We investigate the minimum weight cycle (MWC) problem in the 𝖢𝖮𝖭𝖦𝖤𝖲𝖳 model of distributed computing. For undirected weighted graphs, we design a randomized algorithm that achieves a (k+1)-approximation, for any real number k ≥ 1. The round complexity of algorithm is Õ( n^k+1/2k+1 + n^1/k + D n^1/2(2k+1) + D^2/5 n^2/5+1/2(2k+1)). where n denotes the number of nodes and D is the unweighted diameter of the graph. This result yields a smooth trade-off between approximation ratio and round complexity. In particular, when k ≥ 2 and D = Õ(n^1/4), the bound simplifies to Õ( n^k+1/2k+1) On the lower bound side, assuming the Erdős girth conjecture, we prove that for every integer k ≥ 1, any randomized (k+1-ε)-approximation algorithm for MWC requires ( n^k+1/2k+1) rounds. This lower bound holds for both directed unweighted and undirected weighted graphs, and applies even to graphs with small diameter D = Θ(log n). Taken together, our upper and lower bounds match up to polylogarithmic factors for graphs of sufficiently small diameter D = Õ(n^1/4) (when k ≥ 2), yielding a nearly tight bound on the distributed complexity of the problem. Our results improve upon the previous state of the art: Manoharan and Ramachandran (PODC 2024) demonstrated a (2+ε)-approximation algorithm for undirected weighted graphs with round complexity Õ(n^2/3+D), and proved that for any arbitrarily large number α, any α-approximation algorithm for directed unweighted or undirected weighted graphs requires Ω(√(n)/log n) rounds.

In this paper, we present a new randomized O(1)-approximation algorithm for the All-Pairs Shortest Paths (APSP) problem in weighted undirected graphs that runs in just O(logloglog n) rounds in the Congested-Clique model. Before our work, the fastest algorithms achieving an O(1)-approximation for APSP in weighted undirected graphs required poly(log n) rounds, as shown by Censor-Hillel, Dory, Korhonen, and Leitersdorf (PODC 2019 Distributed Computing 2021). In the unweighted undirected setting, Dory and Parter (PODC 2020 Journal of the ACM 2022) obtained O(1)-approximation in poly(loglog n) rounds. By terminating our algorithm early, for any given parameter t ≥ 1, we obtain an O(t)-round algorithm that guarantees an O(log^1/2^t n) approximation in weighted undirected graphs. This tradeoff between round complexity and approximation factor offers flexibility, allowing the algorithm to adapt to different requirements. In particular, for any constant ε > 0, an O(log^ε n)-approximation can be obtained in O(1) rounds. Previously, O(1)-round algorithms were only known for O(log n)-approximation, as shown by Chechik and Zhang (PODC 2022). A key ingredient in our algorithm is a lemma that, under certain conditions, allows us to improve an a-approximation for APSP to an O(√(a))-approximation in O(1) rounds. To prove this lemma, we develop several new techniques, including an O(1)-round algorithm for computing the k-nearest nodes, as well as new types of hopsets and skeleton graphs based on the notion of k-nearest nodes.

We show that almost stable matching can be solved in constant distributed rounds on general bipartite graphs G=(V,E) using only a few shared random bits. Specifically, in the model, we compute a matching whose expected number of blocking pairs is at most ε |E| in O(log(1/ε)/ε^4) rounds using O(log(1/ε)) shared random bits. Thus, for every constant ε>0, the round complexity is O(1), independent of the number of vertices and the maximum degree. Previous algorithms achieve constant round complexity only for bounded-degree or almost-regular graphs; on general graphs, their round complexity depends polylogarithmically on n. Our main technical idea is a degree-guarded freezing rule that allows widely varying degrees to be handled by a single global charging argument, avoiding the Θ(log n) successive degree thresholds used in previous work. The shared random bits are used only to select a common random output iteration. As consequences, we obtain an O( log(1/ε)/ε^4 + log n/ε)-round algorithm without pre-shared randomness, via a low-diameter decomposition, and an O(log(1/ε)/ε^4)-round algorithm in the fully-scalable Massively Parallel Computation () model with linear total memory.

