Optimal (Degree+1)-Coloring in Congested Clique

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 (?+1)-coloring and (?+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

Optimal (degree+1)-Coloring in Congested Clique

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 (Δ + 1)-coloring and (Δ + 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

Deterministic massively parallel connectivity

We consider the problem of designing fundamental graph algorithms on the model of Massive Parallel Computation (MPC). The input to the problem is an undirected grap

On parallel k-center clustering

We consider the classic k-center problem in a parallel setting, on the low-local-space Massively Parallel Computation (MPC) model, with local space per machine of O (nδ), where δ ∈ (0,1) is an arbitrary constant. As a central clustering problem, the k-center problem has been studied extensively. Still, until very recently, all parallel MPC algorithms have been requiring Ω(k) or even Ω(knδ) local space per machine. While this setting covers the case of small values of k, for a large number of clusters these algorithms require large local memory, making them poorly scalable. The case of large k,k ≥ Ω(nδ), has been considered recently for the low-local-space MPC model by Bateni et al. (2021), who gave an O (log log n)-round MPC algorithm that produces k(1 + ο (1)) centers whose cost has multiplicative approximation of O (log log log n). In this paper we extend the algorithm of Bateni et al. and design a low-local-space MPC algorithm that in O (log log n) rounds returns a clustering with k(1 + ο(1)) clusters that is an O (log*n)-approximation for k-center

Optimal (degree+1)-coloring in congested clique

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 (Δ + 1)-coloring and (Δ + 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

Parallel derandomization for coloring

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

Near-Shortest Path Routing in Hybrid Communication Networks

Hybrid networks, i.e., networks that leverage different means of communication, become ever more widespread. To allow theoretical study of such networks, [Augustine et al., SODA\u2720] introduced the HYBRID model, which is based on the concept of synchronous message passing and uses two fundamentally different principles of communication: a local mode, which allows every node to exchange one message per round with each neighbor in a local communication graph; and a global mode where any pair of nodes can exchange messages, but only few such exchanges can take place per round. A sizable portion of the previous research for the HYBRID model revolves around basic communication primitives and computing distances or shortest paths in networks. In this paper, we extend this study to a related fundamental problem of computing compact routing schemes for near-shortest paths in the local communication graph. We demonstrate that, for the case where the local communication graph is a unit-disc graph with n nodes that is realized in the plane and has no radio holes, we can deterministically compute a routing scheme that has constant stretch and uses labels and local routing tables of size O(log n) bits in only O(log n) rounds

Routing schemes for hybrid communication networks

We consider the problem of computing routing schemes in the HYBRID model of distributed computing where nodes have access to two fundamentally different communication modes. In this problem nodes have to compute small labels and routing tables that allow for efficient routing of messages in the local network, which typically offers the majority of the throughput. Recent work has shown that using the HYBRID model admits a significant speed-up compared to what would be possible if either communication mode were used in isolation. Nonetheless, if general graphs are used as the input graph the computation of routing schemes still takes polynomial rounds in the HYBRID model. We bypass this lower bound by restricting the local graph to unit-disc-graphs and solve the problem deterministically with running time O(|H|2+log⁡n), label size O(log⁡n), and size of routing tables O(|H|2⋅log⁡n) where |H| is the number of “radio holes” in the network. Our work builds on recent work by Coy et al., who obtain this result in the much simpler setting where the input graph has no radio holes. We develop new techniques to achieve this, including a decomposition of the local graph into path-convex regions, where each region contains a shortest path for any pair of nodes in it

Analysis of Neptune's 2017 Bright Equatorial Storm

We report the discovery of a large ($\sim$8500 km diameter) infrared-bright storm at Neptune's equator in June 2017. We tracked the storm over a period of 7 months with high-cadence infrared snapshot imaging, carried out on 14 nights at the 10 meter Keck II telescope and 17 nights at the Shane 120 inch reflector at Lick Observatory. The cloud feature was larger and more persistent than any equatorial clouds seen before on Neptune, remaining intermittently active from at least 10 June to 31 December 2017. Our Keck and Lick observations were augmented by very high-cadence images from the amateur community, which permitted the determination of accurate drift rates for the cloud feature. Its zonal drift speed was variable from 10 June to at least 25 July, but remained a constant $237.4 \pm 0.2$ m s$^{-1}$ from 30 September until at least 15 November. The pressure of the cloud top was determined from radiative transfer calculations to be 0.3-0.6 bar; this value remained constant over the course of the observations. Multiple cloud break-up events, in which a bright cloud band wrapped around Neptune's equator, were observed over the course of our observations. No "dark spot" vortices were seen near the equator in HST imaging on 6 and 7 October. The size and pressure of the storm are consistent with moist convection or a planetary-scale wave as the energy source of convective upwelling, but more modeling is required to determine the driver of this equatorial disturbance as well as the triggers for and dynamics of the observed cloud break-up events.Comment: 42 pages, 14 figures, 6 tables; Accepted to Icaru