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Register a new userDeterministic Distributed Vertex Coloring: Simpler, Faster, and without Network Decomposition
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Fabian Kuhn
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We present a simple deterministic distributed algorithm that computes a $(Delta+1)$-vertex coloring in $O(log^2 Delta cdot log n)$ 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 $(Delta+1)$-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 Rozhov{n} and Ghaffari [STOC20]. 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 $(Delta+1)$-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 [STOC18], this improved deterministic algorithm also leads to an improvement in the complexity of randomized algorithms for $(Delta+1)$-coloring, now reaching the bound of $O(log^3log n)$ 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 $(2+epsilon)a$-vertex coloring can be computed in $O(log^3 acdotlog n)$ rounds. We also show that for $Deltageq 3$, a $Delta$-coloring of a $Delta$-colorable graph $G$ can be computed in $O(log^2 Deltacdotlog^2 n)$ rounds.
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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 Rozhov{n} and Ghaffari [STOC20], 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 [FOCS18]. - One drawback of the algorithm of Rozhov{n} 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 $2^{O(sqrt{loglog n})}$ term, which was a clear leftover of the older network decomposition complexity [Panconesi and Srinivasan STOC92]. 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.
In the decremental $(1+epsilon)$-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph $G=(V,E)$ with $n = |V|, m = |E|$, undergoing edge deletions, and a distinguished source $s in V$, and we are asked to process edge deletions efficiently and answer queries for distance estimates $widetilde{mathbf{dist}}_G(s,v)$ for each $v in V$, at any stage, such that $mathbf{dist}_G(s,v) leq widetilde{mathbf{dist}}_G(s,v) leq (1+ epsilon)mathbf{dist}_G(s,v)$. In the decremental $(1+epsilon)$-approximate All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for distance estimates $widetilde{mathbf{dist}}_G(u,v)$ for every $u,v in V$. In this article, we consider the problems for undirected, unweighted graphs. We present a new emph{deterministic} algorithm for the decremental $(1+epsilon)$-approximate SSSP problem that takes total update time $O(mn^{0.5 + o(1)})$. Our algorithm improves on the currently best algorithm for dense graphs by Chechik and Bernstein [STOC 2016] with total update time $tilde{O}(n^2)$ and the best existing algorithm for sparse graphs with running time $tilde{O}(n^{1.25}sqrt{m})$ [SODA 2017] whenever $m = O(n^{1.5 - o(1)})$. In order to obtain this new algorithm, we develop several new techniques including improved decremental cover data structures for graphs, a more efficient notion of the heavy/light decomposition framework introduced by Chechik and Bernstein and the first clustering technique to maintain a dynamic emph{sparse} emulator in the deterministic setting. As a by-product, we also obtain a new simple deterministic algorithm for the decremental $(1+epsilon)$-approximate APSP problem with near-optimal total running time $tilde{O}(mn /epsilon)$ matching the time complexity of the sophisticated but rather involved algorithm by Henzinger, Forster and Nanongkai [FOCS 2013].
We present a randomized distributed algorithm that computes a $Delta$-coloring in any non-complete graph with maximum degree $Delta geq 4$ in $O(log Delta) + 2^{O(sqrt{loglog n})}$ rounds, as well as a randomized algorithm that computes a $Delta$-coloring in $O((log log n)^2)$ rounds when $Delta in [3, O(1)]$. Both these algorithms improve on an $O(log^3 n/log Delta)$-round algorithm of Panconesi and Srinivasan~[STOC1993], which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an $Omega(loglog n)$ round lower bound of Brandt et al.~[STOC16].
In this paper we present a deterministic CONGEST algorithm to compute an $O(kDelta)$-vertex coloring in $O(Delta/k)+log^* n$ rounds, where $Delta$ is the maximum degree of the network graph and $1leq kleq O(Delta)$ can be freely chosen. The algorithm is extremely simple: Each node locally computes a sequence of colors and then it tries colors from the sequence in batches of size $k$. Our algorithm subsumes many important results in the history of distributed graph coloring as special cases, including Linials color reduction [Linial, FOCS87], the celebrated locally iterative algorithm from [Barenboim, Elkin, Goldenberg, PODC18], and various algorithms to compute defective and arbdefective colorings. Our algorithm can smoothly scale between these and also simplifies the state of the art $(Delta+1)$-coloring algorithm. At the cost of losing the full algorithms simplicity we also provide a $O(kDelta)$-coloring algorithm in $O(sqrt{Delta/k})+log^* n$ rounds. We also provide improved deterministic algorithms for ruling sets, and, additionally, we provide a tight characterization for one-round color reduction algorithms.
Distributed vertex coloring is one of the classic problems and probably also the most widely studied problems in the area of distributed graph algorithms. We present a new randomized distributed vertex coloring algorithm for the standard CONGEST model, where the network is modeled as an $n$-node graph $G$, and where the nodes of $G$ operate in synchronous communication rounds in which they can exchange $O(log n)$-bit messages over all the edges of $G$. For graphs with maximum degree $Delta$, we show that the $(Delta+1)$-list coloring problem (and therefore also the standard $(Delta+1)$-coloring problem) can be solved in $O(log^5log n)$ rounds. Previously such a result was only known for the significantly more powerful LOCAL model, where in each round, neighboring nodes can exchange messages of arbitrary size. The best previous $(Delta+1)$-coloring algorithm in the CONGEST model had a running time of $O(logDelta + log^6log n)$ rounds. As a function of $n$ alone, the best previous algorithm therefore had a round complexity of $O(log n)$, which is a bound that can also be achieved by a na{i}ve folklore algorithm. For large maximum degree $Delta$, our algorithm hence is an exponential improvement over the previous state of the art.
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