The content-oblivious model, introduced by Censor-Hillel, Cohen, Gelles, and Sela (PODC 2022; Distributed Computing 2023), captures an extremely weak form of communication where nodes can only send asynchronous, content-less pulses. They showed that in 2-edge-connected networks, any distributed algorithm can be simulated in the content-oblivious model, provided that a unique leader is designated a priori. Subsequent works of Frei, Gelles, Ghazy, and Nolin (DISC 2024) and Chalopin et al. (DISC 2025) developed content-oblivious leader election algorithms, first for unoriented rings and then for general 2-edge-connected graphs. These results establish that all graph problems are solvable in content-oblivious, 2-edge-connected networks. Much less is known about networks that are not 2-edge-connected. Censor-Hillel, Cohen, Gelles, and Sela showed that no non-constant function f (x, y) can be computed correctly by two parties using content-oblivious communication over a single edge, where one party holds x and the other holds y. This seemingly ruled out many natural graph problems on non-2-edge-connected graphs. In this work, we show that, with the knowledge of network topology G, leader election is possible in a wide range of graphs. Our main contributions are as follows: Impossibility: Graphs symmetric about an edge admit no randomized terminating leader election algorithm, even when nodes have unique identifiers and full knowledge of G. Leader election algorithms: Trees that are not symmetric about any edge admit a quiescently terminating leader election algorithm with topology knowledge, even in anonymous networks, using O(n(2)) messages, where n is the number of nodes. Moreover, even-diameter trees admit a terminating leader election given only the knowledge of the network diameter D = 2r, with message complexity O(nr). Necessity of topology knowledge: In the family of graphs G = {P-3, P-5}, both the 3-path P-3 and the 5-path P-5 admit a quiescently terminating leader election if nodes know the topology exactly. However, if nodes only know that the underlying topology belongs to G, then terminating leader election is impossible. 2012 ACM Subject Classification Theory of computation -> Distributed algorithms

We study the leader election problem in oriented ring networks under content-oblivious asynchronous message-passing systems, where an adversary may arbitrarily corrupt message contents. Frei et al. (DISC 2024) presented a uniform terminating leader election algorithm for oriented rings in this setting, with message complexity O(n ·𝖨𝖣_max) on a ring of size n, where 𝖨𝖣_max is the largest identifier in the system, this result has been recently extended by Chalopin et al. (DISC 2025) to unoriented rings. In this paper, we investigate the message complexity of leader election on ring networks in the content-oblivious model, showing that no uniform algorithm can solve the problem if each process is limited to sending a constant number of messages in one direction. Interestingly, this limitation hinges on the uniformity assumption. In the non-uniform setting, where processes know an upper bound U ≥ n on the ring size, we present an algorithm with message complexity O(n · U ·𝖨𝖣_min), in which each process sends O(U ·𝖨𝖣_min) messages clockwise and only three messages counter-clockwise. Here, 𝖨𝖣_min is the smallest identifier in the system. This dependence on the identifiers compares favorably with the dependence on 𝖨𝖣_max of Frei et al. We also show a non-uniform algorithm where each process sends O(U ·log𝖨𝖣_min) messages in one direction and O(log𝖨𝖣_min) in the other. The factor log𝖨𝖣_min is optimal, matching the lower bound of Frei et al. Finally, in the anonymous setting, where processes do not have identifiers, we propose a randomized algorithm where each process sends only O(log^2 U) messages, with a success probability of 1 - U^-c.

Censor-Hillel, Cohen, Gelles, and Sela (PODC 2022 Distributed Computing 2023) studied fully-defective asynchronous networks, where communication channels may suffer an extreme form of alteration errors, rendering messages completely corrupted. The model is equivalent to content-oblivious computation, where nodes communicate solely via pulses. They showed that if the network is 2-edge-connected, then any algorithm for a noiseless setting can be simulated in the fully-defective setting; otherwise, no non-trivial computation is possible in the fully-defective setting. However, their simulation requires a predesignated leader, which they conjectured to be necessary for any non-trivial content-oblivious task. Recently, Frei, Gelles, Ghazy, and Nolin (DISC 2024) refuted this conjecture for the special case of oriented ring topology. They designed two asynchronous content-oblivious leader election algorithms with message complexity O(n ·𝖨𝖣_𝗆𝖺𝗑) , where n is the number of nodes and 𝖨𝖣_𝗆𝖺𝗑 is the maximum . The first algorithm stabilizes in unoriented rings without termination detection. The second algorithm quiescently terminates in oriented rings, thus enabling the execution of the simulation algorithm after leader election. In this work, we present two results:

