arXiv subjects

Mohsen Ghaffari

Publications and source records attributed to Mohsen Ghaffari.

At least 19 recordsLinked to original sources

arXiv2026

True Work-Efficiency in Parallel Derandomization

A longstanding limitation of known techniques for parallel derandomization was that they incurred at least polylogarithmic overhead in work. For instance, for fundamental and frequently used problems such as maximal independent set, maximal matching, and $(Δ+1)$-coloring, where $Δ$ denotes the maximum degree of the graph, the best-known deterministic parallel algorithms with polylogarithmic depth used $Ω((m+n)\mathrm{poly}(\log n))$ work on $n$-vertex, $m$-edge graphs; see, e.g., Luby [FOCS '88]. Consequently, at least $\mathrm{poly}(\log n)$ processors were needed for these algorithms to outperform straightforward single-processor algorithms. Recently, Ghaffari and Grunau [FOCS '25] introduced a new parallel derandomization method that substantially reduced the overhead from $\mathrm{poly}(\log n)$ to $\mathrm{poly}(\log\log n)$, achieving work bounds of $O((m+n)\mathrm{poly}(\log\log n))$. In this paper, we settle this line of research by obtaining linear work bounds of $O(m+n)$, thereby achieving truly work-efficient parallel derandomization.

cs.DS↗

arXiv2026

Dynamic Graph Coloring: Sequential, Parallel, and Distributed

We present a simple randomized algorithm that can efficiently maintain a $(Δ+1)$ coloring as the graph undergoes edge insertion and deletion updates, where $Δ$ denotes an upper bound on the maximum degree. A key advantage is the algorithm's ability to process many updates simultaneously, which makes it naturally adaptable to parallel and distributed computations. Concretely, it gives a unified framework across the models, leading to the following results: - In the sequential setting, the algorithm processes each update in $O(1)$ expected time, worst-case. This matches and strengthens the results of Henzinger and Peng [TALG 2022] and Bhattacharya et al. [TALG 2022], who achieved an $O(1)$ bound but amortized over the sequence of updates (in expectation and with high probability, respectively), whose work improved the $O(\log Δ)$ expected amortized bound of Bhattacharya et al. [SODA'18]. - In the parallel setting, the algorithm processes each batch of updates using $O(1)$ work per update in the batch in expectation, and in $\text{poly}(\log n)$ depth with high probability. This is, in a sense, an ideal parallelization of the above results. - In the distributed setting, the algorithm can maintain a coloring of the network graph as (potentially many) edges are added or deleted. The maintained coloring is always proper; it may become partial upon updates, i.e., some nodes may temporarily lose their colors, but quickly converges to a full, proper coloring. Concretely, each insertion and deletion causes at most $O(1)$ nodes to become uncolored, but this is resolved within $\tilde{O}(\log n)$ rounds with high probability (e.g., in the absence of further updates nearby--the precise guarantee is stronger, but technical). Importantly, the algorithm incurs only $O(1)$ expected message complexity and computation per update.

cs.DS↗

arXiv2026

Robust Shattering Arguments

Graph shattering is a central technique underlying sublogarithmic-time distributed algorithms in the LOCAL model. Its analysis typically relies on bounding the probability that large sets of distant nodes remain unresolved, often via independence assumptions justified by locality. We show that these assumptions fail for pre-shattering procedures that run for super-constant rounds, where dependencies accumulate over time. As a result, several standard shattering arguments in the literature are incomplete, including those for maximal independent set, $(Δ+1)$-coloring, and the distributed Lovász Local Lemma (LLL). We provide a systematic repair of these analyses. Our main contribution is a corrected shattering analysis of the Fischer--Ghaffari LLL algorithm. In addition, we develop general tools that capture common patterns in modern algorithms and yield the required decay bounds without relying on independence. We also present explicit counterexamples to commonly used shattering lemmas. Overall, we establish a robust and reusable foundation for shattering arguments in the presence of long-range dependencies.

