arXiv subjects

Ami Paz

Publications and source records attributed to Ami Paz.

At least 19 recordsLinked to original sources

arXiv2026

Polynomial Time Local Decision Revisited

We consider three classification systems for distributed decision tasks: With unbounded computation and certificates, defined by Balliu, D'Angelo, Fraigniaud, and Olivetti [JCSS'18], and with (two flavors of) polynomially bounded local computation and certificates, defined in recent works by Aldema Tshuva and Oshman [OPODIS'23], and by Reiter [PODC'24]. The latter two differ in the way they evaluate the polynomial bounds: the former considers polynomials with respect to the size of the graph, while the latter refers to being polynomial in the size of each node's local neighborhood. We start by revisiting decision without certificates. For this scenario, we show that the latter two definitions coincide: roughly, a node cannot know the graph size, and thus can only use a running time dependent on its neighborhood. We then consider decision with certificates. With existential certificates ($\Sigma_1$-type classes), a larger running time defines strictly larger classes of languages: when it grows from being polynomial in each node's view, through polynomial in the graph's size, and to unbounded, the derived classes strictly contain each other. With universal certificates ($\Pi_1$-type classes), on the other hand, we prove a surprising incomparability result: having running time bounded by the graph size sometimes allows us to decide languages undecidable even with unbounded certificates. We complement these results with other containment and separation results, which together portray a surprisingly complex lattice of strict containment relations between the classes at the base of the three classification systems.

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arXiv2025

Lower Bounds for $k$-Set Agreement in Fault-Prone Networks

We develop a new lower bound for k-set agreement in synchronous message-passing systems connected by an arbitrary directed communication network, where up to t processes may crash. Our result thus generalizes the t/k+1 lower bound for complete networks in the t-resilient model by Chaudhuri, Herlihy, Lynch, and Tuttle [JACM'00]. Moreover, it generalizes two lower bounds for oblivious algorithms in synchronous systems connected by an arbitrary undirected communication network known to the processes, namely, the domination number-based lower bound by Castaneda, Fraigniaud, Paz, Rajsbaum, Roy, and Travers [TCS'21] for failure-free processes, and the radius-based lower bound in the t-resilient model by Fraigniaud, Nguyen, and Paz [STACS'24]. Our topological proof non-trivially generalizes and extends the connectivity-based approach for the complete network, as presented in the book by Herlihy, Kozlov, and Rajsbaum (2013). It is based on a sequence of shellable carrier maps that, starting from a shellable input complex, determine the evolution of the protocol complex: During the first t/k rounds, carrier maps that crash exactly k processes per round are used, ensuring high connectivity of their images. A Sperner's lemma style argument is used to prove that k-set agreement is still impossible by that round. From round t/k+1 up to our lower bound, we employ a novel carrier map that maintains high connectivity. Our proof also provides a strikingly simple lower bound for k-set agreement in synchronous systems with an arbitrary communication network with initial crashes. We express the resulting additional agreement overhead via an appropriately defined radius of the communication graphs. Finally, we prove that the usual input pseudosphere complex for k-set agreement can be replaced by an exponentially smaller input complex based on Kuhn triangulations, which we prove to be also shellable.

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arXiv2025

Semi-Streaming Algorithms for Graph Property Certification

We introduce the {\em certification} of solutions to graph problems when access to the input is restricted. This topic has received a lot of attention in the distributed computing setting, and we introduce it here in the context of \emph{streaming} algorithms, where the input is too large to be stored in memory. Given a graph property $\mbox{P}$, a \emph{streaming certification scheme} for $\mbox{P}$ is a \emph{prover-verifier} pair where the prover is a computationally unlimited but non-trustable oracle, and the verifier is a streaming algorithm. For any input graph, the prover provides the verifier with a \emph{certificate}. The verifier then receives the input graph as a stream of edges in an adversarial order, and must check whether the certificate is indeed a \emph{proof} that the input graph satisfies $\mbox{P}$. The main complexity measure for a streaming certification scheme is its \emph{space complexity}, defined as the sum of the size of the certificate provided by the oracle, and of the memory space required by the verifier. We give streaming certification schemes for several graph properties, including maximum matching, diameter, degeneracy, and coloring, with space complexity matching the requirement of \emph{semi-streaming}, i.e., with space complexity $O(n\,\mbox{polylog}\, n)$ for $n$-node graphs. All these problems do {\em not} admit semi-streaming algorithms, showing that also in the (semi) streaming setting, certification is sometimes easier than calculation (like $NP$). For each of these properties, we provide upper and lower bounds on the space complexity of the corresponding certification schemes, many being tight up to logarithmic multiplicative factors. We also show that some graph properties are hard for streaming certification, in the sense that they cannot be certified in semi-streaming, as they require $\Omega(n^2)$-bit certificates.