In the content-oblivious (CO) model (proposed by Censor-Hillel et al.), processes inhabit an asynchronous network and communicate only by exchanging pulses. A series of works has clarified the computational power of this model. In particular, it was shown that, when a leader is present and the network is 2-edge-connected, content-oblivious communication can simulate classical asynchronous message passing. Subsequent results extended this equivalence to leaderless oriented and unoriented rings, and, under non-uniform assumptions, to general 2-edge-connected networks. The simulator of Censor-Hillel et al. requires O(n^3b+n^3log n) pulses to emulate the send of a single b-bit message, making it impractical even on modest-size networks. We focus on message-efficient computation in CO networks. We study the fundamental problem of counting in ring topologies, both because knowing the exact network size is a basic prerequisite for many distributed tasks and because counting immediately implies a broad class of aggregation primitives. We give an algorithm that counts using O(n^1.5) pulses in anonymous rings with a leader, an O(nlog^2 n) algorithm for counting in rings with IDs. Moreover, we show that any counting algorithm in CO requires Ω(nlog n) pulses. Interestingly, in the course of this investigation, we design a simulator for classic message passing: in one simulated round, each process can send a b-bit message to each of its neighbors using only O(b) pulses per process. The simulator extends to general 2-edge-connected networks, after a pre-processing step that requires O(n^8log n) pulses, where n is the number of processes, allowing thus efficient simulation of asynchronous message passing in general 2-edge-connected networks.

We study the aggregation problem in synchronous multi-hop radio networks with O(log n)-bit messages and no collision detection. Each node initially holds a value, and the goal is to compute a global aggregate such as the sum of all values. Aggregation tasks arise naturally in wireless sensor networks, where nodes are often battery-powered and radio activity is the dominant source of energy consumption. Accordingly, our main objective is to minimize the energy complexity, defined as the maximum number of rounds in which any node is awake. Our main result is a randomized distributed algorithm that, with high probability, constructs and executes an aggregation schedule in O(n polylog n) rounds and using O(Δ^∗polylog n) energy, where Δ^∗ is the minimum possible maximum degree of a spanning tree of the network graph. This guarantee is nearly optimal: for any aggregation schedule and any graph, there exists a node that must be awake for at least Δ^∗ rounds. As a by-product, the algorithm also computes a spanning tree whose maximum degree is within an O(log n) factor of Δ^∗, with the same round and energy guarantees. For every tree edge, both endpoints learn that the edge belongs to the tree.

We introduce a graph decomposition primitive called introvert clustering, which strengthens standard low-diameter clustering by guaranteeing that every clustered vertex keeps at least a (1/2-ε)-fraction of its relevant neighbors in its own cluster. Repeatedly applying this primitive yields a layered introvert network decomposition with O(log n) layers and weak diameter O(log n). We give two applications in the 𝖫𝖮𝖢𝖠𝖫 model. For every constant ε>0, we obtain a O(log^2 n)-round deterministic algorithm for list (3/2+ε)Δ-edge coloring on graphs of maximum degree Δ_0(ε); for bipartite graphs, the result holds for all Δ. For every constant 0<ε<1/4, we also obtain a O(log^2 n)-round deterministic algorithm for a (1/4-ε)-locally balanced cut, where every vertex has at least a (1/4-ε)-fraction of its neighbors on the opposite side. The resulting algorithms are remarkably simple: edge coloring processes the layers in reverse order and colors each cluster, while locally balanced cut processes them forward and computes a locally maximum cut within each cluster. The introvert guarantee enables these procedures beyond the usual greedy regime of network decomposition. We construct the decomposition in O(log^2 n) randomized rounds using Miller–Peng–Xu low-diameter clustering and a simple trimming procedure, and deterministically in O(log^2 n) rounds via a white-box adaptation of the recursive network decomposition algorithm of Ghaffari and Grunau [FOCS 2024].

