arXiv subjects

Moti Medina

Publications and source records attributed to Moti Medina.

At least 19 recordsLinked to original sources

arXiv2026

Metastability-Containing Turing Machines

Metastability is a spurious mode of operation in digital signals, where an electrical signal fails to settle into a stable state within a specified time, leading to uncertainty and potentially failing downstream hardware. A system that computes the closure over all possibilities, given an uncertain input, is called a Metastability-containing system. While prior work has addressed metastability-containing systems in the context of combinational and clocked circuits, state machines, and logic formulas, its implications for general-purpose computation remain largely unexplored. In this work, we study the metastability-containing systems within an abstract computational model: The Turing Machine. This approach allows us to investigate the computational limits and capabilities of Turing Machines operating under uncertain inputs. Specifically, we prove that in general the metastable closure of a Turing Machine is non-computable. Then we discuss cases where the meta-stable closure is computable: For EXPTIME problems, we prove that resolving even a single uncertain bit is EXPTIME-complete. In contrast, we prove that for polynomial time problems, the meta-stable closure is polynomial time computable for a logarithmic number of uncertain bits, but coNP-complete, when the number of undefined inputs is arbitrary. Finally, we describe a hardware-realizable Universal Turning Machine that computes the metastable closure of any given bounded-time Turing Machine with at most an exponential blowup in time.

cs.CC↗

arXiv2026

Codes for Metastability-Containing Addition

We investigate the fundamental task of addition under uncertainty, namely, addends that are represented as intervals of numbers rather than single values. One potential source of such uncertainty can occur when obtaining discrete-valued measurements of analog values, which are prone to metastability. Naturally, unstable bits impact gate-level and, consequently, circuit-level computations. Using Binary encoding for such an addition produces a sum with an amplified imprecision. Hence, the challenge is to devise an encoding that does not amplify the imprecision caused by unstable bits. We call such codes recoverable. While this challenge is easily met for unary encoding, no suitable codes of high rates are known. In this work, we prove an upper bound on the rate of preserving and recoverable codes for a given bound on the addends' combined uncertainty. We then design an asymptotically optimal code that preserves the addends' combined uncertainty. We then discuss how to obtain adders for our code. The approach can be used with any known or future construction for containing metastability of the inputs. We conjecture that careful design based on existing techniques can lead to significant latency reduction.

cs.IT↗

arXiv2023

PALS: Distributed Gradient Clocking on Chip

Consider an arbitrary network of communicating modules on a chip, each requiring a local signal telling it when to execute a computational step. There are three common solutions to generating such a local clock signal: (i) by deriving it from a single, central clock source, (ii) by local, free-running oscillators, or (iii) by handshaking between neighboring modules. Conceptually, each of these solutions is the result of a perceived dichotomy in which (sub)systems are either clocked or asynchronous. We present a solution and its implementation that lies between these extremes. Based on a distributed gradient clock synchronization algorithm, we show a novel design providing modules with local clocks, the frequency bounds of which are almost as good as those of free-running oscillators, yet neighboring modules are guaranteed to have a phase offset substantially smaller than one clock cycle. Concretely, parameters obtained from a 15nm ASIC simulation running at 2GHz yield mathematical worst-case bounds of 20ps on the phase offset for a $32 \times 32$ node grid network.

cs.AR↗

arXiv2023

Distributed CONGEST Algorithm for Finding Hamiltonian Paths in Dirac Graphs and Generalizations

We study the problem of finding a Hamiltonian cycle under the promise that the input graph has a minimum degree of at least $n/2$, where $n$ denotes the number of vertices in the graph. The classical theorem of Dirac states that such graphs (a.k.a. Dirac graphs) are Hamiltonian, i.e., contain a Hamiltonian cycle. Moreover, finding a Hamiltonian cycle in Dirac graphs can be done in polynomial time in the classical centralized model. This paper presents a randomized distributed CONGEST algorithm that finds w.h.p. a Hamiltonian cycle (as well as maximum matching) within $O(\log n)$ rounds under the promise that the input graph is a Dirac graph. This upper bound is in contrast to general graphs in which both the decision and search variants of Hamiltonicity require $\tildeΩ(n^2)$ rounds, as shown by Bachrach et al. [PODC'19]. In addition, we consider two generalizations of Dirac graphs: Ore graphs and Rahman-Kaykobad graphs [IPL'05]. In Ore graphs, the sum of the degrees of every pair of non-adjacent vertices is at least $n$, and in Rahman-Kaykobad graphs, the sum of the degrees of every pair of non-adjacent vertices plus their distance is at least $n+1$. We show how our algorithm for Dirac graphs can be adapted to work for these more general families of graphs.

