Networks are a mathematical abstraction of the interactions between a set of entities, with extensive applications in social science, epidemiology, bioinformatics, and cybersecurity, among others. There are many fundamental problems when analyzing network data, such as anomaly detection, dense subgraph mining, motif finding, information diffusion, and epidemic spread. A common underlying task in all these problems is finding an "interesting subgraph"; that is, finding a part of the graph---usually small relative to the whole---that optimizes a score function and has some property of interest, such as connectivity or a minimum density. Finding subgraphs that satisfy common constraints of interest, such as the ones above, is computationally hard in general, and state-of-the-art algorithms for many problems in network analysis are heuristic in nature. These methods are fast and usually easy to implement. However, they come with no theoretical guarantees on the quality of the solution, which makes it difficult to assess how the discovered subgraphs compare to an optimal solution, which in turn affects the data mining task at hand. For instance, in anomaly detection, solutions with low anomaly score lead to sub-optimal detection power. On the other end of the spectrum, there have been significant advances on approximation algorithms for these challenging graph problems in the theoretical computer science community. However, these algorithms tend to be slow, difficult to implement, and they do not scale to the large datasets that are common nowadays. The goal of this dissertation is developing scalable algorithms with theoretical guarantees for various network analysis problems, where the underlying task is to find subgraphs with constraints. We find interesting subgraphs with guarantees by adapting techniques from parameterized complexity, convex optimization, and submodularity optimization. These techniques are well-known in the algorithm design literature, but they lead to slow and impractical algorithms. One unifying theme in the problems that we study is that our methods are scalable without sacrificing the theoretical guarantees of these algorithm design techniques. We accomplish this combination of scalability and rigorous bounds by exploiting properties of the problems we are trying to optimize, decomposing or compressing the input graph to a manageable size, and parallelization. We consider problems on network analysis for both static and dynamic network models. And we illustrate the power of our methods in applications, such as public health, sensor data analysis, and event detection using social media data. / Ph. D.

Recent advances in wireless communications along with developments in low-power circuit design and micro-electro mechanical systems (MEMS) have heralded the advent of compact and inexpensive wireless micro-sensor devices. A large network of such sensor nodes capable of communicating with each other provides significant new capabilities for automatically collecting and analyzing data from physical environments. A notable feature of these networks is that more nodes than are strictly necessary may be deployed to cover a given region. This permits the system to provide reliable information, tolerate many types of faults, and prolong the effective service time. Like most wireless systems, achieving low power consumption is a key consideration in the design of these networks. This thesis presents algorithms for managing power at the distributed system level, rather than just at the individual node level. These distributed algorithms allocate work based on user requests to the individual sensor nodes that comprise the network. The primary goal of the algorithms is to provide a robust and scalable approach for tasking nodes that prolongs the effective life of the network. Theoretical analysis and simulation results are presented to characterize the behavior of these algorithms. Results obtained from simulation experiments indicate that the algorithms can achieve a significant increase in the life of the network. In some cases this may be by an order of magnitude. The algorithms are also shown to ensure a good quality of sensor coverage while improving the network life. Finally, they are shown to be robust to faults and scale to large numbers of nodes. / Master of Science

[pt] Neste trabalho propomos uma arquitetura para a verificação formal de protocolos e algoritmos distribuídos. Esta pode ser vista como uma camada mais abstrata sobre o processo tradicional de verificação formal, onde temos a especificação e propriedade a serem verificadas, o verificador e o resultado retornado por este. O objetivo é simplificar o processo de especificação e verificação formal de protocolos e algoritmos distribuídos através de um ambiente mais dedicado. A parte principal desta arquitetura é a linguagem de especificação LEP, que contém construções de domínio-especifíco para simplificar a especificação destes sistemas. Outra característica desta linguagem é separar as especificações da topologia e do protocolo propriamente dito. Acreditamos que esta separação é válida pois torna mais clara a intenção das partes e ainda permite, por exemplo, o reuso de uma topologia entre diferentes especificações de protocolos. Assim, visamos oferecer uma linguagem cujos exemplos de especificações devem se assemelhar às descrições de algoritmos encontradas nos livros didáticos. Além disso, de forma a se ter a entrada e a saída dos verificadores formais de forma a obter a saída no nível de abstração de LEP. / [en] In this work we propose an architecture for the formal verification of protocols and distribued algoritms. This can be see as a more abstract layer over the ordinary process of formal verification, where we have just the specification of the protocol and properties to be verified, and the formal tool. Our goal is to simplifu the specification and formal verification of protocols and distributed algorithms through a dedicated environment. The core of the architecture is its input specification language (Lep), which provides domain-specific constructions for simplifying the specification of those systems. With LEP the specification of the protocol and the specification of the topology to be referred to protocol are given separetely. We feel that this division improves the legibility of both and allows the reuse of the specification of a topology among distinct protocols. Using this approach we try to offer a language whose specifications should be similar to the descriptions of the algorithms found on the didactic books. Moreover, in order to have the input and output of the architecture compatible, we also propose a way of processing the result of the formal verification tool. Then we could have the result on the abstract level of LEP.

