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We consider the distributed complexity of the (degree+1)-list coloring problem, in which each node $u$ of degree $d(u)$ is assigned a palette of $d(u)+1$ colors, and the goal is to find a proper coloring using these color palettes. The (degree+1)-list coloring problem is a natural generalization of the classical $(\Delta+1)$-coloring and $(\Delta+1)$-list coloring problems, both being benchmark problems extensively studied in distributed and parallel computing. In this paper we settle the complexity of the (degree+1)-list coloring problem in the Congested Clique model by showing that it can be solved deterministically in a constant number of rounds., Comment: 27 pages. Appeared at ICALP 2023

In the study of radio networks, the tasks of broadcasting (propagating a message throughout the network) and leader election (having the network agree on a node to designate `leader') are two of the most fundamental global problems, and have a long history of work devoted to them. This work has two divergent strands: some works focus on exploiting the geometric properties of wireless networks based in physical space, while others consider general graphs. Algorithmic results in each of these avenues have often used quite different techniques, and produced bounds using incomparable parametrizations. In this work, we unite the study of general-graph and geometric-based radio networks, by adapting the broadcast and leader election algorithm of Czumaj and Davies (JACM '21) to achieve a running-time parametrized by the independence number of the network (i.e., the size of the maximum independent set). This parametrization preserves the running time on general graphs, matching the best known, but also improves running times to near-optimality across a wide range of geometric-based graph classes. As part of this algorithm, we also provide the first algorithm for computing a maximal independent set in general-graph radio networks. This algorithm runs in $O(\log^3 n)$ time-steps, only a $\log n$ factor away from the $\Omega(\log^2 n)$ lower bound., Comment: To appear at PODC 2023

Beeping models are models for networks of weak devices, such as sensor networks or biological networks. In these networks, nodes are allowed to communicate only via emitting beeps: unary pulses of energy. Listening nodes only the capability of {\it carrier sensing}: they can only distinguish between the presence or absence of a beep, but receive no other information. The noisy beeping model further assumes listening nodes may be disrupted by random noise. Despite this extremely restrictive communication model, it transpires that complex distributed tasks can still be performed by such networks. In this paper we provide an optimal procedure for simulating general message passing in the beeping and noisy beeping models. We show that a round of \textsf{Broadcast CONGEST} can be simulated in $O(\Delta\log n)$ round of the noisy (or noiseless) beeping model, and a round of \textsf{CONGEST} can be simulated in $O(\Delta^2\log n)$ rounds (where $\Delta$ is the maximum degree of the network). We also prove lower bounds demonstrating that no simulation can use asymptotically fewer rounds. This allows a host of graph algorithms to be efficiently implemented in beeping models. As an example, we present an $O(\log n)$-round \textsf{Broadcast CONGEST} algorithm for maximal matching, which, when simulated using our method, immediately implies a near-optimal $O(\Delta \log^2 n)$-round maximal matching algorithm in the noisy beeping model., Comment: To appear at PODC 2023

Graph coloring problems are among the most fundamental problems in parallel and distributed computing, and have been studied extensively in both settings. In this context, designing efficient deterministic algorithms for these problems has been found particularly challenging. In this work we consider this challenge, and design a novel framework for derandomizing algorithms for coloring-type problems in the Massively Parallel Computation (MPC) model with sublinear space. We give an application of this framework by showing that a recent $(degree+1)$-list coloring algorithm by Halldorsson et al. (STOC'22) in the LOCAL model of distributed computation can be translated to the MPC model and efficiently derandomized. Our algorithm runs in $O(\log \log \log n)$ rounds, which matches the complexity of the state of the art algorithm for the $(\Delta + 1)$-coloring problem., Comment: 26 Pages. The paper will appear in IPDPS 2024

The Lov\'asz 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 Lov\'asz 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 Lov\'asz 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 $O(\frac{d}{\log d}+\log^{O(1)}\log n)$. 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 $\Delta+o(\Delta)$-edge coloring a graph of maximum degree $\Delta$. This is an almost exponential improvement over previous results: no prior $\log^{o(1)} n$-round algorithm was known even for $2\Delta-2$-edge coloring., Comment: Accepted at SODA 2023