cs.DC↗

arXiv2023

Robust Routing Made Easy: Reinforcing Networks Against Non-Benign Faults

With the increasing scale of communication networks, the likelihood of failures grows as well. Since these networks form a critical backbone of our digital society, it is important that they rely on robust routing algorithms which ensure connectivity despite such failures. While most modern communication networks feature robust routing mechanisms, these mechanisms are often fairly complex to design and verify, as they need to account for the effects of failures and rerouting on communication. This paper conceptualizes the design of robust routing mechanisms, with the aim to avoid such complexity. In particular, we showcase \emph{simple} and generic blackbox transformations that increase resilience of routing against independently distributed failures, which allows to simulate the routing scheme on the original network, even in the presence of non-benign node failures (henceforth called faults). This is attractive as the system specification and routing policy can simply be preserved. We present a scheme for constructing such a reinforced network, given an existing (synchronous) network and a routing scheme. We prove that this algorithm comes with small constant overheads, and only requires a minimal amount of additional node and edge resources; in fact, if the failure probability is smaller than $1/n$, the algorithm can come without any overhead at all. At the same time, it allows to tolerate a large number of independent random (node) faults, asymptotically almost surely. We complement our analytical results with simulations on different real-world topologies.

cs.DC↗

arXiv2023

Graph Ranking and the Cost of Sybil Defense

Ranking functions such as PageRank assign numeric values (ranks) to nodes of graphs, most notably the web graph. Node rankings are an integral part of Internet search algorithms, since they can be used to order the results of queries. However, these ranking functions are famously subject to attacks by spammers, who modify the web graph in order to give their own pages more rank. We characterize the interplay between rankers and spammers as a game. We define the two critical features of this game, spam resistance and distortion, based on how spammers spam and how rankers protect against spam. We observe that all the ranking functions that are well-studied in the literature, including the original formulation of PageRank, have poor spam resistance, poor distortion, or both. Finally, we study Min-PPR, the form of PageRank used at Google itself, but which has received no (theoretical or empirical) treatment in the literature. We prove that Min-PPR has low distortion and high spam resistance. A secondary benefit is that Min-PPR comes with an explicit cost function on nodes that shows how important they are to the spammer; thus a ranker can focus their spam-detection capacity on these vulnerable nodes. Both Min-PPR and its associated cost function are straightforward to compute.

cs.DS↗

arXiv2021

Robust Routing Made Easy

With the increasing scale of communication networks, the likelihood of failures grows as well. Since these networks form a critical backbone of our digital society, it is important that they rely on robust routing algorithms which ensure connectivity despite such failures. While most modern communication networks feature robust routing mechanisms, these mechanisms are often fairly complex to design and verify, as they need to account for the effects of failures and rerouting on communication. This paper revisits the design of robust routing mechanisms, with the aim to avoid such complexity. In particular, we showcase simple and generic blackbox transformations that increase resilience of routing against independently distributed failures, which allows to simulate the routing scheme on the original network, even in the presence of non-benign node failures (henceforth called faults). This is attractive as the system specification and routing policy can simply be preserved. We present a scheme for constructing such a reinforced network, given an existing (synchronous) network and a routing scheme. We prove that this algorithm comes with small constant overheads, and only requires a minimal amount of additional node and edge resources. At the same time, it allows to tolerate a large number of independent random (node) faults, asymptotically almost surely. We complement our analytical results with simulations on different real-world topologies.

cs.DC↗

arXiv2020

Synchronizer-Free Digital Link Controller

This work presents a producer-consumer link between two independent clock domains. The link allows for metastability-free, low-latency, high-throughput communication by slight adjustments to the clock frequencies of the producer and consumer domains steered by a controller circuit. Any such controller cannot deterministically avoid, detect, nor resolve metastability. Typically, this is addressed by synchronizers, incurring a larger dead time in the control loop. We follow the approach of Friedrichs et al. (TC 2018) who proposed metastability-containing circuits. The result is a simple control circuit that may become metastable, yet deterministically avoids buffer underrun or overflow. More specifically, the controller output may become metastable, but this may only affect oscillator speeds within specific bounds. In contrast, communication is guaranteed to remain metastability-free. We formally prove correctness of the producer-consumer link and a possible implementation that has only small overhead. With SPICE simulations of the proposed implementation we further substantiate our claims. The simulation uses 65nm process running at roughly 2GHz.

