Recent improvements on the deterministic complexities of fundamental graph problems in the LOCAL model of distributed computing have yielded state-of-the-art upper bounds of \(\tilde{O}(\log^{5/3} n)\) rounds for maximal independent set (MIS) and \((\Delta + 1)\)-coloring [Ghaffari, Grunau, FOCS’24], and \(\tilde{O}(\log^{19/9} n)\) rounds for the more restrictive \(\Delta\)-coloring problem [Ghaffari, Kuhn, FOCS’21; Ghaffari, Grunau, FOCS’24; Bourreau, Brandt, Nolin, STOC’25]. In our work, we show that \(\Delta\)-coloring can be solved deterministically in \(\tilde{O}(\log^{5/3} n)\) rounds as well, matching the currently best bound for \((\Delta + 1)\)-coloring.

In a rendezvous task, a set of mobile agents initially dispersed in a network have to gather at an arbitrary common site. We consider the rendezvous problem on the infinite labeled line, with 2 initially asleep agents, without communication, and a synchronous notion of time. Each node on the line is labeled with a unique positive integer. The initial distance between the two agents is denoted by D. Time is divided into rounds and measured from the moment an agent first wakes up. We denote by τ the delay between the two agents' wake up times. If awake in a given round T, an agent at a node v has three options: stay at the node v, take port 0, or take port 1. If it decides to stay, the agent will still be at node v in round T+1. Otherwise, it will be at one of the two neighbors of v on the infinite line, depending on the port it chose. The agents achieve rendezvous in T rounds if they are at the same node in round T. We aim for a deterministic algorithm for this problem. The problem was recently considered by Miller and Pelc [Distributed Computing, 2025]. With 𝓁_{max} the largest label of the two starting nodes, they showed that no algorithm can guarantee rendezvous in o(D log^* 𝓁_{max}) rounds. The lower bound follows from a connection with the LOCAL model of distributed computing, and holds even if the agents are guaranteed simultaneous wake-up (τ = 0) and are told their initial distance D. Miller and Pelc also gave an algorithm of optimal matching complexity O(D log^* 𝓁_{max}) when the agents know D, but only obtained the higher bound of O(D² (log^* 𝓁_{max})³) when D is unknown to the agents. In this paper, we improve this second complexity to a tight O(D log^* 𝓁_{max}), closing the gap between the best known lower and upper bounds. In fact, our algorithm achieves rendezvous in O(D log^* 𝓁_{min}) rounds, where 𝓁_{min} is the smallest label within distance O(D) of the two starting positions. We obtain this result by having the agents compute sparse subsets of the nodes to gather at (formally, ruling sets over the line), as well as some general observations about the setting of rendezvous on labeled graphs.

In a rendezvous task, a set of mobile agents initially dispersed in a network have to gather at an arbitrary common site. We consider the rendezvous problem on the infinite labeled line, with 2 initially asleep agents, without communication, and a synchronous notion of time. Each node on the line is labeled with a unique positive integer. The initial distance between the two agents is denoted by D. Time is divided into rounds. We count time from the first moment that an agent wakes up, and denote by τ the delay in two agents' wake up times. If awake in a given round T, an agent at a node υ has three options: stay at the node υ, take port 0, or take port 1. If it decides to stay, the agent will still be at node υ in round T + 1. Otherwise, it will be at one of the two neighbors of υ on the infinite line, depending on the port it chose. The agents achieve rendezvous in T rounds if they are at the same node in round T. We aim for a deterministic algorithm for this problem.

Brooks’ theorem states that all connected graphs but odd cycles and cliques can be colored with Δ colors, where Δ is the maximum degree of the graph. Such colorings have been shown to admit non-trivial distributed algorithms [Panconesi and Srinivasan, Combinatorica 1995] and have been studied intensively in the distributed literature. In particular, it is known that any deterministic algorithm computing a Δ-coloring requires Ω(logn) rounds in the LOCAL model [Chang, Kopelowitz, and Pettie, FOCS 2016], and that this lower bound holds already on constant-degree graphs. In contrast, the best upper bound in this setting is given by an O(log2 n)-round deterministic algorithm that can be inferred already from the works of [Awerbuch, Goldberg, Luby, and Plotkin, FOCS 1989] and [Panconesi and Srinivasan, Combinatorica 1995] roughly three decades ago, raising the fundamental question about the true complexity of Δ-coloring in the constant-degree setting. We answer this long-standing question almost completely by providing an almost-optimal deterministic O(logn log* n)-round algorithm for Δ-coloring, matching the lower bound up to a log* n-factor. Similarly, in the randomized LOCAL model, we provide an O(loglogn log* n)-round algorithm, improving over the state-of-the-art upper bound of O(log2 logn) [Ghaffari, Hirvonen, Kuhn, and Maus, Distributed Computing 2021] and almost matching the Ω(loglogn)-round lower bound by [BFHKLRSU, STOC 2016]. Our results make progress on several important open problems and conjectures. One key ingredient for obtaining our results is the introduction of ruling subgraph families as a novel tool for breaking symmetry between substructures of a graph, which we expect to be of independent interest.

Given a graph $G$ and a positive integer $k$, the Graphlet Sampling problem asks to sample a connected induced $k$-vertex subgraph of $G$ uniformly at random. Graphlet sampling enhances machine learning applications by transforming graph structures into feature vectors for tasks such as graph classification and subgraph identification, boosting neural network performance, and supporting clustered federated learning by capturing local structures and relationships. A recent work has shown that the problem admits an algorithm that preprocesses $G$ in time $O(nk^2 \log k + m)$, and draws one sample in expected time $k^{O(k)} \log n$, where $n=|V(G)|$ and $m=|E(G)|$. Such an algorithm relies on the assumption that the input graph fits into main memory and it does not seem to be straightforward to adapt it to very large graphs. We consider Graphlet Sampling in the semi-streaming setting, where we have a memory of $M = \Omega(n \log n)$ words, and $G$ can be only read through sequential passes over the edge list. We develop a semi-streaming algorithm that preprocesses $G$ in $p={O}(\log n)$ passes and samples $\Theta(M k^{-O(k)})$ independent uniform $k$-graphlets in $O(k)$ passes. For constant $k$, both phases run in time $O((n+m)\log n)$. We also show that the tradeoff between memory and number of passes of our algorithms is near-optimal. Our extensive evaluation on very large graphs shows the effectiveness of our algorithms.