arXiv subjects

Yi-Jun Chang

Publications and source records attributed to Yi-Jun Chang.

At least 37 records · Page 2Linked to original sources

arXiv2025

Optimal Distributed Replacement Paths

We study the replacement paths problem in the $\mathsf{CONGEST}$ model of distributed computing. Given an $s$-$t$ shortest path $P$, the goal is to compute, for every edge $e$ in $P$, the shortest-path distance from $s$ to $t$ avoiding $e$. For unweighted directed graphs, we establish the tight randomized round complexity bound for this problem as $\widetildeΘ(n^{2/3} + D)$ by showing matching upper and lower bounds. Our upper bound extends to $(1+ε)$-approximation for weighted directed graphs. Our lower bound applies even to the second simple shortest path problem, which asks only for the smallest replacement path length. These results improve upon the very recent work of Manoharan and Ramachandran (SIROCCO 2024), who showed a lower bound of $\widetildeΩ(n^{1/2} + D)$ and an upper bound of $\widetilde{O}(n^{2/3} + \sqrt{n h_{st}} + D)$, where $h_{st}$ is the number of hops in the given $s$-$t$ shortest path $P$.

cs.DS↗

arXiv2025

Femtojoule-per-operation photonic computer for the subset sum problem

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.

physics.optics↗

arXiv2025

The Complexity Landscape of Dynamic Distributed Subgraph Finding

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 $Ω(\log \log n)$ bandwidth lower bound for one-round membership-detection under edge insertions only and an $Ω(\log \log \log n)$ 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 \emph{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.

cs.DS↗

arXiv2025

Content-Oblivious Leader Election in 2-Edge-Connected Networks

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. 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 $O(m \cdot N \cdot \mathsf{ID}_{\min})$, where $m$ is the number of edges, $N$ is a known upper bound on the number of nodes, and $\mathsf{ID}_{\min}$ is the smallest $\mathsf{ID}$. 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 $O(n \cdot \mathsf{ID}_{\max})$ that matches the previous bound. Consequently, this result constitutes a strict improvement over the previous leader election in oriented rings by Frei, Gelles, Ghazy, and Nolin (DISC 2024) and shows that, on rings, fully-defective and noiseless communication are computationally equivalent, with no additional assumptions.

cs.DC↗

arXiv2025

Bounded Memory in Distributed Networks

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.

cs.DC↗

arXiv2025

Low-Distortion Clustering in Bounded Growth Graphs

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 $Ω((\log n)^{1/3})$ 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 LOCAL, CONGEST, and RADIO-CONGEST models, as well as efficient approximate solutions to distributed combinatorial optimization problems. We complement these results with matching or nearly matching lower bounds.

cs.DC↗

arXiv2025

Round and Communication Efficient Graph Coloring

In the context of communication complexity, we explore protocols for graph coloring, focusing on the vertex and edge coloring problems in $n$-vertex graphs $G$ with a maximum degree $Δ$. 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 $(Δ+ 1)$-vertex coloring of $G$, utilizing $O(n)$ bits of communication in expectation and completing in $O(\log \log n \cdot \log Δ)$ rounds in the worst case. This advancement represents a significant improvement over the work of Flin and Mittal [Distributed Computing 2025], who achieved the same communication cost but required $O(n)$ rounds in expectation, thereby making a significant reduction in the round complexity. Our second contribution is a deterministic protocol to compute a $(2Δ- 1)$-edge coloring of $G$, which maintains the same $O(n)$ bits of communication and uses only $O(1)$ rounds. We complement the result with a tight $Ω(n)$-bit lower bound on the communication complexity of the $(2Δ-1)$-edge coloring problem, while a similar $Ω(n)$ lower bound for the $(Δ+1)$-vertex coloring problem has been established by Flin and Mittal [Distributed Computing 2025]. Our result implies a space lower bound of $Ω(n)$ bits for $(2Δ- 1)$-edge coloring in the $W$-streaming model, which is the first non-trivial space lower bound for edge coloring in the $W$-streaming model.