cs.AR↗

arXiv2020

A Constant Approximation Algorithm for Scheduling Packets on Line Networks

In this paper we improve the approximation ratio for the problem of scheduling packets on line networks with bounded buffers, where the aim is that of maximizing the throughput. Each node in the network has a local buffer of bounded size $B$, and each edge (or link) can transmit a limited number, $c$, of packets in every time unit. The input to the problem consists of a set of packet requests, each defined by a source node, a destination node, and a release time. We denote by $n$ the size of the network. A solution for this problem is a schedule that delivers (some of the) packets to their destinations without violating the capacity constraints of the network (buffers or edges). Our goal is to design an efficient algorithm that computes a schedule that maximizes the number of packets that arrive to their respective destinations. We give a randomized approximation algorithm with constant approximation ratio for the case where $B=Θ(c)$. This improves over the previously best result of $O(\log^* n)$ (Räcke and Rosén, Theory Comput. Syst., 49(4), 2011). Our improvement is based on a new combinatorial lemma that we prove, stating, roughly speaking, that if packets are allowed to stay put in buffers only a limited number of time steps, $2d$, where $d$ is the longest source-destination distance of any input packet, then the cardinality of the optimal solution is decreased by only a constant factor. This claim was not previously known in the directed integral (i.e., unsplittable, zero-one) case, and may find additional applications for routing and scheduling algorithms.

cs.DS↗

arXiv2020

PALS: Plesiochronous and Locally Synchronous Systems

Consider an arbitrary network of communicating modules on a chip, each requiring a local signal telling it when to execute a computational step. There are three common solutions to generating such a local clock signal: (i) by deriving it from a single, central clock source, (ii) by local, free-running oscillators, or (iii) by handshaking between neighboring modules. Conceptually, each of these solutions is the result of a perceived dichotomy in which (sub)systems are either clocked or fully asynchronous, suggesting that the designer's choice is limited to deciding where to draw the line between synchronous and asynchronous design. In contrast, we take the view that the better question to ask is how synchronous the system can and should be. Based on a distributed clock synchronization algorithm, we present a novel design providing modules with local clocks whose frequency bounds are almost as good as those of corresponding free-running oscillators, yet neighboring modules are guaranteed to have a phase offset substantially smaller than one clock cycle. Concretely, parameters obtained from a 15nm ASIC implementation running at 2GHz yield mathematical worst-case bounds of 30ps on phase offset for a 32x32 node grid network.

cs.DC↗

arXiv2020

Distributed Testing of Graph Isomorphism in the CONGEST model

In this paper we study the problem of testing graph isomorphism (GI) in the CONGEST distributed model. In this setting we test whether the distributive network, $G_U$, is isomorphic to $G_K$ which is given as an input to all the nodes in the network, or alternatively, only to a single node. We first consider the decision variant of the problem in which the algorithm distinguishes $G_U$ and $G_K$ which are isomorphic from $G_U$ and $G_K$ which are not isomorphic. We provide a randomized algorithm with $O(n)$ rounds for the setting in which $G_K$ is given only to a single node. We prove that for this setting the number of rounds of any deterministic algorithm is $\tildeΩ(n^2)$ rounds, where $n$ denotes the number of nodes, which implies a separation between the randomized and the deterministic complexities of deciding GI. We then consider the \emph{property testing} variant of the problem, where the algorithm is only required to distinguish the case that $G_U$ and $G_K$ are isomorphic from the case that $G_U$ and $G_K$ are \emph{far} from being isomorphic (according to some predetermined distance measure). We show that every algorithm requires $Ω(D)$ rounds, where $D$ denotes the diameter of the network. This lower bound holds even if all the nodes are given $G_K$ as an input, and even if the message size is unbounded. We provide a randomized algorithm with an almost matching round complexity of $O(D+(ε^{-1}\log n)^2)$ rounds that is suitable for dense graphs. We also show that with the same number of rounds it is possible that each node outputs its mapping according to a bijection which is an \emph{approximated} isomorphism. We conclude with simple simulation arguments that allow us to obtain essentially tight algorithms with round complexity $\tilde{O}(D)$ for special families of sparse graphs.