In the \(\textsf{LOCAL}\) model of distributed computing, low-diameter decomposition is a fundamental tool for algorithm design, as it enables a reduction from general graphs to low-diameter graphs where brute-force information gathering can be performed efficiently. Chang and Su [PODC 2022] showed that any high-conductance network excluding a fixed minor contains a high-degree vertex \(v^{\star}\) , allowing the entire graph topology to be gathered at \(v^{\star}\) efficiently in the \(\textsf{CONGEST}\) model via expander routing . Consequently, in such networks, many problems that admit efficient \(\textsf{LOCAL}\) algorithms via low-diameter decomposition can also be solved efficiently in \(\textsf{CONGEST}\) using expander decomposition . In this work, we present improved decomposition and routing algorithms for networks excluding a fixed minor. We define an \((\epsilon,D,T)\) -decomposition of a graph \(G=(V,E)\) as a partition of \( V \) into clusters of diameter at most \( D \) , with at most \(\epsilon|E|\) inter-cluster edges, such that information gathering within each cluster can be completed in \( T \) rounds in parallel. We show that an \((\epsilon,D,T)\) -decomposition with \(\begin{align*} D=O(\epsilon^{-1})\quad\text{and}\quad T=\min\left\{2^{O\left(\log^{2}\frac{1} {\epsilon}\right)}\cdot O(\log\Delta),\ \operatorname{poly}(\epsilon^{-1},\log \Delta)\right\} \nonumber\end{align*}\) can be computed deterministically in \(\begin{align*} O(\epsilon^{-1}\log^{\ast}n)+\min\left\{2^{O\left(\log^{2}\frac{1}{\epsilon} \right)}\cdot O(\log\Delta),\ \operatorname{poly}(\epsilon^{-1},\log\Delta)\right\}\nonumber \end{align*}\) rounds in the \(\textsf{CONGEST}\) model for networks excluding a fixed minor. Our algorithm has a wide range of applications, including the following results in \(\textsf{CONGEST}\) : — A \((1-\epsilon)\) -approximate maximum independent set in networks excluding a fixed minor can be computed deterministically in \(O(\epsilon^{-1}\log^{\ast}n)+\operatorname{poly}(\epsilon^{-1})\) rounds, nearly matching the \(\Omega(\epsilon^{-1}\log^{\ast}n)\) lower bound of Lenzen and Wattenhofer [DISC 2008]. — Property testing of any additive minor-closed property can be performed deterministically in \(O(\log n)\) rounds for constant \(\epsilon\) , or in \(O(\epsilon^{-1}\log n)+\operatorname{poly}(\epsilon^{-1})\) rounds for constant \(\Delta\) , nearly matching the \(\Omega(\epsilon^{-1}\log n)\) lower bound of Levi et al. [PODC 2018].

Local certification is a framework for verifying global graph properties using only local information. In this model, a prover assigns short labels, called certificates, to the vertices of a graph. Each vertex then exchanges certificates with its neighbors and performs a purely local check to determine whether the graph satisfies the desired property. This line of research has led to efficient certification schemes for a broad range of graph classes, including minor-closed families, topological graph classes, and graphs defined by forbidden subgraphs. In this paper, we study the local certification of graph connectivity. Prior work by Bousquet, Feuilloley, and Pierron (JPDC 2024) showed that 2-vertex-connectivity, 2-edge-connectivity, and 3-vertex-connectivity admit O(log n)-bit certificates, leveraging structural characterizations such as ear decompositions. We go substantially beyond these cases and investigate general k-vertex-connectivity and k-edge-connectivity. We develop new approaches that exploit connections between connectivity and combinatorial structures, including branchings, Eulerian subgraphs, and independent spanning trees. For k-edge-connectivity, we obtain an O_k(log n)-bit certification scheme and prove a matching Ω_k(log n) lower bound for every k≥ 3. The lower bound also applies to k-vertex-connectivity. For k-vertex-connectivity, we obtain Õ_k(√(n))-bit certificates for every k under a conjecture of Itai and Zehavi. We further show that, for k=2, the logarithmic barrier can be broken on sparse graph classes: 2-edge-connectivity admits constant-size certificates in bounded-expansion graphs, and 2-vertex-connectivity admits constant-size certificates in bounded-degree graphs. In contrast, for 2-vertex-connectivity in general graphs, we prove an Ω(log(log^∗ n))-bit lower bound.

