The problem of edge coloring has been extensively studied over the years. Recently, this problem has received significant attention in the dynamic setting, where we are given a dynamic graph evolving via a sequence of edge insertions and deletions and our objective is to maintain an edge coloring of the graph.
Currently, it is not known whether it is possible to maintain a $(Δ+ O(Δ^{1 - μ}))$-edge coloring in $\tilde{O}(1)$ update time, for any constant $μ> 0$, where $Δ$ is the maximum degree of the graph. In this paper, we show how to efficiently maintain a $(Δ+ O(α))$-edge coloring in $\tilde O(1)$ amortized update time, where $α$ is the arboricty of the graph. Thus, we answer this question in the affirmative for graphs of sufficiently small arboricity.
Graph coloring is a fundamental problem in computer science. We study the fully dynamic version of the problem in which the graph is undergoing edge insertions and deletions and we wish to maintain a vertex-coloring with small update time after each insertion and deletion. We show how to maintain an $O(α\lg n)$-coloring with polylogarithmic update time, where $n$ is the number of vertices in the graph and $α$ is the current arboricity of the graph. This improves upon a result by Solomon and Wein (ESA'18) who maintained an $O(α_{\max}\lg^2 n)$-coloring, where $α_{\max}$ is the maximum arboricit…
In this paper we consider the problem of testing whether a graph has bounded arboricity. The family of graphs with bounded arboricity includes, among others, bounded-degree graphs, all minor-closed graph classes (e.g. planar graphs, graphs with bounded treewidth) and randomly generated preferential attachment graphs. Graphs with bounded arboricity have been studied extensively in the past, in particular since for many problems they allow for much more efficient algorithms and/or better approximation ratios. We present a tolerant tester in the sparse-graphs model. The sparse-graphs model allows…
We consider the problem of approximating the arboricity of a graph $G= (V,E)$, which we denote by $\mathsf{arb}(G)$, in sublinear time, where the arboricity of a graph is the minimal number of forests required to cover its edges. An algorithm for this problem may perform degree and neighbor queries, and is allowed a small error probability. We design an algorithm that outputs an estimate $\hatα$, such that with probability $1-1/\textrm{poly}(n)$, $\mathsf{arb}(G)/c\log^2 n \leq \hatα \leq \mathsf{arb}(G)$, where $n=|V|$ and $c$ is a constant. The expected query complexity and running time of t…
In the implicit dynamic colouring problem, the task is to maintain a representation of a proper colouring as a dynamic graph is subject to insertions and deletions of edges, while facilitating interspersed queries to the colours of vertices. The goal is to use few colours, while still efficiently handling edge-updates and responding to colour-queries. For an n-vertex dynamic graph of arboricity $α$, we present an algorithm that maintains an implicit vertex colouring with $4\cdot2^α$ colours, in amortised poly-$(\log n)$ update time, and with $O(α log n)$ worst-case query time. The previous bes…
We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the out-degree of each vertex is bounded. On one hand, we show how to orient the edges such that the out-degree of each vertex is proportional to the arboricity $α$ of the graph, in a worst-case update time of $O(\log^3 n \log α)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off, namely the improved worst case update time of $O(\log ^2 n \log α)$ for the problem of maintaining an edge-orientation with at most $O(α+ \log n)$ out-edges…
The arboricity $Γ$ of a graph is the minimum number of forests its edge set can be partitioned into. Previous approximation schemes were nonconstructive, i.e., they only approximated the arboricity as a value without computing a corresponding forest partition. This is because they operate on the related pseudoforest partitions or the dual problem of finding dense subgraphs. We propose an algorithm for converting a partition of $k$ pseudoforests into a partition of $k+1$ forests in $O(mk\log k + m \log n)$ time with a data structure by Brodal and Fagerberg that stores graphs of arboricity $k$. …
We study the problem of testing triangle freeness in the general graph model. This problem was first studied in the general graph model by Alon et al. (SIAM J. Discret. Math. 2008) who provided both lower bounds and upper bounds that depend on the number of vertices and the average degree of the graph. Their bounds are tight only when $d_{\rm max} = O(d)$ and $\bar{d} \leq \sqrt{n}$ or when $\bar{d} = Θ(1)$, where $d_{\rm max}$ denotes the maximum degree and $\bar{d}$ denotes the average degree of the graph. In this paper we provide bounds that depend on the arboricity of the graph and the ave…
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 …