cs.DS↗

arXiv2025

Ortho-Radial Drawing in Near-Linear Time

An orthogonal drawing is an embedding of a plane graph into a grid. In a seminal work of Tamassia (SIAM Journal on Computing 1987), a simple combinatorial characterization of angle assignments that can be realized as bend-free orthogonal drawings was established, thereby allowing an orthogonal drawing to be described combinatorially by listing the angles of all corners. The characterization reduces the need to consider certain geometric aspects, such as edge lengths and vertex coordinates, and simplifies the task of graph drawing algorithm design. Barth, Niedermann, Rutter, and Wolf (SoCG 2017) established an analogous combinatorial characterization for ortho-radial drawings, which are a generalization of orthogonal drawings to cylindrical grids. The proof of the characterization is existential and does not result in an efficient algorithm. Niedermann, Rutter, and Wolf (SoCG 2019) later addressed this issue by developing quadratic-time algorithms for both testing the realizability of a given angle assignment as an ortho-radial drawing without bends and constructing such a drawing. In this paper, we further improve the time complexity of these tasks to near-linear time. We establish a new characterization for ortho-radial drawings based on the concept of a good sequence. Using the new characterization, we design a simple greedy algorithm for constructing ortho-radial drawings.

cs.CG↗

arXiv2025

Optimal local certification on graphs of bounded pathwidth

We present proof labeling schemes for graphs with bounded pathwidth that can decide any graph property expressible in monadic second-order (MSO) logic using $O(\log n)$-bit vertex labels. Examples of such properties include planarity, Hamiltonicity, $k$-colorability, $H$-minor-freeness, admitting a perfect matching, and having a vertex cover of a given size. Our proof labeling schemes improve upon a recent result by Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024), which achieved the same result for graphs of bounded treewidth but required $O(\log^2 n)$-bit labels. Our improved label size $O(\log n)$ is optimal, as it is well-known that any proof labeling scheme that accepts paths and rejects cycles requires labels of size $Ω(\log n)$. Our result implies that graphs with pathwidth at most $k$ can be certified using $O(\log n)$-bit labels for any fixed constant $k$. Applying the Excluding Forest Theorem of Robertson and Seymour, we deduce that the class of $F$-minor-free graphs can be certified with $O(\log n)$-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).

cs.DC↗

arXiv2024

Universally Optimal Information Dissemination and Shortest Paths in the HYBRID Distributed Model

In this work we consider the HYBRID model of distributed computing, introduced recently by Augustine, Hinnenthal, Kuhn, Scheideler, and Schneider (SODA 2020), where nodes have access to two different communication modes: high-bandwidth local communication along the edges of the graph and low-bandwidth all-to-all communication, capturing the non-uniform nature of modern communication networks. Prior work in HYBRID has focused on showing existentially optimal algorithms, meaning there exists a pathological family of instances on which no algorithm can do better. This neglects the fact that such worst-case instances often do not appear or can be actively avoided in practice. In this work, we focus on the notion of universal optimality, first raised by Garay, Kutten, and Peleg (FOCS 1993). Roughly speaking, a universally optimal algorithm is one that, given any input graph, runs as fast as the best algorithm designed specifically for that graph. We show the first universally optimal algorithms in HYBRID. We present universally optimal solutions for fundamental information dissemination tasks, such as broadcasting and unicasting multiple messages in HYBRID. Furthermore, we apply these tools to obtain universally optimal solutions for various shortest paths problems in HYBRID. A main conceptual contribution of this work is the conception of a new graph parameter called neighborhood quality that captures the inherent complexity of many fundamental graph problems in HYBRID. We also show new existentially optimal shortest paths algorithms in HYBRID, which are utilized as key subroutines in our universally optimal algorithms and are of independent interest. Our new algorithms for $k$-source shortest paths match the existing $\tildeΩ(\sqrt{k})$ lower bound for all $k$. Previously, the lower bound was only known to be tight when $k \in \tildeΩ(n^{2/3})$.