cs.DS↗

arXiv2026

Density-Dependent Graph Orientation and Coloring in Scalable MPC

This paper presents massively parallel computation (MPC) algorithms in the strongly sublinear memory regime (aka, scalable MPC) for orienting and coloring graphs as a function of its subgraph density. Our algorithms run in $poly(\log\log n)$ rounds and compute an orientation of the edges with maximum outdegree $O(α\log\log n)$ as well as a coloring of the vertices with $O(α\log\log n)$ colors. Here, $α$ denotes the density of the densest subgraph. Our algorithm's round complexity is notable because it breaks the $\tildeΘ(\sqrt{\log n})$ barrier, which applied to the previously best known density-dependent orientation algorithm [Ghaffari, Lattanzi, and Mitrovic ICML'19] and is common to many other scalable MPC algorithms.

cs.DS↗

arXiv2025

Constant Approximation of Arboricity in Near-Optimal Sublinear Time

We present a randomized algorithm that computes a constant approximation of a graph's arboricity, using $\tilde{O}(n/λ)$ queries to adjacency lists and in the same time bound. Here, $n$ and $λ$ denote the number of nodes and the graph's arboricity, respectively. The $\tilde{O}(n/λ)$ query complexity of our algorithm is nearly optimal. Our constant approximation settles a question of Eden, Mossel, and Ron [SODA'22], who achieved an $O(\log^2 n)$ approximation with the same query and time complexity and asked whether a better approximation can be achieved using near-optimal query complexity. A key technical challenge in the problem is due to recursive algorithms based on probabilistic samplings, each with a non-negligible error probability. In our case, many of the recursions invoked could have bad probabilistic samples and result in high query complexities. The particular difficulty is that those bad recursions are not easy or cheap to detect and discard. Our approach runs multiple recursions in parallel, to attenuate the error probability, using a careful \textit{scheduling mechanism} that manages the speed at which each of them progresses and makes our overall query complexity competitive with the single good recursion. We find this usage of parallelism and scheduling in a sublinear algorithm remarkable, and we are hopeful that similar ideas may find applications in a wider range of sublinear algorithms that rely on probabilistic recursions.

cs.DS↗

arXiv2025

Parallel Batch-Dynamic Algorithms for Spanners, and Extensions

This paper presents the first parallel batch-dynamic algorithms for computing spanners and sparsifiers. Our algorithms process any batch of edge insertions and deletions in an $n$-node undirected graph, in $\text{poly}(\log n)$ depth and using amortized work near-linear in the batch size. Our concrete results are as follows: - Our base algorithm maintains a spanner with $(2k-1)$ stretch and $\tilde{O}(n^{1+1/k})$ edges, for any $k\geq 1$. - Our first extension maintains a sparse spanner with only $O(n)$ edges, and $\tilde{O}(\log n)$ stretch. - Our second extension maintains a $t$-bundle of spanners -- i.e., $t$ spanners, each of which is the spanner of the graph remaining after removing the previous ones -- and allows us to maintain cut/spectral sparsifiers with $\tilde{O}(n)$ edges.

cs.DS↗

arXiv2025

Parallel Batch-Dynamic Coreness Decomposition with Worst-Case Guarantees

We present the first parallel batch-dynamic algorithm for approximating coreness decomposition with worst-case update times. Given any batch of edge insertions and deletions, our algorithm processes all these updates in $ \text{poly}(\log n)$ depth, using a worst-case work bound of $b\cdot \text{poly}(\log n)$ where $b$ denotes the batch size. This means the batch gets processed in $\tilde{O}(b/p)$ time, given $p$ processors, which is optimal up to logarithmic factors. Previously, an algorithm with similar guarantees was known by the celebrated work of Liu, Shi, Yu, Dhulipala, and Shun [SPAA'22], but with the caveat of the work bound, and thus the runtime, being only amortized.

