Distributed algorithms for ultrasparse spanners and linear size skeletons

We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an ${O(2^{{\rm log}^{*} n} {\rm log} n)}We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an O(2^{log* n} log n){O(2^{{\rm log}^{*} n} {\rm log} n)} -spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)-size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)-spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size, namely O(n(loglogn)^f){O(n(\log \log n)^{\phi})} , where f = (1 + ?5)/2{\phi = (1 + \sqrt{5})/2} is the golden ratio. As the distance increases the multiplicative distortion of a Fibonacci spanner passes through four discrete stages, moving from logarithmic to log-logarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion.     

Distributed MST for constant diameter graphs

This paper considers the problem of distributively constructing a minimum-weight spanning tree (MST) for graphs of constant diameter in the bounded-messages model, where each message can contain at most B bits for some parameter B. It is shown that the number of communication rounds necessary to compute an MST for graphs of diameter 4 or 3 can be as high as and , respectively. The asymptotic lower bounds hold for randomized algorithms as well. On the other hand, we observe that O(log n) communication rounds always suffice to compute an MST deterministically for graphs with diameter 2, when B = O(log n). These results complement a previously known lower bound of for graphs of diameter Ω(log n). An extended abstract of this work appears in Proceedings of 20th ACM Symposium on Principles of Distributed Computing, August 2001.     

The space complexity of unbounded timestamps

The timestamp problem captures a fundamental aspect of asynchronous distributed computing. It allows processes to label events throughout the system with timestamps that provide information about the real-time ordering of those events. We consider the space complexity of wait-free implementations of timestamps from shared read-write registers in a system of n processes. We prove an lower bound on the number of registers required. If the timestamps are elements of a nowhere dense set, for example the integers, we prove a stronger, and tight, lower bound of n. However, if timestamps are not from a nowhere dense set, this bound can be beaten: we give an implementation that uses n − 1 (single-writer) registers. We also consider the special case of anonymous implementations, where processes are programmed identically and do not have unique identifiers. In contrast to the general case, we prove anonymous timestamp implementations require n registers. We also give an implementation to prove that this lower bound is tight. This is the first anonymous timestamp implementation that uses a finite number of registers.     

A simple population protocol for fast robust approximate majority

We describe and analyze a 3-state one-way population protocol to compute approximate majority in the model in which pairs of agents are drawn uniformly at random to interact. Given an initial configuration of x’s, y’s and blanks that contains at least one non-blank, the goal is for the agents to reach consensus on one of the values x or y. Additionally, the value chosen should be the majority non-blank initial value, provided it exceeds the minority by a sufficient margin. We prove that with high probability n agents reach consensus in O(n log n) interactions and the value chosen is the majority provided that its initial margin is at least . This protocol has the additional property of tolerating Byzantine behavior in of the agents, making it the first known population protocol that tolerates Byzantine agents.     

An improved algorithm for finding the median distributively

Given two processes, each having a total-ordered set ofn elements, we present a distributed algorithm for finding median of these 2n elements using no more than logn +O(logn) messages, but if the elements are distinct, only logn +O(1) messages will be required. The communication complexity of our algorithm is better than the previously known result which takes 2 logn messages.     

Time efficient k-shot broadcasting in known topology radio networks

The paper considers broadcasting protocols in radio networks with known topology that are efficient in both time and energy. The radio network is modelled as an undirected graph G = (V, E) where |V| = n. It is assumed that during execution of the communication task every node in V is allowed to transmit at most once. Under this assumption it is shown that any radio broadcast protocol requires transmission rounds, where D is the diameter of G. This lower bound is complemented with an efficient construction of a deterministic protocol that accomplishes broadcasting in rounds. Moreover, if we allow each node to transmit at most k times, the lower bound on the number of transmission rounds holds. We also provide a randomised protocol that accomplishes broadcasting in rounds. The paper concludes with a number of open problems in the area. The research of L. Gąsieniec, D.R. Kowalski and C. Su supported in part by the Royal Society grant Algorithmic and Combinatorial Aspects of Radio Communication, IJP - 2006/R2. The research of E. Kantor and D. Peleg supported in part by grants from the Minerva Foundation and the Israel Ministry of Science.     