Many combinatorial optimization problems, including maximum weighted matching and maximum independent set, can be approximated within (1 ± ε) factors in poly(log n, 1/ε) rounds in the LOCAL model via network decompositions [Ghaffari, Kuhn, and Maus, STOC 2018]. These approaches, however, require sending messages of unlimited size, so they do not extend to the more realistic CONGEST model, which restricts the message size to be O(log n) bits. For example, despite the long line of research devoted to the distributed matching problem, it still remains a major open problem whether an (1-ε)-approximate maximum weighted matching can be computed in poly(log n, 1/ε) rounds in the CONGEST model. In this paper, we develop a generic framework for obtaining poly(log n, 1/ε)-round (1 ± ε)-approximation algorithms for many combinatorial optimization problems, including maximum weighted matching, maximum independent set, and correlation clustering, in graphs excluding a fixed minor in the CONGEST model. This class of graphs covers many sparse network classes that have been studied in the literature, including planar graphs, bounded-genus graphs, and bounded-treewidth graphs. Furthermore, we show that our framework can be applied to give an efficient distributed property testing algorithm for an arbitrary minor-closed graph property that is closed under taking disjoint union, significantly generalizing the previous distributed property testing algorithm for planarity in [Levi, Medina, and Ron, PODC 2018 & Distributed Computing 2021]. Our framework uses distributed expander decomposition algorithms [Chang and Saranurak, FOCS 2020] to decompose the graph into clusters of high conductance. We show that any graph excluding a fixed minor admits small edge separators. Using this result, we show the existence of a high-degree vertex in each cluster in an expander decomposition, which allows the entire graph topology of the cluster to be routed to a vertex. Similar to the use of network decompositions in the LOCAL model, the vertex will be able to perform any local computation on the subgraph induced by the cluster and broadcast the result over the cluster.

We present proof labeling schemes for graphs with bounded pathwidth that can decide any graph property expressible in monadic second-order (MSO2) logic using Omicron ( log n)-bit vertex labels. Examples of such properties include planarity, Hamiltonicity, kappa colorability, H-minor-freeness, admitting a perfect matching, and having a vertex cover of a given size. Our proof labeling schemes improve a recent result by Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024), which achieved the same result for graphs of bounded treewidth but required Omicron( log(2) n)-bit labels. Our improved label size Omicron (log n) is optimal, as it is known that any proof labeling scheme that accepts paths and rejects cycles requires labels of size Omega(log n). Our result implies that graphs with pathwidth at most kappa can be certified using Omicron( log n)-bit labels for any fixed constant kappa. Applying the Excluding Forest Theorem of Robertson and Seymour, we deduce that the class of.. -minor-free graphs can be certified with Omicron (log n)-bit labels for any fixed forest F, thereby providing an affirmative answer to an open question posed by Bousquet, Feuilloley, and Pierron (Journal of Parallel and Distributed Computing 2024).