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arXiv2025

Smoothed Analysis of Dynamic Graph Algorithms

Recent years have seen significant progress in the study of dynamic graph algorithms, and most notably, the introduction of strong lower bound techniques for them (e.g., Henzinger, Krinninger, Nanongkai and Saranurak, STOC 2015; Larsen and Yu, FOCS 2023). As worst-case analysis (adversarial inputs) may lead to the necessity of high running times, a natural question arises: in which cases are high running times really necessary, and in which cases these inputs merely manifest unique pathological cases? Early attempts to tackle this question were made by Nikoletseas, Reif, Spirakis and Yung (ICALP 1995) and by Alberts and Henzinger (Algorithmica 1998), who considered models with very little adversarial control over the inputs, and showed fast algorithms exist for them. The question was then overlooked for decades, until Henzinger, Lincoln and Saha (SODA 2022) recently addressed uniformly random inputs, and presented algorithms and impossibility results for several subgraph counting problems. To tackle the above question more thoroughly, we employ smoothed analysis, a celebrated framework introduced by Spielman and Teng (J. ACM, 2004). An input is proposed by an adversary but then a noisy version of it is processed by the algorithm instead. Parameterized by the amount of adversarial control, this input model fully interpolates between worst-case inputs and a uniformly random input. Doing so, we extend impossibility results for some problems to the smoothed model with only a minor quantitative loss. That is, we show that partially-adversarial inputs suffice to impose high running times for certain problems. In contrast, we show that other problems become easy even with the slightest amount of noise. In addition, we study the interplay between the adversary and the noise, leading to three natural models of smoothed inputs, for which we show a hierarchy of increasing complexity.

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arXiv2025

Distributed Non-Interactive Zero-Knowledge Proofs

Distributed certification is a set of mechanisms that allows an all-knowing prover to convince the units of a communication network that the network's state has some desired property, such as being 3-colorable or triangle-free. Classical mechanisms, such as proof labeling schemes (PLS), consist of a message from the prover to each unit, followed by one round of communication between each unit and its neighbors. Later works consider extensions, called distributed interactive proofs, where the prover and the units can have multiple rounds of communication before the communication among the units. Recently, Bick, Kol, and Oshman (SODA '22) defined a zero-knowledge version of distributed interactive proofs, where the prover convinces the units of the network's state without revealing any other information about the network's state or structure. In their work, they propose different variants of this model and show that many graph properties of interest can be certified with them. In this work, we define and study distributed non-interactive zero-knowledge proofs (dNIZK); these can be seen as a non-interactive version of the aforementioned model, and also as a zero-knowledge version of PLS. We prove the following: - There exists a dNIZK protocol for 3-coloring with O(log n)-bit messages from the prover and O(log n)-size messages among neighbors. - There exists a family of dNIZK protocols for triangle-freeness, that presents a trade-off between the size of the messages from the prover and the size of the messages among neighbors. - There exists a dNIZK protocol for any graph property in NP in the random oracle models, which is secure against an arbitrary number of malicious parties.

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arXiv2025

A Simple Lower Bound for Set Agreement in Dynamic Networks

Given a positive integer $k$, $k$-set agreement is the distributed task in which each process $i\in [n]$ in a group of $n$ processing nodes starts with an input value $x_i$ in the set $\{0,\dots,k\}$, and must output a value $y_i$ such that (1) for every $i \in [n]$, $y_i$ is the input value of some process, and (2)$|\{y_i : i\in [n]\}|\leq k$. That is, at most $k$ different values in total must be outputted by the processes. The case $k=1$ correspond to (binary) consensus, arguably the most studied problem in distributed computing. While lower bounds for consensus have been obtained for most of the standard distributed computing models, the design of lower bounds for $k$-set agreement with $k>1$ is notoriously known to be much more difficult, and remains open for many models. The main techniques for designing lower bounds for k-set agreement with $k>1$ use tools from algebraic topology. The algebraic topology tools are difficult to manipulate, and require a lot of care for avoiding mistakes. This difficulty increases when the communications are mediated by a network of arbitrary structure. Recently, the KNOWALL model has been specifically designed as a first attempt to understand the LOCAL model through the lens of algebraic topology, and Casta\~neda et al.(2021) have designed lower bounds for $k$-set agreement in the KNOWALL model, with applications to dynamic networks. In this work, we re-prove the same lower bound for $k$-set agreement in the KNOWALL model. This new proof stands out in its simplicity, which makes it accessible to a broader audience, and increases confidence in the result.

