^{f(ε)} emulators of size Õ(n^{1+ε}), for f(ε) = 1/5-3ε/5. This should be compared with the current bound of f(ε) = 1/4-3ε/4 by [Pettie, ICALP 2007].
The new emulators are based on an extended and optimized toolkit for computing weighted additive emulators with sublinear distance error. Our key construction provides a weighted modification of the well-known Thorup and Zwick emulators [SODA 2006]. We believe that this TZ variant might be of independent interest, especially for providing improved stretch for distant pairs
Giving Some Slack: Shortcuts and Transitive Closure Compressions
Full text linkWe consider the fundamental problems of reachability shortcuts and compression schemes of the transitive closure (TC) of n-vertex directed acyclic graphs (DAGs) G when we are allowed to neglect the distance (or reachability) constraints for an ε fraction of the pairs in the transitive closure of G, denoted by TC(G).
Shortcuts with Slack. For a directed graph G = (V,E), a d-reachability shortcut is a set of edges H ⊆ TC(G), whose addition decreases the directed diameter of G to be at most d. We introduce the notion of shortcuts with slack which provide the desired distance bound d for all but a small fraction ε of the vertex pairs in TC(G). For ε ∈ (0,1), a (d,ε)-shortcut H ⊆ TC(G) is a subset of edges with the property that dist_{G ∪ H}(u,v) ≤ d for at least (1-ε) fraction of the (u,v) pairs in TC(G). Our constructions hold for any DAG G and their size bounds are parameterized by the width of the graph G defined by the smallest number of directed paths in G that cover all vertices in G.
- For every ε ∈ (0,1] and integer d ≥ 5, every n-vertex DAG G of width {ω} admits a (d,ε)-shortcut of size Õ({ω}²/(ε d)+n). A more delicate construction yields a (3,ε)-shortcut of size Õ({ω}²/(ε d)+n/ε), hence of linear size for {ω} ≤ √n. We show that without a slack (i.e., for ε = 0), graphs with {ω} ≤ √n cannot be shortcut to diameter below n^{1/6} using a linear number of shortcut edges.
- There exists an n-vertex DAG G for which any (3,ε = 1/2^{√{log ω}})-shortcut set has Ω({ω}²/2^{√{log ω}}+n) edges. Hence, for d = Õ(1), our constructions are almost optimal.
Approximate TC Representations. A key application of our shortcut’s constructions is a (1-ε)-approximate all-successors data structure which given a vertex v, reports a list containing (1-ε) fraction of the successors of v in the graph. We present a Õ({ω}²/ε+n)-space data structure with a near linear (in the output size) query time. Using connections to Error Correcting Codes, we also present a near-matching space lower bound of Ω({ω}²+n) bits (regardless of the query time) for constant ε. This improves upon the state-of-the-art space bounds of O({ω} ⋅ n) for ε = 0 by the prior work of Jagadish [ACM Trans. Database Syst., 1990]
Optimal Short Cycle Decomposition in Almost Linear Time
Full text linkShort cycle decomposition is an edge partitioning of an unweighted graph into edge-disjoint short cycles, plus a small number of extra edges not in any cycle. This notion was introduced by Chu et al. [FOCS'18] as a fundamental tool for graph sparsification and sketching. Clearly, it is most desirable to have a fast algorithm for partitioning the edges into as short as possible cycles, while omitting few edges.
The most naïve procedure for such decomposition runs in time O(m * n) and partitions the edges into O(log n)-length edge-disjoint cycles plus at most 2n edges. Chu et al. improved the running time considerably to m^{1+o(1)}, while increasing both the length of the cycles and the number of omitted edges by a factor of n^{o(1)}. Even more recently, Liu-Sachdeva-Yu [SODA'19] showed that for every constant delta in (0,1] there is an O(m * n^{delta})-time algorithm that provides, w.h.p., cycles of length O(log n)^{1/delta} and O(n) extra edges.
In this paper, we significantly improve upon these bounds. We first show an m^{1+o(1)}-time deterministic algorithm for computing nearly optimal cycle decomposition, i.e., with cycle length O(log^2 n) and an extra subset of O(n log n) edges not in any cycle. This algorithm is based on a reduction to low-congestion cycle covers, introduced by the authors in [SODA'19].
We also provide a simple deterministic algorithm that computes edge-disjoint cycles of length 2^{1/epsilon} with n^{1+epsilon}* 2^{1/epsilon} extra edges, for every epsilon in (0,1]. Combining this with Liu-Sachdeva-Yu [SODA'19] gives a linear time randomized algorithm for computing cycles of length poly(log n) and O(n) extra edges, for every n-vertex graphs with n^{1+1/delta} edges for some constant delta.
These decomposition algorithms lead to improvements in all the algorithmic applications of Chu et al. as well as to new distributed constructions
Beating Matrix Multiplication for n^{1/3}-Directed Shortcuts
Full text linkFor an n-vertex digraph G = (V,E) and integer parameter D, a D-shortcut is a small set H of directed edges taken from the transitive closure of G, satisfying that the diameter of G ∪ H is at most D. A recent work [Kogan and Parter, SODA 2022] presented shortcutting algorithms with improved diameter vs. size tradeoffs. Most notably, obtaining linear size D-shortcuts for D = Õ(n^{1/3}), breaking the √n-diameter barrier. These algorithms run in O(n^{ω}) time, as they are based on the computation of the transitive closure of the graph.
We present a new algorithmic approach for D-shortcuts, that matches the bounds of [Kogan and Parter, SODA 2022], while running in o(n^{ω}) time for every D ≥ n^{1/3}. Our approach is based on a reduction to the min-cost max-flow problem, which can be solved in Õ(m+n^{3/2}) time due to the recent breakthrough result of [Brand et al., STOC 2021].
We also demonstrate the applicability of our techniques to computing the minimal chain covers and dipath decompositions for directed acyclic graphs. For an n-vertex m-edge digraph G = (V,E), our key results are:
- An Õ(n^{1/3}⋅ m+n^{3/2})-time algorithm for computing D-shortcuts of linear size for D = Õ(n^{1/3}), and an Õ(n^{1/4}⋅ m+n^{7/4})-time algorithm for computing D-shortcuts of Õ(n^{3/4}) edges for D = Õ(n^{1/2}).
- For a DAG G, we provide Õ(m+n^{3/2})-time algorithms for computing its minimum chain covers, maximum antichain, and decomposition into dipaths and independent sets. This improves considerably over the state-of-the-art bounds by [Caceres et al., SODA 2022] and [Grandoni et al., SODA 2021].
Our results also provide a new connection between shortcutting sets and the seemingly less related problems of minimum chain covers and the maximum antichains in DAGs
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