cs.DS↗

arXiv2025

Fully Scalable MPC Algorithms for Euclidean k-Center

The $k$-center problem is a fundamental optimization problem with numerous applications in machine learning, data analysis, data mining, and communication networks. The $k$-center problem has been extensively studied in the classical sequential setting for several decades, and more recently there have been some efforts in understanding the problem in parallel computing, on the Massively Parallel Computation (MPC) model. For now, we have a good understanding of $k$-center in the case where each local MPC machine has sufficient local memory to store some representatives from each cluster, that is, when one has $Ω(k)$ local memory per machine. While this setting covers the case of small values of $k$, for a large number of clusters these algorithms require undesirably large local memory, making them poorly scalable. The case of large $k$ has been considered only recently for the fully scalable low-local-memory MPC model for the Euclidean instances of the $k$-center problem. However, the earlier works have been considering only the constant dimensional Euclidean space, required a super-constant number of rounds, and produced only $k(1+o(1))$ centers whose cost is a super-constant approximation of $k$-center. In this work, we significantly improve upon the earlier results for the $k$-center problem for the fully scalable low-local-memory MPC model. In the low dimensional Euclidean case in $\mathbb{R}^d$, we present the first constant-round fully scalable MPC algorithm for $(2+\varepsilon)$-approximation. We push the ratio further to $(1 + \varepsilon)$-approximation albeit using slightly more $(1 + \varepsilon)k$ centers. All these results naturally extends to slightly super-constant values of $d$. In the high-dimensional regime, we provide the first fully scalable MPC algorithm that in a constant number of rounds achieves an $O(\log n/ \log \log n)$-approximation for $k$-center.

cs.DS↗

arXiv2025

Towards True Work-Efficiency in Parallel Derandomization: MIS, Maximal Matching, and Hitting Set

Derandomization is one of the classic topics studied in the theory of parallel computations, dating back to the early 1980s. Despite much work, all known techniques lead to deterministic algorithms that are not work-efficient. For instance, for the well-studied problem of maximal independent set -- e.g., [Karp, Wigderson STOC'84; Luby STOC' 85; Luby FOCS'88] -- state-of-the-art deterministic algorithms require at least $m \cdot poly(\log n)$ work, where $m$ and $n$ denote the number of edges and vertices. Hence, these deterministic algorithms will remain slower than their trivial sequential counterparts unless we have at least $poly(\log n)$ processors. In this paper, we present a generic parallel derandomization technique that moves exponentially closer to work-efficiency. The method iteratively rounds fractional solutions representing the randomized assignments to integral solutions that provide deterministic assignments, while maintaining certain linear or quadratic objective functions, and in an \textit{essentially work-efficient} manner. As example end-results, we use this technique to obtain deterministic algorithms with $m \cdot poly(\log \log n)$ work and $poly(\log n)$ depth for problems such as maximal independent set, maximal matching, and hitting set.

cs.DS↗

arXiv2025

Symbolic State Partitioning for Reinforcement Learning

Tabular reinforcement learning methods cannot operate directly on continuous state spaces. One solution for this problem is to partition the state space. A good partitioning enables generalization during learning and more efficient exploitation of prior experiences. Consequently, the learning process becomes faster and produces more reliable policies. However, partitioning introduces approximation, which is particularly harmful in the presence of nonlinear relations between state components. An ideal partition should be as coarse as possible, while capturing the key structure of the state space for the given problem. This work extracts partitions from the environment dynamics by symbolic execution. We show that symbolic partitioning improves state space coverage with respect to environmental behavior and allows reinforcement learning to perform better for sparse rewards. We evaluate symbolic state space partitioning with respect to precision, scalability, learning agent performance and state space coverage for the learnt policies.

cs.LG↗

arXiv2024

A Cut-Matching Game for Constant-Hop Expanders

This paper extends and generalizes the well-known cut-matching game framework and provides a novel cut-strategy that produces constant-hop expanders. Constant-hop expanders are a significant strengthening of regular expanders with the additional guarantee that any demand can be (obliviously) routed along constant-hop flow-paths - in contrast to the $Ω(\log n)$-hop paths in expanders. Cut-matching games for expanders are key tools for obtaining linear-time approximation algorithms for many hard problems, including finding (balanced or approximately-largest) sparse cuts, certifying the expansion of a graph by embedding an (explicit) expander, as well as computing expander decompositions, hierarchical cut decompositions, oblivious routings, multi-cuts, and multi-commodity flows. The cut-matching game of this paper is crucial in extending this versatile and powerful machinery to constant-hop and length-constrained expanders and has been already been extensively used. For example, as a key ingredient in several recent breakthroughs, including, computing constant-approximate $k$-commodity (min-cost) flows in $(m+k)^{1+ε}$ time as well as the optimal constant-approximate deterministic worst-case fully-dynamic APSP-distance oracle - in all applications the constant-approximation factor directly traces to and crucially relies on the expanders from a cut-matching game guaranteeing constant-hop routing paths.