ℓ 2 2 Spreading Metrics for Vertex Ordering Problems

We design approximation algorithms for the vertex ordering problems Minimum Linear Arrangement, Minimum Containing Interval Graph, and Minimum Storage-Time Product, achieving approximation factors of $O(\sqrt{\log n}\log\log n)We design approximation algorithms for the vertex ordering problems Minimum Linear Arrangement, Minimum Containing Interval Graph, and Minimum Storage-Time Product, achieving approximation factors of O(?{logn}loglogn)O(\sqrt{\log n}\log\log n) , O(?{logn}loglogn)O(\sqrt{\log n}\log\log n) , and O(?{logT}loglogT)O(\sqrt{\log T}\log\log T) , respectively, the last running in time polynomial in T (T being the sum of execution times). The technical contribution of our paper is to introduce “ℓ ₂² spreading metrics” (that can be computed by semidefinite programming) as relaxations for both undirected and directed “permutation metrics,” which are induced by permutations of {1,2,…,n}. The techniques introduced in the recent work of Arora, Rao and Vazirani (Proc. of 36th STOC, pp. 222–231, 2004) can be adapted to exploit the geometry of such ℓ ₂² spreading metrics, giving a powerful tool for the design of divide-and-conquer algorithms. In addition to their applications to approximation algorithms, the study of such ℓ ₂² spreading metrics as relaxations of permutation metrics is interesting in its own right. We show how our results imply that, in a certain sense we make precise, ℓ ₂² spreading metrics approximate permutation metrics on n points to a factor of O(?{logn}loglogn)O(\sqrt{\log n}\log\log n) .     

The Network as a Storage Device: Dynamic Routing with Bounded Buffers

We study dynamic routing in store-and-forward packet networks where each network link has bounded buffer capacity for receiving incoming packets and is capable of transmitting a fixed number of packets per unit of time. At any moment in time, packets are injected at various network nodes with each packet specifying its destination node. The goal is to maximize the throughput, defined as the number of packets delivered to their destinations. In this paper, we make some progress on throughput maximization in various network topologies. Let n and m denote the number of nodes and links in the network, respectively. For line networks, we show that Nearest-to-Go (NTG), a natural distributed greedy algorithm, is -competitive, essentially matching a known lower bound on the performance of any greedy algorithm. We also show that if we allow the online routing algorithm to make centralized decisions, there is a randomized polylog(n)-competitive algorithm for line networks as well as for rooted tree networks, where each packet is destined for the root of the tree. For grid graphs, we show that NTG has a competitive ratio of while no greedy algorithm can achieve a ratio better than . Finally, for arbitrary network topologies, we show that NTG is -competitive, improving upon an earlier bound of O(mn). An extended abstract appeared in the Proceedings of the 8th Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005, Berkeley, CA, USA, pp. 1–13, Lecture Notes in Computer Science, vol. 1741, Springer, Berlin. S. Angelov is supported in part by NSF Career Award CCR-0093117, NSF Award ITR 0205456 and NIGMS Award 1-P20-GM-6912-1. S. Khanna is supported in part by an NSF Career Award CCR-0093117, NSF Award CCF-0429836, and a US-Israel Binational Science Foundation Grant. K. Kunal is supported in part by an NSF Career Award CCR-0093117 and NSF Award CCF-0429836.     