Censor-Hillel, Cohen, Gelles, and Sela (PODC 2022 & Distributed Computing 2023) studied fullydefective asynchronous networks, where communication channels may arbitrarily corrupt messages. The model is equivalent to content-oblivious computation, where nodes communicate solely via pulses. They showed that if the network is 2-edge-connected, then any algorithm for a noiseless setting can be simulated in the fully-defective setting; otherwise, no non-trivial computation is possible in the fully-defective setting. However, their simulation requires a predesignated leader, which they conjectured to be necessary for any non-trivial content-oblivious task. Recently, Frei, Gelles, Ghazy, and Nolin (DISC 2024) refuted this conjecture for the special case of oriented ring topology. They designed two asynchronous content-oblivious leader election algorithms with message complexity O( n center dot IDmax), where n is the number of nodes and IDmax is the maximum ID. The first algorithm stabilizes in unoriented rings without termination detection. The second algorithm quiescently terminates in oriented rings, thus enabling the execution of the simulation algorithm after leader election. In this work, we present two results: General 2-edge-connected topologies: First, we show an asynchronous content-oblivious leader election algorithm that quiescently terminates in any 2-edge-connected network with message complexity O( m center dot N center dot IDmin), where m is the number of edges, N is a known upper bound on the number of nodes, and IDmin is the smallest ID. Combined with the above simulation, this result shows that whenever a size bound N is known, any noiseless algorithm can be simulated in the fully-defective model without a preselected leader, fully refuting the conjecture. Unoriented rings: We then show that the knowledge of N can be dropped in unoriented ring topologies by presenting a quiescently terminating election algorithm with message complexity O( n center dot IDmax) that matches the previous bound. Consequently, this result constitutes a strict improvement over the previous state of the art and shows that, on rings, fully-defective and noiseless communication are computationally equivalent, with no additional assumptions. 2012 ACM Subject Classification Theory of computation -> Distributed algorithms

The recent advent of programmable switches makes distributed algorithms readily deployable in real-world datacenter networks. However, there are still gaps between theory and practice that prevent the smooth adaptation of CONGEST algorithms to these environments. In this paper, we focus on the memory restrictions that arise in real-world deployments. We introduce the μ-CONGEST model where on top of the bandwidth restriction, the memory of nodes is also limited to μ words, in line with real-world systems. We provide fast algorithms of two main flavors. First, we observe that many algorithms in the CONGEST model are memory-intensive and do not work in μ-CONGEST. A prime example of a family of algorithms that use large memory is clique-listing algorithms. We show that the memory issue that arises here cannot be resolved without incurring a cost in the round complexity, by establishing a lower bound on the round complexity of listing cliques in μ-CONGEST. We introduce novel techniques to overcome these issues and generalize the algorithms to work within a given memory bound. Combined with our lower bound, these provide tight tradeoffs between the running time and memory of nodes. Second, we show that it is possible to efficiently simulate various families of streaming algorithms in μ-CONGEST. These include fast simulations of p-pass algorithms, random order streams, and various types of mergeable streaming algorithms. Combining our contributions, we show that we can use streaming algorithms to efficiently generate statistics regarding combinatorial structures in the network. An example of an end result of this type is that we can efficiently identify and provide the per-color frequencies of the frequent monochromatic triangles in μ-CONGEST.

In this paper, we study the leader election problem in oriented ring networks under content-oblivious asynchronous message-passing systems, where an adversary may arbitrarily corrupt message contents. Frei et al. (DISC 2024) recently presented a uniform terminating leader election algorithm for oriented rings in this setting, with message complexity O(nID(max)) on a ring of size n, where IDmax is the largest identifier in the system. In this paper, we investigate the message complexity of leader election in this model, showing that no uniform algorithm can solve the problem if each process is limited to sending a constant number of messages in one direction. Interestingly, this limitation hinges on the uniformity assumption. In the non-uniform setting - where processes know an upper bound U >= n on the ring size - we present an algorithm with message complexity O(nUID(min)), in which each process sends O(UIDmin) messages clockwise and only three messages counter-clockwise. Here, IDmin is the smallest identifier in the system. This dependence on the identifiers compares favorably with the dependence on IDmax of Frei et al. (DISC 2024). We also show a non-uniform algorithm where each process sends O( U log IDmin) messages in one direction and O(log IDmin) in the other. The factor log IDmin is optimal, matching the lower bound of Frei et al. (DISC 2024). Finally, in the anonymous setting, we propose a randomized algorithm where each process sends only O(log(2) U) messages, with a success probability of 1 - U-c. 2012 ACM Subject Classification Theory of computation -> Distributed algorithms