cs.DS↗

arXiv2024

Near-Optimal Deterministic Network Decomposition and Ruling Set, and Improved MIS

This paper improves and in two cases nearly settles, up to logarithmically lower-order factors, the deterministic complexity of some of the most central problems in distributed graph algorithms, which have been studied for over three decades: Near-Optimal Network Decomposition: We present a deterministic distributed algorithm that computes a network decomposition in approximately O(log^2 n) rounds, with O(log n) diameter and O(log n) colors. This round complexity is near-optimal in the following sense: even given an ideal network decomposition, using it (in the standard way) requires round complexity equal to the product of diameter and number of colors, which is known to be approximately Omega(log^2 n). This near-optimality is remarkable, considering the rarity of optimal deterministic distributed algorithms and that for network decomposition, the first polylogarithmic-round algorithm was achieved only recently, by Rozhon and Ghaffari [STOC 2020], after three decades. Near-Optimal Ruling Set: We present a deterministic distributed algorithm that computes an O(log log n) ruling set in approximately O(log n) rounds. This is an exponential improvement over the O(log n) ruling set of Awerbuch, Goldberg, Luby, and Plotkin [FOCS 1989], while almost matching their O(log n) round complexity. Our result's round complexity nearly matches the approximately Omega(log n) lower bound established by Balliu, Brandt, Kuhn, and Olivetti [STOC 2022], which applies to any poly(log log n) ruling set. Improved Maximal Independent Set (MIS): We present a deterministic distributed algorithm for computing an MIS in approximately O(log^(5/3) n) rounds. This improves upon the approximately O(log^2 n) complexity achieved by Ghaffari and Grunau [STOC 2023] and breaks the log-squared barrier necessary for any method based on network decomposition.

cs.DS↗

arXiv2024

Dynamic O(arboricity) coloring in polylogarithmic worst-case time

A recent work by Christiansen, Nowicki, and Rotenberg provides dynamic algorithms for coloring sparse graphs, concretely as a function of the arboricity alpha of the input graph. They give two randomized algorithms: O({alpha} log {alpha}) implicit coloring in poly(log n) worst-case update and query times, and O(min{{alpha} log {alpha}, {alpha} log log log n}) implicit coloring in poly(log n) amortized update and query times (against an oblivious adversary). We improve these results in terms of the number of colors and the time guarantee: First, we present an extremely simple algorithm that computes an O({alpha})-implicit coloring with poly(log n) amortized update and query times. Second, and as the main technical contribution of our work, we show that the time complexity guarantee can be strengthened from amortized to worst-case. That is, we give a dynamic algorithm for implicit O({alpha})-coloring with poly(log n) worst-case update and query times (against an oblivious adversary).

cs.DS↗

arXiv2024

A Near-Optimal Low-Energy Deterministic Distributed SSSP with Ramifications on Congestion and APSP

We present a low-energy deterministic distributed algorithm that computes exact Single-Source Shortest Paths (SSSP) in near-optimal time: it runs in $\tilde{O}(n)$ rounds and each node is awake during only $poly(\log n)$ rounds. When a node is not awake, it performs no computations or communications and spends no energy. The general approach we take along the way to this result can be viewed as a novel adaptation of Dijkstra's classic approach to SSSP, which makes it suitable for the distributed setting. Notice that Dijkstra's algorithm itself is not efficient in the distributed setting due to its need for repeatedly computing the minimum-distance unvisited node in the entire network. Our adapted approach has other implications, as we outline next. As a step toward the above end-result, we obtain a simple deterministic algorithm for exact SSSP with near-optimal time and message complexities of $\tilde{O}(n)$ and $\tilde{O}(m)$, in which each edge communicates only $poly(\log n)$ messages. Therefore, one can simultaneously run $n$ instances of it for $n$ sources, using a simple random delay scheduling. That computes All Pairs Shortest Paths (APSP) in the near-optimal time complexity of $\tilde{O}(n)$. This algorithm matches the complexity of the recent APSP algorithm of Bernstein and Nanongkai [STOC 2019] using a completely different method (and one that is more modular, in the sense that the SSSPs are solved independently). It also takes a step toward resolving the open problem on a deterministic $\tilde{O}(n)$-time APSP, as the only randomness used now is in the scheduling.