Deterministic broadcasting in ad hoc radio networks

We consider the problem of distributed deterministic broadcasting in radio networks of unknown topology and size. The network is synchronous. If a node u can be reached from two nodes which send messages in the same round, none of the messages is received by u. Such messages block each other and node u either hears the noise of interference of messages, enabling it to detect a collision, or does not hear anything at all, depending on the model. We assume that nodes know neither the topology nor the size of the network, nor even their immediate neighborhood. The initial knowledge of every node is limited to its own label. Such networks are called ad hoc multi-hop networks. We study the time of deterministic broadcasting under this scenario. For the model without collision detection, we develop a linear-time broadcasting algorithm for symmetric graphs, which is optimal, and an algorithm for arbitrary n-node graphs, working in time . Next we show that broadcasting with acknowledgement is not possible in this model at all. For the model with collision detection, we develop efficient algorithms for broadcasting and for acknowledged broadcasting in strongly connected graphs. Received: January 2000 / Accepted: June 2001     

Store-and-Forward Multicast Routing on the Mesh

We study the complexity of routing a set of messages with multiple destinations (multicast routing) on an n-node square mesh under the store-and-forward model. A standard argument proves that time is required to route n messages, where each message is generated by a distinct node and at most c messages are to be delivered to any individual node. The obvious approach of simply replicating each message into the appropriate number of unicast (single-destination) messages and routing these independently does not yield an optimal algorithm. We provide both randomized and deterministic algorithms for multicast routing, which use constant-size buffers at each node. The randomized algorithm attains optimal performance, while the deterministic algorithm is slower by a factor of O( log ² n). We also describe an optimal deterministic algorithm that, however, requires large buffers of size O(c). A preliminary version of this paper appeared in Proceedings of the 13th Annual ACM Symposium on Parallel Algorithms and Architectures, Crete, Greece, 2001. This work was supported, in part, by MIUR under project ALGO-NEXT.     

All-to-All Optical Routing in Chordal Rings of Degree 4

We consider the problem of routing in networks employing all-optical routing technology. In such networks, information between nodes of the network is transmitted as light on fiber-optic lines without being converted to electronic form in between. We consider switched optical networks that use the wavelength-division multiplexing (or WDM) approach. A WDM network consists of nodes connected by point-to-point fiber-optic links, each of which can support a fixed number of wavelengths. The switches are capable of redirecting incoming streams based on wavelengths, without changing the wavelengths. Different messages may use the same link concurrently if they are assigned distinct wavelengths. However, messages assigned the same wavelength must be assigned edge-disjoint paths. Given a communication instance in a network, the optical routing problem is the assignment of {routes} to communication requests of the instance, as well as wavelengths to routes so that the number of wavelengths used by the instance is minimal. We focus on the all-to-all communication instance I _A in a widely studied family of chordal rings of degree 4, called optimal chordal rings . For these networks, we prove exact bounds on the optimal load induced on an edge for I _A , over all shortest-path routing schemes. We show an approximation algorithm that solves the optical routing problem for I _A using at most 1.006 times the lower bound on the number of wavelengths. The previous best approximation algorithm has a performance ratio of 8. Furthermore, we use a variety of novel techniques to achieve this result, which are applicable to other communication instances and may be applicable to other networks. Received July 22, 1998; revised October 14, 1999.     

Distributed discovery of large near-cliques

Given an undirected graph and 0 ￡ e ￡ 1{0\le\epsilon\le1}, a set of nodes is called an e{\epsilon}-near clique if all but an e{\epsilon} fraction of the pairs of nodes in the set have a link between them. In this paper we present a fast synchronous network algorithm that uses small messages and finds a near-clique. Specifically, we present a constant-time algorithm that finds, with constant probability of success, a linear size e{\epsilon}-near clique if there exists an e³{\epsilon^3}-near clique of linear size in the graph. The algorithm uses messages of O(log n) bits. The failure probability can be reduced to n ^−Ω(1) by increasing the time complexity by a logarithmic factor, and the algorithm also works if the graph contains a clique of size Ω(n/(log log n) ^α ) for some a ? (0,1){\alpha \in (0,1)}. Our approach is based on a new idea of adapting property testing algorithms to the distributed setting.     