Bonne and Censor-Hillel (ICALP 2019) initiated the study of distributed subgraph finding in dynamic networks of limited bandwidth. For the case where the target subgraph is a clique, they determined the tight bandwidth complexity bounds in nearly all settings. However, several open questions remain, and very little is known about finding subgraphs beyond cliques. In this work, we consider these questions and explore subgraphs beyond cliques in the deterministic setting. For finding cliques, we establish an O(log log n) bandwidth lower bound for one-round membershipdetection under edge insertions only and an O(log log log n) bandwidth lower bound for one-round detection under both edge insertions and node insertions. Moreover, we demonstrate new algorithms to show that our lower bounds are tight in bounded-degree networks when the target subgraph is a triangle. Prior to our work, no lower bounds were known for these problems. For finding subgraphs beyond cliques, we present a complete characterization of the bandwidth complexity of the membership-listing problem for every target subgraph, every number of rounds, and every type of topological change: node insertions, node deletions, edge insertions, and edge deletions. We also show partial characterizations for one-round membership-detection and listing. 2012 ACM Subject Classification Theory of computation. Distributed algorithms

The well-known clustering algorithm of Miller, Peng, and Xu (SPAA 2013) is useful for many applications, including low-diameter decomposition and low-energy distributed algorithms. One nice property of their clustering, shown in previous work by Chang, Dani, Hayes, and Pettie (PODC 2020), is that distances in the cluster graph are rescaled versions of distances in the original graph, up to an O(log n) distortion factor and rounding issues. Minimizing this distortion factor is important for efficiency in computing the clustering, as well as in further applications, once the clustering has been constructed. We prove that there exist graphs for which an Ω( log ^1/3 n ) distortion factor is necessary for any clustering. We also consider a class of nice graphs which we call uniformly bounded independence graphs. These include, for example, paths, lattice graphs, and “dense” unit disk graphs. For these graphs, we prove that clusterings of constant distortion always exist, and moreover, we give an efficient distributed algorithm to construct them. Our clustering algorithm is based on Voronoi cells centered at the vertices of a maximal independent set in a suitable power graph. Applications of our new clustering include low-energy simulation of distributed algorithms in the , , and - models, as well as efficient approximate solutions to distributed combinatorial optimization problems. We complement these results with matching or nearly matching lower bounds.

In the context of communication complexity, we explore randomized protocols for graph coloring, focusing specifically on the vertex and edge coloring problems in $n$-vertex graphs $G$ with a maximum degree $\Delta$. We consider a scenario where the edges of $G$ are partitioned between two players. Our first contribution is a randomized protocol that efficiently finds a $(\Delta + 1)$-vertex coloring of $G$, utilizing $O(n)$ bits of communication in expectation and completing in $O(\log \log n \cdot \log \Delta)$ rounds in the worst case. This advancement represents a significant improvement over the work of Flin and Mittal [PODC 2024], who achieved the same communication cost but required $O(n)$ rounds in expectation, thereby making a significant reduction in the round complexity. We also present a randomized protocol for a $(2\Delta - 1)$-edge coloring of $G$, which maintains the same $O(n)$ bits of communication in expectation over $O(\log^\ast \Delta)$ rounds in the worst case. We complement the result with a tight $\Omega(n)$-bit lower bound on the communication complexity of the $(2\Delta-1)$-edge coloring, while a similar $\Omega(n)$ lower bound for the $(\Delta+1)$-vertex coloring has been established by Flin and Mittal [PODC 2024].

Energy-efficient computing is becoming increasingly important in the information era. However, electronic computers with von Neumann architecture can hardly meet the challenge due to the inevitable energy-intensive data movement, especially when tackling computationally hard problems or complicated tasks. Here, we experimentally demonstrate an energy-efficient photonic computer that solves intractable subset sum problem (SSP) by making use of the extremely low energy level of photons ( 10^(-19) J) and a time-of-flight storage technique. We show that the energy consumption of the photonic computer maintains no larger than 10^(-15) J per operation at a reasonably large problem size N=33, and it consumes 10^(8) times less energy than the most energy-efficient supercomputer for a medium-scale problem. In addition, when the photonic computer is applied to deal with real-life problems that involves iterative computation of the SSP, the photonic advantage in energy consumption is further enhanced and massive energy can be saved. Our results indicate the superior competitiveness of the photonic computer in the energy costs of complex computation, opening a possible path to green computing.