cs.DS↗

arXiv2019

Optimal Metastability-Containing Sorting via Parallel Prefix Computation

Friedrichs et al. (TC 2018) showed that metastability can be contained when sorting inputs arising from time-to-digital converters, i.e., measurement values can be correctly sorted without resolving metastability using synchronizers first. However, this work left open whether this can be done by small circuits. We show that this is indeed possible, by providing a circuit that sorts Gray code inputs (possibly containing a metastable bit) and has asymptotically optimal depth and size. Our solution utilizes the parallel prefix computation (PPC) framework (JACM 1980). We improve this construction by bounding its fan-out by an arbitrary $f \geq 3$, without affecting depth and increasing circuit size by a small constant factor only. Thus, we obtain the first PPC circuits with asymptotically optimal size, constant fan-out, and optimal depth. To show that applying the PPC framework to the sorting task is feasible, we prove that the latter can, despite potential metastability, be decomposed such that the core operation is associative. We obtain asymptotically optimal metastability-containing sorting networks. We complement these results with simulations, independently verifying the correctness as well as small size and delay of our circuits.

cs.DC↗

arXiv2019

Property Testing of Planarity in the CONGEST model

We give a distributed algorithm in the {\sf CONGEST} model for property testing of planarity with one-sided error in general (unbounded-degree) graphs. Following Censor-Hillel et al. (DISC 2016), who recently initiated the study of property testing in the distributed setting, our algorithm gives the following guarantee: For a graph $G = (V,E)$ and a distance parameter $ε$, if $G$ is planar, then every node outputs {\sf accept\/}, and if $G$ is $ε$-far from being planar (i.e., more than $ε\cdot |E|$ edges need to be removed in order to make $G$ planar), then with probability $1-1/{\rm poly}(n)$ at least one node outputs {\sf reject}. The algorithm runs in $O(\log|V|\cdot{\rm poly}(1/ε))$ rounds, and we show that this result is tight in terms of the dependence on $|V|$. Our algorithm combines several techniques of graph partitioning and local verification of planar embeddings. Furthermore, we show how a main subroutine in our algorithm can be applied to derive additional results for property testing of cycle-freeness and bipartiteness, as well as the construction of spanners, in minor-free (unweighted) graphs.

cs.DC↗

arXiv2018

Small Hazard-free Transducers

Ikenmeyer et al. (JACM'19) proved an unconditional exponential separation between the hazard-free complexity and (standard) circuit complexity of explicit functions. This raises the question: which classes of functions permit efficient hazard-free circuits? In this work, we prove that circuit implementations of transducers with small state space are such a class. A transducer is a finite state machine that transcribes, symbol by symbol, an input string of length $n$ into an output string of length $n$. We present a construction that transforms any function arising from a transducer into an efficient circuit of size $\mathcal{O}(n)$ computing the hazard-free extension of the function. More precisely, given a transducer with $s$ states, receiving $n$ input symbols encoded by $l$ bits, and computing $n$ output symbols encoded by $m$ bits, the transducer has a hazard-free circuit of size $2^{\mathcal{O}(s+\ell)} m n$ and depth $\mathcal{O}(s\log n + \ell)$; in particular, if $s, \ell,m\in \mathcal{O}(1)$, size and depth are asymptotically optimal. In light of the strong hardness results by Ikenmeyer et al. (JACM'19), we consider this a surprising result.

cs.DS↗

arXiv2018

Optimal Metastability-Containing Sorting Networks

When setup/hold times of bistable elements are violated, they may become metastable, i.e., enter a transient state that is neither digital 0 nor 1. In general, metastability cannot be avoided, a problem that manifests whenever taking discrete measurements of analog values. Metastability of the output then reflects uncertainty as to whether a measurement should be rounded up or down to the next possible measurement outcome. Surprisingly, Lenzen and Medina (ASYNC 2016) showed that metastability can be contained, i.e., measurement values can be correctly sorted without resolving metastability first. However, both their work and the state of the art by Bund et al. (DATE 2017) leave open whether such a solution can be as small and fast as standard sorting networks. We show that this is indeed possible, by providing a circuit that sorts Gray code inputs (possibly containing a metastable bit) and has asymptotically optimal depth and size. Concretely, for 10-channel sorting networks and 16-bit wide inputs, we improve by 48.46% in delay and by 71.58% in area over Bund et al. Our simulations indicate that straightforward transistor-level optimization is likely to result in performance on par with standard (non-containing) solutions.

cs.DC↗

arXiv2017

Faster and Simpler Distributed Algorithms for Testing and Correcting Graph Properties in the CONGEST-Model