Fairneß (d.h. faire Konfliktlösung), Randomisierung (d.h. Münzwürfe) und partielle Synchronie sind verschiedene Konzepte, die häufig zur Lösung zentraler Synchronisations- und Koordinationsprobleme in verteilten Systemen verwendet werden. Beispiele für solche Probleme sind das Problem des wechselseitigen Ausschlusses (kurz: Mutex-Problem) sowie das Konsens-Problem. Für einige solcher Probleme wurde bewiesen, daß ohne die oben genannten Konzepte keine Lösung für das betrachtete Problem existiert. Unmöglichkeitsresultate dieser Art verbessern unser Verständnis der Wirkungsweise verteilter Algorithmen sowie das Verständnis des Trade-offs zwischen einem leicht analysierbaren und einem ausdrucksstarken Modell für verteiltes Rechnen. In dieser Arbeit stellen wir zwei neue Unmöglichkeitsresultate vor. Darüberhinaus beleuchten wir ihre Hintergründe. Wir betrachten dabei Modelle, die Randomisierung einbeziehen, da bisher wenig über die Grenzen der Ausdrucksstärke von Randomisierung bekannt ist. Mit einer Lösung eines Problems durch Randomisierung meinen wir, daß das betrachtete Problem mit Wahrscheinlichkeit 1 gelöst wird. Im ersten Teil der Arbeit untersuchen wir die Beziehung von Fairneß und Randomisierung. Einerseits ist bekannt, daß einige Probleme (z.B. das Konsens- Problem) durch Randomisierung nicht aber durch Fairneß lösbar sind. Wir zeigen nun, daß es andererseits auch Probleme gibt (nämlich das Mutex-Problem), die durch Fairneß, nicht aber durch Randomisierung lösbar sind. Daraus folgt, daß Fairneß nicht durch Randomisierung implementiert werden kann. Im zweiten Teil der Arbeit verwenden wir ein Modell, das Fairneß und Randomisierung vereint. Ein solches Modell ist relativ ausdrucksstark: Es erlaubt Lösungen für das Mutex-Problem, das Konsens-Problem, sowie eine Lösung für das allgemeine Mutex-Problem. Beim allgemeinen Mutex-Problem (auch bekannt als Problem der speisenden Philosophen) ist eine Nachbarschaftsrelation auf den Agenten gegeben und ein Algorithmus gesucht, der das Mutex-Problem für jedes Paar von Nachbarn simultan löst. Schließlich betrachten wir das ausfalltolerante allgemeine Mutex-Problem -- eine Variante des allgemeinen Mutex-Problems, bei der Agenten ausfallen können. Wir zeigen, daß sogar die Verbindung von Fairneß und Randomisierung nicht genügt, um eine Lösung für das ausfalltolerante allgemeine Mutex-Problem zu konstruieren. Ein Hintergrund für dieses Unmöglichkeitsresultat ist ein unerwünschtes Phänomen, für das in der Literatur der Begriff Konspiration geprägt wurde. Konspiration wurde bisher nicht adäquat charakterisiert. Wir charakterisieren Konspiration auf der Grundlage nicht-sequentieller Abläufe. Desweiteren zeigen wir, daß Konspiration für eine große Klasse von Systemen durch die zusätzliche Annahme von partieller Synchronie verhindert werden kann, d.h. ein konspirationsbehaftetes System kann zu einem randomisierten System verfeinert werden, das unter Fairneß und partieller Synchronie mit Wahrscheinlichkeit 1 konspirationsfrei ist. Partielle Synchronie fordert, daß alle relativen Geschwindigkeiten im System durch eine Konstante beschränkt sind, die jedoch den Agenten nicht bekannt ist. Die Darstellung der Unmöglichkeitsresultate und die Charakterisierung von Konspiration wird erst durch die Verwendung nicht-sequentieller Abläufe möglich. Ein nicht-sequentieller Ablauf repräsentiert im Gegensatz zu einem sequentiellen Ablauf kausale Ordnung und nicht zeitliche Ordnung von Ereignissen. Wir entwickeln in dieser