cs.DC↗

arXiv2024

Deterministic Expander Routing: Faster and More Versatile

We consider the expander routing problem formulated by Ghaffari, Kuhn, and Su (PODC 2017), where the goal is to route all the tokens to their destinations given that each vertex is the source and the destination of at most $°(v)$ tokens. They developed $\textit{randomized algorithms}$ that solve this problem in $\text{poly}(ϕ^{-1}) \cdot 2^{O(\sqrt{\log n \log \log n})}$ rounds in the $\textsf{CONGEST}$ model, where $ϕ$ is the conductance of the graph. Later, Ghaffari and Li (DISC 2018) gave an improved algorithm. However, both algorithms are randomized, which means that all the resulting applications are also randomized. Recently, Chang and Saranurak (FOCS 2020) gave a deterministic algorithm that solves an expander routing instance in $2^{O(\log^{2/3} n \cdot \log^{1/3} \log n)}$ rounds. The deterministic algorithm is less efficient and does not allow preprocessing/query tradeoffs, which precludes the de-randomization of algorithms that require this feature, such as the $k$-clique enumeration algorithm in general graphs. The main contribution of our work is a new deterministic expander routing algorithm that not only matches the randomized bound of [GKS 2017] but also allows preprocessing/query tradeoffs. Our algorithm solves a single instance of routing query in $2^{{O}(\sqrt{\log n \cdot \log \log n})}$ rounds. Our algorithm achieves the following preprocessing and query tradeoffs: For $0 < ε< 1$, we can answer every routing query in $\log^{O(1/ε)} n$ rounds at the cost of a $(n^{O(ε)} + \log^{O(1/ε)} n)$-round preprocessing procedure. Combining this with the approach of Censor-Hillel, Leitersdorf, and Vulakh (PODC 2022), we obtain a near-optimal $\tilde{O}(n^{1-2/k})$-round deterministic algorithm for $k$-clique enumeration in general graphs, improving the previous state-of-the-art $n^{1-2/k+o(1)}$.

cs.DC↗

arXiv2024

A Tight Lower Bound for 3-Coloring Grids in the Online-LOCAL Model

Recently, \citeauthor*{akbari2021locality}~(ICALP 2023) studied the locality of graph problems in distributed, sequential, dynamic, and online settings from a {unified} point of view. They designed a novel $O(\log n)$-locality deterministic algorithm for proper 3-coloring bipartite graphs in the $\mathsf{Online}$-$\mathsf{LOCAL}$ model. In this work, we establish the optimality of the algorithm by showing a \textit{tight} deterministic $Ω(\log n)$ locality lower bound, which holds even on grids. To complement this result, we have the following additional results: \begin{enumerate} \item We show a higher and {tight} $Ω(\sqrt{n})$ lower bound for 3-coloring toroidal and cylindrical grids. \item Considering the generalization of $3$-coloring bipartite graphs to $(k+1)$-coloring $k$-partite graphs, %where $k \geq 2$ is a constant, we show that the problem also has $O(\log n)$ locality when the input is a $k$-partite graph that admits a \emph{locally inferable unique coloring}. This special class of $k$-partite graphs covers several fundamental graph classes such as $k$-trees and triangular grids. Moreover, for this special class of graphs, we show a {tight} $Ω(\log n)$ locality lower bound. \item For general $k$-partite graphs with $k \geq 3$, we prove that the problem of $(2k-2)$-coloring $k$-partite graphs exhibits a locality of $Ω(n)$ in the $\onlineLOCAL$ model, matching the round complexity of the same problem in the $\LOCAL$ model recently shown by \citeauthor*{coiteux2023no}~(STOC 2024). Consequently, the problem of $(k+1)$-coloring $k$-partite graphs admits a locality lower bound of $Ω(n)$ when $k\geq 3$, contrasting sharply with the $Θ(\log n)$ locality for the case of $k=2$. \end{enumerate}

cs.DS↗

arXiv2024

On Homomorphism Graphs

We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for the construction comes from the adaptation of this method to the LOCAL model of distributed computing. Our approach unifies the previous results in the area, as well as produces new ones. In particular, we show that for $Δ>2$ it is impossible to give a simple characterization of acyclic $Δ$-regular Borel graphs with Borel chromatic number at most $Δ$: such graphs form a $\mathbfΣ^1_2$-complete set. This implies a strong failure of Brooks'-like theorems in the Borel context.