The searching over separators strategy to solve some NP-hard problems in subexponential time

In this paper we propose a new strategy for designing algorithms, called the searching over separators strategy. Suppose that we have a problem where the divide-and-conquer strategy can not be applied directly. Yet, also suppose that in an optimal solution to this problem, there exists a separator which divides the input points into two parts,A _d andC _d, in such a way that after solving these two subproblems withA _d andC _d as inputs, respectively, we can merge the respective subsolutions into an optimal solution. Let us further assume that this problem is an optimization problem. In this case our searching over separators strategy will use a separator generator to generate all possible separators. For each separator, the problem is solved by the divide-and-conquer strategy. If the separator generator is guaranteed to generate the desired separator existing in an optimal solution, our searching over separators strategy will always produce an optimal solution. The performance of our approach will critically depend upon the performance of the separator generator. It will perform well if the total number of separators generated is relatively small. We apply this approach to solve the discrete EuclideanP-median problem (DEPM), the discrete EuclideanP-center problem (DEPC), the EuclideanP-center problem (EPC), and the Euclidean traveling salesperson problem (ETSP). We propose algorithms for the DEPM problem, the DEPC problem, and the EPC problem, and we propose an algorithm for the ETSP problem, wheren is the number of input points.This research work was partially supported by the National Science Council of the Republic of China under Grant NSC 79-0408-E007-04.     

Fast message dissemination in random geometric networks

We analyze information dissemination in random geometric networks, which consist of n nodes placed uniformly at random in the square ${[0,\sqrt{n}]^{2}}$ . In the corresponding graph two nodes u and v are connected by a (directed) edge, i.e., u is an (incoming) neighbor of v, if and only if the distance between u and v is smaller than the transmission radius assigned to u. In order to study the performance of distributed communication algorithms in such networks, we adopt here the ad-hoc radio communication model with no collision detection mechanism available. In this model the topology of network connections is not known in advance. Also a node v is capable of receiving a message from its neighbor u if u is the only (incoming) neighbor transmitting in a given step. Otherwise a collision occurs prompting interference that is not distinguishable from the background noise in the network. First, we consider networks modeled by random geometric graphs in which all nodes have the same radius ${r > \delta \sqrt{\log n}}$ , where δ is a sufficiently large constant. In such networks, we provide a rigorous study of the classical communication problem of distributed gossiping (all-to-all communication). We examine various scenarios depending on initial local knowledge and capabilities of network nodes. We show that in many cases an asymptotically optimal distributed O(D)-time gossiping is feasible, where D stands for the diameter of the network. Later, we consider networks in which the transmission radii of the nodes vary according to a power law distribution, i.e., any node is assigned a transmission radius r > r _min according to probability density function ρ(r) ~ r ^?α . More precisely, ${\rho(r) = (\alpha-1)r_{\min}^{\alpha-1} r^{-\alpha}}$ , where ${\alpha \in (1, 3)}$ and ${r_{\min} > \delta \sqrt{\log n}}$ with δ being a large constant. In this case, we develop a simple broadcasting algorithm that runs in time O(log log n) (i.e., O(D)) always surely, and we show that this result is asymptotically optimal. Finally, we consider networks in which any node is assigned a transmission radius r > c according to the probability density function ρ(r) =  (α?1)c ^α-1 r ^?α , where α is a constant from the same range as before and c is a constant. In this model the graph is usually not strongly connected, however, there is one giant component with Ω(n) nodes, and there is a directed path from each node of this giant component to every other node in the graph. We assume that the message which has to be disseminated is placed initially in one of the nodes of the giant component, and every node is aware of its own position in ${[0,\sqrt{n}] \times [0,\sqrt{n}]}$ . Then, we show that there exists a randomized algorithm which delivers the broadcast message to all nodes in the network in time O(D . (log log n)²), almost always surely, where D stands for the diameter of the giant component of the graph. One can conclude from our studies that setting the transmission radii of the nodes according to a power law distribution brings clear advantages. In particular, one can design energy efficient radio networks with low average transmission radius, in which broadcasting can be performed exponentially faster than in the (extensively studied) case where all nodes have the uniform low transmission power.     

Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification

This paper introduces the notion of distributed verification without preprocessing. It focuses on the Minimum-weight Spanning Tree (MST) verification problem and establishes tight upper and lower bounds for the time and message complexities of this problem. Specifically, we provide an MST verification algorithm that achieves simultaneously $\tilde{O}(m)$ messages and $\tilde{O}(\sqrt{n} + D)$ time, where m is the number of edges in the given graph G, n is the number of nodes, and D is G’s diameter. On the other hand, we show that any MST verification algorithm must send $\tilde{\varOmega}(m)$ messages and incur $\tilde{\varOmega}(\sqrt{n} + D)$ time in worst case. Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of $\tilde{\varOmega}(m)$ messages and $\tilde{\varOmega}(\sqrt{n} + D)$ time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously $\tilde{O}(m)$ messages and $\tilde{O}(\sqrt{n} + D)$ time. Specifically, the best known time-optimal algorithm (using ${\tilde{O}}(\sqrt {n} + D)$ time) requires O(m+n ^3/2) messages, and the best known message-optimal algorithm (using ${\tilde{O}}(m)$ messages) requires O(n) time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction.     

Fast arithmetic in algorithmic self-assembly

In this paper we consider the time complexity of adding two n-bit numbers together within the tile self-assembly model. The (abstract) tile assembly model is a mathematical model of self-assembly in which system components are square tiles with different glue types assigned to tile edges. Assembly is driven by the attachment of singleton tiles to a growing seed assembly when the net force of glue attraction for a tile exceeds some fixed threshold. Within this frame work, we examine the time complexity of computing the sum of two n-bit numbers, where the input numbers are encoded in an initial seed assembly, and the output sum is encoded in the final, terminal assembly of the system. We show that this problem, along with multiplication, has a worst case lower bound of \(\varOmega ( \sqrt{n} )\) in 2D assembly, and \(\varOmega (\root 3 \of {n})\) in 3D assembly. We further design algorithms for both 2D and 3D that meet this bound with worst case run times of \(O(\sqrt{n})\) and \(O(\root 3 \of {n})\) respectively, which beats the previous best known upper bound of O(n). Finally, we consider average case complexity of addition over uniformly distributed n-bit strings and show how we can achieve \(O(\log n)\) average case time with a simultaneous \(O(\sqrt{n})\) worst case run time in 2D. As additional evidence for the speed of our algorithms, we implement our algorithms, along with the simpler O(n) time algorithm, into a probabilistic run-time simulator and compare the timing results.     

Termination detection for dynamically distributed systems with non-first-in-first-out communication

We propose a new algorithm for detecting termination of distributed systems. The algorithm works correctly whether the system is static or dynamic, whether the interprocess communication is synchronous or asynchronous, and whether the communication channels are first-in-first-out or not. The algorithm requires, in the worst case, O(nm) control messages in all, where n is the number of processes in the system and m is the total number of messages transmitted during the operation of the system. After the system terminates, the algorithm is able to detect the termination using only O(n) control messages; it is optimal if the system concerned is static.     

An efficient distributed algorithm for maximum matching in general graphs

We present a distributed algorithm for maximum cardinality matching in general graphs. On a general graph withn vertices, our algorithm requiresO(n ^5/2) messages in the worst case. On trees, our algorithm computes a maximum matching usingO(n) messages after the election of a leader.Work on this paper has been supported by the Office of Naval Research under Contract N00014-85-K-0570.     

An Incremental Distributed Algorithm for Computing Biconnected Components in Dynamic Graphs

This paper describes a distributed algorithm for computing the biconnected components of a dynamically changing graph. Our algorithm has a worst-case communication complexity of O(b+c) messages for an edge insertion and O(b'+c) messages for an edge removal, and a worst-case time complexity of O(c) for both operations, where c is the maximum number of biconnected components in any of the connected components during the operation, b is the number of nodes in the biconnected component containing the new edge, and b' is the number of nodes in the biconnected component just before the deletion. The algorithm is presented in two stages. First, a serial algorithm is presented in which topology updates occur one at a time. Then, building on the serial algorithm, an algorithm is presented in which concurrent update requests are serialized within each connected component. The problem is motivated by the need to implement causal ordering of messages efficiently in a dynamically changing communication structure. Received January 1995; revised February 1997.