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arXiv2024

Agreement Tasks in Fault-Prone Synchronous Networks of Arbitrary Structure

Consensus is arguably the most studied problem in distributed computing as a whole, and particularly in the distributed message-passing setting. In this latter framework, research on consensus has considered various hypotheses regarding the failure types, the memory constraints, the algorithmic performances (e.g., early stopping and obliviousness), etc. Surprisingly, almost all of this work assumes that messages are passed in a \emph{complete} network, i.e., each process has a direct link to every other process. A noticeable exception is the recent work of Casta\~neda et al. (Inf. Comput. 2023) who designed a generic oblivious algorithm for consensus running in $\radius(G,t)$ rounds in every graph~$G$, when up to $t$ nodes can crash by irrevocably stopping, where $t$ is smaller than the node-connectivity $\kappa$ of~$G$. Here, $\radius(G,t)$ denotes a graph parameter called the \emph{radius of~$G$ whenever up to $t$ nodes can crash}. For $t=0$, this parameter coincides with $\radius(G)$, the standard radius of a graph, and, for $G=K_n$, the running time $\radius(K_n,t)=t +1$ of the algorithm exactly matches the known round-complexity of consensus in the clique~$K_n$. Our main result is a proof that $\radius(G,t)$ rounds are necessary for oblivious algorithms solving consensus in $G$ when up to $t$ nodes can crash, thus validating a conjecture of Casta\~neda et al., and demonstrating that their consensus algorithm is optimal for any graph~$G$. We also extend the result of Casta\~neda et al. to two different settings: First, to the case where the number $t$ of failures is not necessarily smaller than the connectivity $\kappa$ of the considered graph; Second, to the $k$-set agreement problem for which agreement is not restricted to be on a single value as in consensus, but on up to $k$ different values.

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arXiv2024

$k$-Center Clustering in Distributed Models

The $k$-center problem is a central optimization problem with numerous applications for machine learning, data mining, and communication networks. Despite extensive study in various scenarios, it surprisingly has not been thoroughly explored in the traditional distributed setting, where the communication graph of a network also defines the distance metric. We initiate the study of the $k$-center problem in a setting where the underlying metric is the graph's shortest path metric in three canonical distributed settings: the LOCAL, CONGEST, and CLIQUE models. Our results encompass constant-factor approximation algorithms and lower bounds in these models, as well as hardness results for the bi-criteria approximation setting.

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arXiv2023

One Step Forward, One Step Back: FLP-Style Proofs and the Round-Reduction Technique for Colorless Tasks

The paper compares two generic techniques for deriving lower bounds and impossibility results in distributed computing. First, we prove a speedup theorem (a-la Brandt, 2019), for wait-free colorless algorithms, aiming at capturing the essence of the seminal round-reduction proof establishing a lower bound on the number of rounds for 3-coloring a cycle (Linial, 1992), and going by backward induction. Second, we consider FLP-style proofs, aiming at capturing the essence of the seminal consensus impossibility proof (Fischer, Lynch, and Paterson, 1985) and using forward induction. We show that despite their very different natures, these two forms of proof are tightly connected. In particular, we show that for every colorless task $\Pi$, if there is a round-reduction proof establishing the impossibility of solving $\Pi$ using wait-free colorless algorithms, then there is an FLP-style proof establishing the same impossibility. For 1-dimensional colorless tasks (for an arbitrary number $n\geq 2$ of processes), we prove that the two proof techniques have exactly the same power, and more importantly, both are complete: if a 1-dimensional colorless task is not wait-free solvable by $n\geq 2$ processes, then the impossibility can be proved by both proof techniques. Moreover, a round-reduction proof can be automatically derived, and an FLP-style proof can be automatically generated from it. Finally, we illustrate the use of these two techniques by establishing the impossibility of solving any colorless covering task of arbitrary dimension by wait-free algorithms.