Arbeit eine nicht-sequentielle Semantik für randomisierte verteilte Algorithmen, da es bisher keine in der Literatur gibt. In dieser Semantik wird kausale Unabhängigkeit durch stochastische Unabhängigkeit widergespiegelt. / Concepts such as fairness (i.e., fair conflict resolution), randomization (i.e., coin flips), and partial synchrony are frequently used to solve fundamental synchronization- and coordination-problems in distributed systems such as the mutual exclusion problem (mutex problem for short) and the consensus problem. For some problems it is proven that, without such concepts, no solution to the particular problem exists. Impossibilty results of that kind improve our understanding of the way distributed algorithms work. They also improve our understanding of the trade-off between a tractable model and a powerful model of distributed computation. In this thesis, we prove two new impossibility results and we investigate their reasons. We are in particular concerned with models for randomized distributed algorithms since little is yet known about the limitations of randomization with respect to the solvability of problems in distributed systems. By a solution through randomization we mean that the problem under consideration is solved with probability 1. In the first part of the thesis, we investigate the relationship between fairness and randomization. On the one hand, it is known that to some problems (e.g. to the consensus problem), randomization admits a solution where fairness does not admit a solution. On the other hand, we show that there are problems (viz. the mutex problem) to which randomization does not admit a solution where fairness does admit a solution. These results imply that fairness cannot be implemented by coin flips. In the second part of the thesis, we consider a model which combines fairness and randomization. Such a model is quite powerful, allowing solutions to the mutex problem, the consensus problem, and a solution to the generalized mutex problem. In the generalized mutex problem (a.k.a. the dining philosophers problem), a neighborhood relation is given and mutual exclusion must be achieved for each pair of neighbors. We finally consider the crash-tolerant generalized mutex problem where every hungry agent eventually becomes critical provided that neither itself nor one of its neighbors crashes. We prove that even the combination of fairness and randomization does not admit a solution to the crash-tolerant generalized mutex problem. We argue that the reason for this impossibility is the inherent occurrence of an undesirable phenomenon known as conspiracy. Conspiracy was not yet properly characterized. We characterize conspiracy on the basis of non-sequential runs, and we show that conspiracy can be prevented by help of the additional assumption of partial synchrony, i.e., we show that every conspiracy-prone system can be refined to a randomized system which is, with probability 1, conspiracy-free under the assumptions of partial synchrony and fairness. Partial synchrony means that each event consumes a bounded amount of time where, however, the bound is not known. We use a non-sequential semantics for distributed algorithms which is essential to some parts of the thesis. In particular, we develop a non-sequential semantics for randomized distributed algorithms since there is no such semantics in the literature. In this non-sequential semantics, causal independence is reflected by stochastic independence.