math.LO↗

arXiv2024

Fast Broadcast in Highly Connected Networks

We revisit the classic broadcast problem, wherein we have $k$ messages, each composed of $O(\log{n})$ bits, distributed arbitrarily across a network. The objective is to broadcast these messages to all nodes in the network. In the distributed CONGEST model, a textbook algorithm solves this problem in $O(D+k)$ rounds, where $D$ is the diameter of the graph. While the $O(D)$ term in the round complexity is unavoidable$\unicode{x2014}$given that $Ω(D)$ rounds are necessary to solve broadcast in any graph$\unicode{x2014}$it remains unclear whether the $O(k)$ term is needed in all graphs. In cases where the minimum cut size is one, simply transmitting messages from one side of the cut to the other would require $Ω(k)$ rounds. However, if the size of the minimum cut is larger, it may be possible to develop faster algorithms. This motivates the exploration of the broadcast problem in networks with high edge connectivity. In this work, we present a simple randomized distributed algorithm for performing $k$-message broadcast in $O(((n+k)/λ)\log n)$ rounds in any $n$-node simple graph with edge connectivity $λ$. When $k = Ω(n)$, our algorithm is universally optimal, up to an $O(\log n)$ factor, as its complexity nearly matches an information-theoretic $Ω(k/λ)$ lower bound that applies to all graphs, even when the network topology is known to the algorithm. The setting $k = Ω(n)$ is particularly interesting because several fundamental problems can be reduced to broadcasting $Ω(n)$ messages. Our broadcast algorithm finds several applications in distributed computing, enabling $O(1)$-approximation for all distances and $(1+ε)$-approximation for all cut sizes in $\tilde{O}(n/λ)$ rounds.

cs.DC↗

arXiv2023

The Distributed Complexity of Locally Checkable Labeling Problems Beyond Paths and Trees

We consider locally checkable labeling LCL problems in the LOCAL model of distributed computing. Since 2016, there has been a substantial body of work examining the possible complexities of LCL problems. For example, it has been established that there are no LCL problems exhibiting deterministic complexities falling between $ω(\log^* n)$ and $o(\log n)$. This line of inquiry has yielded a wealth of algorithmic techniques and insights that are useful for algorithm designers. While the complexity landscape of LCL problems on general graphs, trees, and paths is now well understood, graph classes beyond these three cases remain largely unexplored. Indeed, recent research trends have shifted towards a fine-grained study of special instances within the domains of paths and trees. In this paper, we generalize the line of research on characterizing the complexity landscape of LCL problems to a much broader range of graph classes. We propose a conjecture that characterizes the complexity landscape of LCL problems for an arbitrary class of graphs that is closed under minors, and we prove a part of the conjecture. Some highlights of our findings are as follows. 1. We establish a simple characterization of the minor-closed graph classes sharing the same deterministic complexity landscape as paths, where $O(1)$, $Θ(\log^* n)$, and $Θ(n)$ are the only possible complexity classes. 2. It is natural to conjecture that any minor-closed graph class shares the same complexity landscape as trees if and only if the graph class has bounded treewidth and unbounded pathwidth. We prove the "only if" part of the conjecture. 3. In addition to the well-known complexity landscapes for paths, trees, and general graphs, there are infinitely many different complexity landscapes among minor-closed graph classes.