In this paper we present distributed testing algorithms of graph properties in the CONGEST-model [Censor-Hillel et al. 2016]. We present one-sided error testing algorithms in the general graph model. We first describe a general procedure for converting $ε$-testers with a number of rounds $f(D)$, where $D$ denotes the diameter of the graph, to $O((\log n)/ε)+f((\log n)/ε)$ rounds, where $n$ is the number of processors of the network. We then apply this procedure to obtain an optimal tester, in terms of $n$, for testing bipartiteness, whose round complexity is $O(ε^{-1}\log n)$, which improves over the $poly(ε^{-1} \log n)$-round algorithm by Censor-Hillel et al. (DISC 2016). Moreover, for cycle-freeness, we obtain a \emph{corrector} of the graph that locally corrects the graph so that the corrected graph is acyclic. Note that, unlike a tester, a corrector needs to mend the graph in many places in the case that the graph is far from having the property. In the second part of the paper we design algorithms for testing whether the network is $H$-free for any connected $H$ of size up to four with round complexity of $O(ε^{-1})$. This improves over the $O(ε^{-2})$-round algorithms for testing triangle freeness by Censor-Hillel et al. (DISC 2016) and for testing excluded graphs of size $4$ by Fraigniaud et al. (DISC 2016). In the last part we generalize the global tester by Iwama and Yoshida (ITCS 2014) of testing $k$-path freeness to testing the exclusion of any tree of order $k$. We then show how to simulate this algorithm in the CONGEST-model in $O(k^{k^2+1}\cdotε^{-k})$ rounds.

cs.DC↗

arXiv2016

Sublinear Random Access Generators for Preferential Attachment Graphs

We consider the problem of sampling from a distribution on graphs, specifically when the distribution is defined by an evolving graph model, and consider the time, space and randomness complexities of such samplers. In the standard approach, the whole graph is chosen randomly according to the randomized evolving process, stored in full, and then queries on the sampled graph are answered by simply accessing the stored graph. This may require prohibitive amounts of time, space and random bits, especially when only a small number of queries are actually issued. Instead, we propose to generate the graph on-the-fly, in response to queries, and therefore to require amounts of time, space, and random bits which are a function of the actual number of queries. We focus on two random graph models: the Barab{\'{a}}si-Albert Preferential Attachment model (BA-graphs) and the random recursive tree model. We give on-the-fly generation algorithms for both models. With probability $1-1/\mbox{poly}(n)$, each and every query is answered in $\mbox{polylog}(n)$ time, and the increase in space and the number of random bits consumed by any single query are both $\mbox{polylog}(n)$, where $n$ denotes the number of vertices in the graph. Our results show that, although the BA random graph model is defined by a sequential process, efficient random access to the graph's nodes is possible. In addition to the conceptual contribution, efficient on-the-fly generation of random graphs can serve as a tool for the efficient simulation of sublinear algorithms over large BA-graphs, and the efficient estimation of their performance on such graphs.

cs.DS↗

arXiv2016

Competitive Path Computation and Function Placement in SDNs

We consider a task of serving requests that arrive in an online fashion in Software-Defined Networks (SDNs) with network function virtualization (NFV). Each request specifies an abstract routing and processing "plan" for a flow. Each processing function can be performed by a specified subset of servers in the system. The algorithm needs to either reject the request or admit it and return detailed routing (a.k.a. "path computation") and processing assignment ("function placement"). Each request also specifies the communication bandwidth and the processing load it requires. Components in the system (links and processors) have bounded capacity; a feasible solution may not violate the capacity constraints. Requests have benefits and the goal is to maximize the total benefit of accepted requests. In this paper we first formalize the problem, and propose a new service model that allows us to cope with requests with unknown duration. The new service model augments the traditional accept/reject schemes with a new possible response of "stand by." Our main result is an online algorithm for path computation and function placement that guarantees, in each time step, throughput of at least $Ω\left(\frac{\text{OPT}^*}{\log n}\right)$, where $n$ is the system size and $\text{OPT}^*$ is an upper bound on the maximal possible throughput. The guarantee holds assuming that requests ask for at most an $O\left(1/{\log n}\right)$-fraction of the capacity of any component in the system. Furthermore, the guarantee holds even though our algorithm serves requests in an all-or-nothing fashion using a single path and never preempts accepted flows, while $\text{OPT}^*$ may serve fractional requests, may split the allocation over multiple paths, and may arbitrarily preempt and resume service of requests.

cs.DS↗