Um Entwurf und Verifikation komplizierter verteilter Algorithmen leichter und verständlicher zu machen, wird oft eine Verfeinerungsmethode verwendet. Dabei wird ein einfacher Algorithmus, der gewünschte Eigenschaften erfüllt, schrittweise zu einem komplizierten Algorithmus verfeinert. In jedem Schritt sollen die gewünschten Eigenschaften erhalten bleiben. Für nachrichtenbasierte verteilte Algorithmen haben wir eine neue Verfeinerungsmethmode entwickelt. Wir beginnen mit einem Anfangsalgorithmus, der Aktionen enthält, die gemeinsame Aufgaben mehrerer Agenten beschreiben. In jedem Schritt verfeinern wir eine dieser Aktionen zu einem Netz, das nur solche Aktionen enthält, die die Aufgaben einzelner Agenten beschreiben. Jeder Schritt ist also eine Verteilung einer unverteilten Aktion. Die Analyse solcher Verfeinerungsschritte wird mit Hilfe eines neuen Verfeinerungsbegriffs - der verteilenden Verfeinerung - durchgeführt. Entscheidend dabei ist das Erhaltenbleiben der Halbordnungen des zu verfeinernden Algorithmus. Dies ist durch Kausalitäten der Aktionen der Agenten im lokalen Verfeinerungsnetz zu erreichen. Die Kausalitäten im lokalen Verfeinerungsnetz lassen sich einerseits beim Entwurf direkt durch Nachrichtenaustausch realisieren. Andererseits kann man bei der Verifikation die Gültigkeit einer Kausalität im lokalen Verfeinerungsnetz direkt vom Netz ablesen. Daher ist diese Methode leicht zu verwenden. Die Anwendung der Methode wird in der Arbeit an verschiedenen nicht trivialen Beispielen demonstriert. / In order to make design and verification of complicated distributed algorithms easier and more understandable, a refinement method is often used. A simple algorithm, which fulfills desired properties, is refined stepwise to a complicated algorithm. In each step the desired properties are preserved. For messages-based distributed algorithms we have developed a new refinement method. We begin with an initial algorithm, which contains actions, which describe common tasks of several agents. In each step we refine one of these actions to a net, which contains only such actions, which describe the tasks of individual agents. Thus, each step is a distribution of an undistributed action. The analysis of such refinement steps is accomplished with the help of a new refinement notation - the distributing refinement. Preservation of the partial order of the refined algorithm is important. This can be achieved by causalities of the actions of the agents in the local refinement net. Causalities in the local refinement net can be realized on the one hand at design directly by messages passing. On the other hand, at verification one can read the validity of causality in the local refinement net directly from the net. Therefore, this method is easy to use. The application of the method is demonstrated by several nontrivial examples in this thesis.

Submitted by Geyciane Santos (geyciane_thamires@hotmail.com) on 2015-06-22T15:27:44Z No. of bitstreams: 1 Tese- Éfren Lopes de Souza.pdf: 7677745 bytes, checksum: 8fe25c4dfc5ccdc0ef44afb8837bd0e0 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-06-24T19:29:19Z (GMT) No. of bitstreams: 1 Tese- Éfren Lopes de Souza.pdf: 7677745 bytes, checksum: 8fe25c4dfc5ccdc0ef44afb8837bd0e0 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-06-24T20:11:47Z (GMT) No. of bitstreams: 1 Tese- Éfren Lopes de Souza.pdf: 7677745 bytes, checksum: 8fe25c4dfc5ccdc0ef44afb8837bd0e0 (MD5) / Made available in DSpace on 2015-06-24T20:11:47Z (GMT). No. of bitstreams: 1 Tese- Éfren Lopes de Souza.pdf: 7677745 bytes, checksum: 8fe25c4dfc5ccdc0ef44afb8837bd0e0 (MD5) Previous issue date: 2014-03-28 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Target tracking in Wireless Sensor Networks (WSNs) is an application in which the nodes cooperate to estimate the position of one or more objects of interest. In this context, the contributions of this work are fourfold. First, a survey the state-of-the-art about target tracking algorithms, in which we identified three formulations of tracking problem, and we classified them according to their characteristics. Furthermore, we divided the target tracking process in components to make the general understanding easier. Second, we propose and evaluate the PRATIQUE algorithm for tracking animals in forests. In this case, the nodes are organized into a grid to make feasible the use of sensor nodes in this kind of area in such a way that each cell of the grid is a region that can be occupied by the target. The algorithm estimates the cell where the target is, and uses predictions and hybrid clustering to reduce the communication cost and ensure the tracking accuracy. The results of the simulations show that prediction errors are approximately one cell. The third contribution is the TATI algorithm, this algorithm guides a tracker to approach the target. The sensor network is organized into faces to make the cooperation among the nodes easier, and reduce the path between the tracker and the target. The results show that energy consumption is reduced by 15%, and the tracker stays about 10m closer to the target, compared to the baseline. The fourth contribution is a scheme for performing localization