cs.DC↗

arXiv2023

Fully Scalable Massively Parallel Algorithms for Embedded Planar Graphs

We consider the massively parallel computation (MPC) model, which is a theoretical abstraction of large-scale parallel processing models such as MapReduce. In this model, assuming the widely believed 1-vs-2-cycles conjecture, solving many basic graph problems in $O(1)$ rounds with a strongly sublinear memory size per machine is impossible. We improve on the recent work of Holm and Tětek [SODA 2023] that bypass this barrier for problems when a planar embedding of the graph is given. In the previous work, on graphs of size $n$ with $O(n/\mathcal{S})$ machines, the memory size per machine needs to be at least $\mathcal{S} = n^{2/3+Ω(1)}$, whereas we extend their work to the fully scalable regime, where the memory size per machine can be $\mathcal{S} = n^δ$ for any constant $0< δ< 1$. We give the first constant round fully scalable algorithms for embedded planar graphs for the problems of (i) connectivity and (ii) minimum spanning tree (MST). Moreover, we show that the $\varepsilon$-emulator of Chang, Krauthgamer, and Tan [STOC 2022] can be incorporated into our recursive framework to obtain constant-round $(1+\varepsilon)$-approximation algorithms for the problems of computing (iii) single source shortest path (SSSP), (iv) global min-cut, and (v) $st$-max flow. All previous results on cuts and flows required linear memory in the MPC model. Furthermore, our results give new algorithms for problems that implicitly involve embedded planar graphs. We give as corollaries constant round fully scalable algorithms for (vi) 2D Euclidean MST using $O(n)$ total memory and (vii) $(1+\varepsilon)$-approximate weighted edit distance using $\widetilde{O}(n^{2-δ})$ memory. Our main technique is a recursive framework combined with novel graph drawing algorithms to compute smaller embedded planar graphs in constant rounds in the fully scalable setting.

cs.DS↗

arXiv2023

The Complexity of Distributed Approximation of Packing and Covering Integer Linear Programs

In this paper, we present a low-diameter decomposition algorithm in the LOCAL model of distributed computing that succeeds with probability $1 - 1/poly(n)$. Specifically, we show how to compute an $\left(ε, O\left(\frac{\log n}ε\right)\right)$ low-diameter decomposition in $O\left(\frac{\log^3(1/ε)\log n}ε\right)$ round Further developing our techniques, we show new distributed algorithms for approximating general packing and covering integer linear programs in the LOCAL model. For packing problems, our algorithm finds an $(1-ε)$-approximate solution in $O\left(\frac{\log^3 (1/ε) \log n}ε\right)$ rounds with probability $1 - 1/poly(n)$. For covering problems, our algorithm finds an $(1+ε)$-approximate solution in $O\left(\frac{\left(\log \log n + \log (1/ε)\right)^3 \log n}ε\right)$ rounds with probability $1 - 1/poly(n)$. These results improve upon the previous $O\left(\frac{\log^3 n}ε\right)$-round algorithm by Ghaffari, Kuhn, and Maus [STOC 2017] which is based on network decompositions. Our algorithms are near-optimal for many fundamental combinatorial graph optimization problems in the LOCAL model, such as minimum vertex cover and minimum dominating set, as their $(1\pm ε)$-approximate solutions require $Ω\left(\frac{\log n}ε\right)$ rounds to compute.

cs.DS↗

arXiv2023

Universally Optimal Deterministic Broadcasting in the HYBRID Distributed Model

In theoretical computer science, it is a common practice to show existential lower bounds for problems, meaning there is a family of pathological inputs on which no algorithm can do better. However, most inputs of interest can be solved much more efficiently, giving rise to the notion of universally optimal algorithms, which run as fast as possible on every input. Questions on the existence of universally optimal algorithms were first raised by Garay, Kutten, and Peleg in FOCS '93. This research direction reemerged recently through a series of works, including the influential work of Haeupler, Wajc, and Zuzic in STOC '21, which resolves some of these decades-old questions in the supported CONGEST model. We work in the HYBRID distributed model, which analyzes networks combining both global and local communication. Much attention has recently been devoted to solving distance related problems, such as All-Pairs Shortest Paths (APSP) in HYBRID, culminating in a $\tilde Θ(n^{1/2})$ round algorithm for exact APSP. However, by definition, every problem in HYBRID is solvable in $D$ (diameter) rounds, showing that it is far from universally optimal. We show the first universally optimal algorithms in HYBRID, by presenting a fundamental tool that solves any broadcasting problem in a universally optimal number of rounds, deterministically. Specifically, we consider the problem in a graph $G$ where a set of $k$ messages $M$ distributed arbitrarily across $G$, requires every node to learn all of $M$. We show a universal lower bound and a matching, deterministic upper bound, for any graph $G$, any value $k$, and any distribution of $M$ across $G$. This broadcasting tool opens a new exciting direction of research into showing universally optimal algorithms in HYBRID. As an example, we use it to obtain algorithms for approximate and exact APSP in general and sparse graphs.

cs.DS↗