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arXiv2023

Topological Characterization of Consensus Solvability in Directed Dynamic Networks

Consensus is one of the most fundamental problems in distributed computing. This paper studies the consensus problem in a synchronous dynamic directed network, in which communication is controlled by an oblivious message adversary. The question when consensus is possible in this model has already been studied thoroughly in the literature from a combinatorial perspective, and is known to be challenging. This paper presents a topological perspective on consensus solvability under oblivious message adversaries, which provides interesting new insights. Our main contribution is a topological characterization of consensus solvability, which also leads to explicit decision procedures. Our approach is based on the novel notion of a communication pseudosphere, which can be seen as the message-passing analog of the well-known standard chromatic subdivision for wait-free shared memory systems. We further push the elegance and expressiveness of the "geometric" reasoning enabled by the topological approach by dealing with uninterpreted complexes, which considerably reduce the size of the protocol complex, and by labeling facets with information flow arrows, which give an intuitive meaning to the implicit epistemic status of the faces in a protocol complex.

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arXiv2023

Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs

A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for general dynamic graphs, yet graph families that arise in practice often exhibit structural properties that the existing lower bound constructions do not possess. We study three specific graph families that are ubiquitous, namely constant-degree graphs, power-law graphs, and expander graphs, and give the first conditional lower bounds for them. Our results show that even when restricting our attention to one of these graph classes, any algorithm for fundamental graph problems such as distance computation or approximation or maximum matching, cannot simultaneously achieve a sub-polynomial update time and query time. For example, we show that the same lower bounds as for general graphs hold for maximum matching and ($s,t$)-distance in constant-degree graphs, power-law graphs or expanders. Namely, in an $m$-edge graph, there exists no dynamic algorithms with both $O(m^{1/2 - ε})$ update time and $ O(m^{1 -ε})$ query time, for any small $ε> 0$. Note that for ($s,t$)-distance the trivial dynamic algorithm achieves an almost matching upper bound of constant update time and $O(m)$ query time. We prove similar bounds for the other graph families and for other fundamental problems such as densest subgraph detection and perfect matching.

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arXiv2022

A Speedup Theorem for Asynchronous Computation with Applications to Consensus and Approximate Agreement

We study two fundamental problems of distributed computing, consensus and approximate agreement, through a novel approach for proving lower bounds and impossibility results, that we call the asynchronous speedup theorem. For a given $n$-process task $Π$ and a given computational model $M$, we define a new task, called the closure of $Π$ with respect to $M$. The asynchronous speedup theorem states that if a task $Π$ is solvable in $t\geq 1$ rounds in $M$, then its closure w.r.t. $M$ is solvable in $t-1$ rounds in $M$. We prove this theorem for iterated models, as long as the model allows solo executions. We illustrate the power of our asynchronous speedup theorem by providing a new proof of the wait-free impossibility of consensus using read/write registers, and a new proof of the wait-free impossibility of solving consensus using registers and test\&set objects for $n>2$. The proof is merely by showing that, in each case, the closure of consensus (w.r.t. the corresponding model) is consensus itself. Our main application is the study of the power of additional objects, namely test\&set and binary consensus, for wait-free solving approximate agreement faster. By analyzing the closure of approximate agreement w.r.t. each of the two models, we show that while these objects are more powerful than read/write registers from the computability perspective, they are not more powerful as far as helping solving approximate agreement faster is concerned.

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arXiv2022

The Augmentation-Speed Tradeoff for Consistent Network Updates

Emerging software-defined networking technologies enable more adaptive communication infrastructures, allowing for quick reactions to changes in networking requirements by exploiting the workload's temporal structure. However, operating networks adaptively is algorithmically challenging, as meeting networks' stringent dependability requirements relies on maintaining basic consistency and performance properties, such as loop freedom and congestion minimization, even during the update process. This paper leverages an augmentation-speed tradeoff to significantly speed up consistent network updates. We show that allowing for a small and short (hence practically tolerable, e.g., using buffering) oversubscription of links allows us to solve many network update instances much faster, as well as to reduce computational complexities (i.e., the running times of the algorithms). We first explore this tradeoff formally, revealing the computational complexity of scheduling updates. We then present and analyze algorithms that maintain logical and performance properties during the update. Using an extensive simulation study, we find that the tradeoff is even more favorable in practice than our analytical bounds suggest. In particular, we find that by allowing just 10% augmentation, update times reduce by more than 32% on average, across a spectrum of real-world networks.