and tracking tasks simultaneously in such a way that errors of range-based localization algorithms are reduced. This algorithm takes advantage of the messages sent to track the target to filter the noise in the distance estimation, reducing localization errors while tracking. The results show that the localization errors can be reduced by up to 70%. / Rastreamento de alvos em Redes de Sensores Sem Fio (RSSFs) é um tipo de aplicação em que os nós cooperam para estimar a posição de um ou mais objetos de interesse. Nesse contexto, este trabalho possui quatro contribuições. A primeira contribuição é um levantamento bibliográfico do estado-da-arte, em que identificamos três diferentes formulações de rastreamento e as classificamos de acordo com suas características. Além disso, dividimos o processo de rastreamento em componentes para facilitar o entendimento geral. A segunda contribuição é a elaboração e avaliação do algoritmo PRATIQUE para rastrear animais em florestas. Nesse caso, os nós são organizados em grade para viabilizar a utilização dos nós sensores nesse tipo de área, de forma que cada célula da grade é uma região que pode ser ocupada pelo alvo. O algoritmo estima a célula em que o alvo está, e usa previsão e um esquema híbrido de agrupamento para reduzir o custo de comunicação e garantir a precisão do rastreamento. Os resultados das simulações mostram que os erros de previsão são de aproximadamente uma célula. A terceira contribuição é o algoritmo TATI, esse algoritmo guia um objeto que visa alcançar o alvo. A rede é estruturada em faces para facilitar a cooperação entre os nós e reduzir o caminho entre o objeto guiado e o alvo. Os resultados mostram que o consumo de energia é reduzido em 15% e o objeto guiado fica cerca de 10m mais próximo do alvo, se comparado com a abordagem relacionada. A quarta contribuição é um esquema para executar as tarefas de localização e rastreamento simultaneamente para reduzir os erros dos algoritmos de localização baseados em alcance. As mensagens enviadas para rastrear o alvo são aproveitadas para filtrar os ruídos presentes nas estimativas de distância, reduzindo o erro de localização enquanto o rastreamento ocorre. Os resultados mostram que os erros de localização podem ser reduzidos em até 70%.

The subject of this thesis is the development of localization algorithms for target localization in wireless sensor networks using received signal strength (RSS) measurements or Quantized RSS (QRSS) measurements. In chapter 3 of the thesis, target localization using RSS measurements is investigated. Many existing works on RSS localization assumes that the shadowing components are uncorrelated. However, here, shadowing is assumed to be spatially correlated. It can be shown that localization accuracy can be improved with the consideration of correlation between pairs of RSS measurements. By linearizing the corresponding Maximum Likelihood (ML) objective function, a weighted least squares (WLS) algorithm is formulated to obtain the target location. An iterative technique based on Newtons method is utilized to give a solution. Numerical simulations show that the proposed algorithms achieves better performance than existing algorithms with reasonable complexity. In chapter 4, target localization with an unknown path loss model parameter is investigated. Most published work estimates location and these parameters jointly using iterative methods with a good initialization of path loss exponent (PLE). To avoid finding an initialization, a global optimization algorithm, particle swarm optimization (PSO) is employed to optimize the ML objective function. By combining PSO with a consensus algorithm, the centralized estimation problem is extended to a distributed version so that can be implemented in distributed WSN. Although suboptimal, the distributed approach is very suitable for implementation in real sensor networks, as it is scalable, robust against changing of network topology and requires only local communication. Numerical simulations show that the accuracy of centralized PSO can attain the Cramer Rao Lower Bound (CRLB). Also, as expected, there is some degradation in performance of the distributed PSO with respect to the centralized PSO. In chapter 5, a distributed gradient algorithm for RSS based target localization using only quantized data is proposed. The ML of the Quantized RSS is derived and PSO is used to provide an initial estimate for the gradient algorithm. A practical quantization threshold designer is presented for RSS data. To derive a distributed algorithm using only the quantized signal, the local estimate at each node is also quantized. The RSS measurements and the local estimate at each sensor node are quantized in different ways. By using a quantization elimination scheme, a quantized distributed gradient method is proposed. In the distributed algorithm, the quantization noise in the local estimate is gradually eliminated with each iteration. Simulations show that the performance of the centralized algorithm can reach the CRLB. The proposed distributed algorithm using a small number of bits can achieve the performance of the distributed gradient algorithm using unquantized data.