cs.DS↗

arXiv2024

Parallel Dynamic Maximal Matching

We present the first (randomized) parallel dynamic algorithm for maximal matching, which can process an arbitrary number of updates simultaneously. Given a batch of edge deletion or insertion updates to the graph, our parallel algorithm adjusts the maximal matching to these updates in $poly(\log n)$ depth and using $poly(\log n)$ amortized work per update. That is, the amortized work for processing a batch of $k$ updates is $kpoly(\log n)$, while all this work is done in $poly(\log n)$ depth, with high probability. This can be seen as a parallel counterpart of the sequential dynamic algorithms for constant-approximate and maximal matching [Onak and Rubinfeld STOC'10; Baswana, Gupta, and Sen FOCS'11; and Solomon FOCS'16]. Our algorithm readily generalizes to maximal matching in hypergraphs of rank $r$ -- where each hyperedge has at most $r$ endpoints -- with a $poly(r)$ increase in work, while retaining the $poly(\log n)$ depth.

cs.DS↗

arXiv2024

Shared Randomness Helps with Local Distributed Problems

By prior work, we have many results related to distributed graph algorithms for problems that can be defined with local constraints; the formal framework used in prior work is locally checkable labeling problems (LCLs), introduced by Naor and Stockmeyer in the 1990s. It is known, for example, that if we have a deterministic algorithm that solves an LCL in $o(\log n)$ rounds, we can speed it up to $O(\log^*n)$ rounds, and if we have a randomized $O(\log^*n)$ rounds algorithm, we can derandomize it for free. It is also known that randomness helps with some LCL problems: there are LCL problems with randomized complexity $Θ(\log\log n)$ and deterministic complexity $Θ(\log n)$. However, so far there have not been any LCL problems in which the use of shared randomness has been necessary; in all prior algorithms it has been enough that the nodes have access to their own private sources of randomness. Could it be the case that shared randomness never helps with LCLs? Could we have a general technique that takes any distributed graph algorithm for any LCL that uses shared randomness, and turns it into an equally fast algorithm where private randomness is enough? In this work we show that the answer is no. We present an LCL problem $Π$ such that the round complexity of $Π$ is $Ω(\sqrt n)$ in the usual randomized \local model with private randomness, but if the nodes have access to a source of shared randomness, then the complexity drops to $O(\log n)$. As corollaries, we also resolve several other open questions related to the landscape of distributed computing in the context of LCL problems. In particular, problem $Π$ demonstrates that distributed quantum algorithms for LCL problems strictly benefit from a shared quantum state. Problem $Π$ also gives a separation between finitely dependent distributions and non-signaling distributions.

cs.DC↗

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

Work-Efficient Parallel Derandomization I: Chernoff-like Concentrations via Pairwise Independence

We present a novel technique for work-efficient parallel derandomization, for algorithms that rely on the concentration of measure bounds such as Chernoff, Hoeffding, and Bernstein inequalities. Our method increases the algorithm's computational work and depth by only polylogarithmic factors. Before our work, the only known method to obtain parallel derandomization with such strong concentrations was by the results of [Motwani, Naor, and Naor FOCS'89; Berger and Rompel FOCS'89], which perform a binary search in a $k$-wise independent space for $k=poly(\log n)$. However, that method blows up the computational work by a high $poly(n)$ factor and does not yield work-efficient parallel algorithms. Their method was an extension of the approach of [Luby FOCS'88], which gave a work-efficient derandomization but was limited to algorithms analyzed with only pairwise independence. Pushing the method from pairwise to the higher $k$-wise analysis resulted in the $poly(n)$ factor computational work blow-up. Our work can be viewed as an alternative extension from the pairwise case, which yields the desired strong concentrations while retaining work efficiency up to logarithmic factors. Our approach works by casting the problem of determining the random variables as an iterative process with $poly(\log n)$ iterations, where different iterations have independent randomness. This is done so that for the desired concentrations, we need only pairwise independence inside each iteration. In particular, we model each binary random variable as a result of a gradual random walk, and our method shows that the desired Chernoff-like concentrations about the endpoints of these walks can be boiled down to some pairwise analysis on the steps of these random walks in each iteration (while having independence across iterations).

cs.DS↗