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arXiv2022

Sinkless Orientation Made Simple

The sinkless orientation problem plays a key role in understanding the foundations of distributed computing. The problem can be used to separate two fundamental models of distributed graph algorithms, LOCAL and SLOCAL: the locality of sinkless orientation is $Ω(\log n)$ in the deterministic LOCAL model and $O(\log \log n)$ in the deterministic SLOCAL model. Both of these results are known by prior work, but here we give new simple, self-contained proofs for them.

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arXiv2022

Time Complexity of Consensus in Dynamic Networks Under Oblivious Message Adversaries

Consensus is a most fundamental task in distributed computing. This paper studies the consensus problem for a set of processes connected by a dynamic directed network, in which computation and communication is lock-step synchronous but controlled by an oblivious message adversary. In this basic model, determining consensus solvability and designing consensus algorithms in the case where it is possible, has been shown to be surprisingly difficult. We present an explicit decision procedure to determine if consensus is possible under a given adversary. This in turn enables us, for the first time, to study the time complexity of consensus in this model. In particular, we derive time complexity upper bounds for consensus solvability both for a centralized decision procedure as well as for solving distributed consensus. We complement these results with time complexity lower bounds. Intriguingly, we find that reaching consensus under an oblivious message adversary can take exponentially longer than broadcasting the input value of some process to all other processes.

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arXiv2021

On the Complexity of Weight-Dynamic Network Algorithms

While operating communication networks adaptively may improve utilization and performance, frequent adjustments also introduce an algorithmic challenge: the re-optimization of traffic engineering solutions is time-consuming and may limit the granularity at which a network can be adjusted. This paper is motivated by question whether the reactivity of a network can be improved by re-optimizing solutions dynamically rather than from scratch, especially if inputs such as link weights do not change significantly. This paper explores to what extent dynamic algorithms can be used to speed up fundamental tasks in network operations. We specifically investigate optimizations related to traffic engineering (namely shortest paths and maximum flow computations), but also consider spanning tree and matching applications. While prior work on dynamic graph algorithms focuses on link insertions and deletions, we are interested in the practical problem of link weight changes. We revisit existing upper bounds in the weight-dynamic model, and present several novel lower bounds on the amortized runtime for recomputing solutions. In general, we find that the potential performance gains depend on the application, and there are also strict limitations on what can be achieved, even if link weights change only slightly.

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arXiv2021

On the Complexity of Load Balancing in Dynamic Networks

In the load balancing problem, each node in a network is assigned a load, and the goal is to equally distribute the loads among the nodes, by preforming local load exchanges. While load balancing was extensively studied in static networks, only recently a load balancing algorithm for dynamic networks with a bounded convergence time was presented. In this paper, we further study the time complexity of load balancing in the context of dynamic networks. First, we show that randomness is not necessary, and present a deterministic algorithm which slightly improves the running time of the previous algorithm, at the price of not being matching-based. Then, we consider integral loads, i.e., loads that cannot be split indefinitely, and prove that no matching-based algorithm can have a bounded convergence time for this case. To circumvent both this impossibility result, and a known one for the non-integral case, we apply the method of smoothed analysis, where random perturbations are made over the worst-case choices of network topologies. We show both impossibility results do not hold under this kind of analysis, suggesting that load-balancing in real world systems might be faster than the lower bounds suggest.

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arXiv2021

Input-Dynamic Distributed Algorithms for Communication Networks

Consider a distributed task where the communication network is fixed but the local inputs given to the nodes of the distributed system may change over time. In this work, we explore the following question: if some of the local inputs change, can an existing solution be updated efficiently, in a dynamic and distributed manner? To address this question, we define the batch dynamic CONGEST model in which we are given a bandwidth-limited communication network and a dynamic edge labelling defines the problem input. The task is to maintain a solution to a graph problem on the labeled graph under batch changes. We investigate, when a batch of $α$ edge label changes arrive, -- how much time as a function of $α$ we need to update an existing solution, and -- how much information the nodes have to keep in local memory between batches in order to update the solution quickly. Our work lays the foundations for the theory of input-dynamic distributed network algorithms. We give a general picture of the complexity landscape in this model, design both universal algorithms and algorithms for concrete problems, and present a general framework for lower bounds. In particular, we derive non-trivial upper bounds for two selected, contrasting problems: maintaining a minimum spanning tree and detecting cliques.

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