In this thesis we look at mobile agent solutions to black hole search and related problems. Mobile agents are computational entities that are autonomous, mobile, and can interact with their environment and each other. The black hole search problem is for a team of these agents to work together to map or explore a graph-like network environment where some elements of the network are dangerous to the agents. Most research into black hole search has focussed on finding a single dangerous node: a black hole. We look at the problem of finding multiple black holes and, in the case of dangerous graph exploration, multiple black links as well. We look at the dangerous graph exploration problem in the network model. The network model is based on a normal static computer network modelled as a simple graph. We give an optimal solution to the dangerous graph exploration problem using agents that start scattered on nodes throughout the network. We then make the problem more difficult by allowing an adversary to delete links during the execution of the algorithm and provide a solution using scattered agents. In the last decade or two, types of networks have emerged, such as ad hoc wireless networks, that are by their nature dynamic. These networks change quickly over time and can make distributed computations difficult. We look at black hole search in one type of dynamic network described by the subway model, which we base on urban subway systems. The model allows us to look at the cost of opportunistic movement by requiring the agents to move using carriers that follow routes among the network's sites, some of which are black holes. We show that there are basic limitations on any solution to black hole search in the subway model and prove lower bounds on any solution's complexity. We then provide two optimal solutions that differ in the agents' starting locations and how they communicate with one another. Our results provide a small window into the cost of deterministic distributed computing in networks that have dynamic elements, but which are not fully random.

We developed a diffusive load balancing scheme that equalizes the available capacities of nodes in a peer-to-peer (P2P) system. These nodes may have different resource capacities, geographic locations, or availabilities (i.e., length of time being part of the peer-to-peer system). The services on these nodes may have different service times and arrival rates of requests. Using the diffusive scheme, the system is able to maintain similar response times for its services. Our scheme is a modification of the diffusive load balancing algorithms proposed for parallel computing systems. This scheme is able to handle services with heterogeneous resource requirements and P2P nodes with heterogeneous capacities. We also adapted the diffusive scheme to clustered peer-to-peer system, where a load balancing operation may move services or nodes between clusters. After a literature survey of this field, this thesis investigates the following issues using analytical reasoning and extensive simulation studies. The load balancing operations equalize the available capacities of the nodes in a neighborhood to their averages. As a result, the available capacities of all nodes in the P2P system converge to a global average. We found that this convergence is faster when the scheme uses neighborhoods defined by the structure of the structured P2P overlay network rather than using randomly selected neighbors. For a system with churn (i.e. nodes joining and leaving), the load balancing operations maintain the standard deviation of the available capacities of nodes within a bound. This bound depends on the amount of churn and the frequency of load balancing operations, as well as on the capacities of the nodes. However, the sizes of the services have little impact on this bound. In a clustered peer-to-peer system, the size of the bound largely depends on the average cluster size. When nodes are moved among clusters for load balancing, the numbers of cluster splits and merges are reduced. This may reduce the maintenance cost of the overlay network.

In this thesis we look at mobile agent solutions to black hole search and related problems. Mobile agents are computational entities that are autonomous, mobile, and can interact with their environment and each other. The black hole search problem is for a team of these agents to work together to map or explore a graph-like network environment where some elements of the network are dangerous to the agents. Most research into black hole search has focussed on finding a single dangerous node: a black hole. We look at the problem of finding multiple black holes and, in the case of dangerous graph exploration, multiple black links as well. We look at the dangerous graph exploration problem in the network model. The network model is based on a normal static computer network modelled as a simple graph. We give an optimal solution to the dangerous graph exploration problem using agents that start scattered on nodes throughout the network. We then make the problem more difficult by allowing an adversary to delete links during the execution of the algorithm and provide a solution using scattered agents. In the last decade or two, types of networks have emerged, such as ad hoc wireless networks, that are by their nature dynamic. These networks change quickly over time and can make distributed computations difficult. We look at black hole search in one type of dynamic network described by the subway model, which we base on urban subway systems. The model allows us to look at the cost of opportunistic movement by requiring the agents to move using carriers that follow routes among the network's sites, some of which are black holes. We show that there are basic limitations on any solution to black hole search in the subway model and prove lower bounds on any solution's complexity. We then provide two optimal solutions that differ in the agents' starting locations and how they communicate with one another. Our results provide a small window into the cost of deterministic distributed computing in networks that have dynamic elements, but which are not fully random.