Given query access to an undirected graph $G$, we consider the problem of computing a $(1\pmε)$-approximation of the number of $k$-cliques in $G$. The standard query model for general graphs allows for degree queries, neighbor queries, and pair queries. Let $n$ be the number of vertices, $m$ be the number of edges, and $n_k$ be the number of $k$-cliques. Previous work by Eden, Ron and Seshadhri (STOC 2018) gives an $O^*(\frac{n}{n^{1/k}_k} + \frac{m^{k/2}}{n_k})$-time algorithm for this problem (we use $O^*(\cdot)$ to suppress $\poly(\log n, 1/ε, k^k)$ dependencies). Moreover, this bound is ne…
In this paper, we revisit the problem of sampling edges in an unknown graph $G = (V, E)$ from a distribution that is (pointwise) almost uniform over $E$. We consider the case where there is some a priori upper bound on the arboriciy of $G$. Given query access to a graph $G$ over $n$ vertices and of average degree $d$ and arboricity at most $α$, we design an algorithm that performs $O\!\left(\fracα{d} \cdot \frac{\log^3 n}{\varepsilon}\right)$ queries in expectation and returns an edge in the graph such that every edge $e \in E$ is sampled with probability $(1 \pm \varepsilon)/m$. The algorithm…
Identifying clusters of similar elements in a set is a common task in data analysis. With the immense growth of data and physical limitations on single processor speed, it is necessary to find efficient parallel algorithms for clustering tasks. In this paper, we study the problem of correlation clustering in bounded arboricity graphs with respect to the Massively Parallel Computation (MPC) model. More specifically, we are given a complete graph where the edges are either positive or negative, indicating whether pairs of vertices are similar or dissimilar. The task is to partition the vertices …
We study the problem of coloring a given graph using a small number of colors in several well-established models of computation for big data. These include the data streaming model, the general graph query model, the massively parallel computation (MPC) model, and the CONGESTED-CLIQUE and the LOCAL models of distributed computation. On the one hand, we give algorithms with sublinear complexity, for the appropriate notion of complexity in each of these models. Our algorithms color a graph $G$ using about $κ(G)$ colors, where $κ(G)$ is the degeneracy of $G$: this parameter is closely related to …
We present trade-offs in the incremental and fully dynamic settings to maintian a proper coloring. For any fully dynamic $2$-coloring algorithm, the maximum of the update time, number of recolorings, and query time is $Ω(\log n)$. We present a deterministic fully dynamic $2$-coloring algorithm with $O(\log^2 n)$ amortized update time, $O(\log n)$ amortized query time, and one recoloring in the worst case. For any incremental $2$-coloring algorithm which explicitly maintains the color of every vertex after each update, the amortized update time and the amortized number of recolorings is $Ω(\log…
Given a dynamic graph subject to edge insertions and deletions, we show how to update an implicit representation of a proper vertex colouring, such that colours of vertices are computable upon query time. We give a deterministic algorithm that uses $O(α^2)$ colours for a dynamic graph of arboricity $α$, and a randomised algorithm that uses $O(\min\{α\log α, α\log \log \log n\})$ colours in the oblivious adversary model. Our deterministic algorithm has update- and query times polynomial in $α$ and $\log n$, and our randomised algorithm has amortised update- and query time that with high probabi…
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 impr…
We provide novel deterministic distributed vertex coloring algorithms. As our main result, we give a deterministic distributed algorithm to compute a $(Δ+1)$-coloring of an $n$-node graph with maximum degree $Δ$ in $2^{O(\sqrt{\logΔ})}\cdot\log n$ rounds. This improves on the best previously known time complexity for a large range of values of $Δ$. For graphs with arboricity $a$, we obtain a deterministic distributed algorithm to compute a $(2+o(1))a$-coloring in time $2^{O(\sqrt{\log a})}\cdot\log^2 n$. Further, for graphs with bounded neighborhood independence, we show that a $(Δ+1)$-colorin…
The problem of coloring the edges of an $n$-node graph of maximum degree $Δ$ with $2Δ- 1$ colors is one of the key symmetry breaking problems in the area of distributed graph algorithms. While there has been a lot of progress towards the understanding of this problem, the dependency of the running time on $Δ$ has been a long-standing open question. Very recently, Kuhn [SODA '20] showed that the problem can be solved in time $2^{O(\sqrt{\logΔ})}+O(\log^* n)$. In this paper, we study the edge coloring problem in the distributed LOCAL model. We show that the $(\mathit{degree}+1)$-list edge colori…
An injective $k$-edge-coloring of a graph $G$ is an assignment of colors, i.e. integers in $\{1, \ldots , k\}$, to the edges of $G$ such that any two edges each incident with one distinct endpoint of a third edge, receive distinct colors. The problem of determining whether such a $k$-coloring exists is called k-INJECTIVE EDGE-COLORING. We show that 3-INJECTIVE EDGE-COLORING is NP-complete, even for triangle-free cubic graphs, planar subcubic graphs of arbitrarily large girth, and planar bipartite subcubic graphs of girth~6. 4-INJECTIVE EDGE-COLORING remains NP-complete for cubic graphs. For an…
