Computation-Aware Data Aggregation

Data aggregation is a fundamental primitive in distributed computing wherein a network computes a function of every nodes\u27 input. However, while compute time is non-negligible in modern systems, standard models of distributed computing do not take compute time into account. Rather, most distributed models of computation only explicitly consider communication time. In this paper, we introduce a model of distributed computation that considers both computation and communication so as to give a theoretical treatment of data aggregation. We study both the structure of and how to compute the fastest data aggregation schedule in this model. As our first result, we give a polynomial-time algorithm that computes the optimal schedule when the input network is a complete graph. Moreover, since one may want to aggregate data over a pre-existing network, we also study data aggregation scheduling on arbitrary graphs. We demonstrate that this problem on arbitrary graphs is hard to approximate within a multiplicative 1.5 factor. Finally, we give an O(log n ? log(OPT/t_m))-approximation algorithm for this problem on arbitrary graphs, where n is the number of nodes and OPT is the length of the optimal schedule

Problems related to broadcasting in graphs

The data transmission delays become the bottleneck on modern high speed interconnection networks utilized by high performance computing or enterprise data centers. This motivates the study directed towards finding more efficient interconnection topologies as well as more efficient algorithms for information exchange between the nodes of the given network. Broadcasting is the process of distributing a message from a node, called the originator, to all other nodes of a communication network. Broadcasting is used as a basic communication primitive by many higher level network operations, which involve a set of nodes in distributed systems. Therefore, it is one the most important operations, which can determine the total efficiency of a given distributed system. We study interconnection networks via modeling them as graphs. The results described in this work can be used for efficient message routing algorithms in switch based interconnection networks as well as in the choice of the interconnection topologies of such networks. This thesis is divided into six chapters. Chapter 1 gives a general introduction to the research area and literature overview. Chapter 2 studies the family of graphs for which the broadcast time is equal to the diameter. Chapter 3 studies the routing and broadcasting problem in the Knodel graph. Chapter 4 studies the possible vertex degrees and the possible connections between vertices of different degrees in a broadcast graph. Using this, a new lower bound is obtained on broadcast function. Chapter 5 presents some miscellaneous results. Chapter 6 summarizes the thesis

Near-Optimal Schedules for Simultaneous Multicasts

We study the store-and-forward packet routing problem for simultaneous multicasts, in which multiple packets have to be forwarded along given trees as fast as possible. This is a natural generalization of the seminal work of Leighton, Maggs and Rao, which solved this problem for unicasts, i.e. the case where all trees are paths. They showed the existence of asymptotically optimal O(C +D)-length schedules, where the congestion C is the maximum number of packets sent over an edge and the dilation D is the maximum depth of a tree. This improves over the trivial O(CD) length schedules. We prove a lower bound for multicasts, which shows that there do not always exist schedules of non-trivial length, o(CD). On the positive side, we construct O(C + D + log2 n)-length schedules in any n-node network. These schedules are near-optimal, since our lower bound shows that this length cannot be improved to O(C + D) + o(log n).ISSN:1868-896

Broadcasting in highly connected graphs

Throughout history, spreading information has been an important task. With computer networks expanding, fast and reliable dissemination of messages became a problem of interest for computer scientists. Broadcasting is one category of information dissemination that transmits a message from a single originator to all members of the network. In the past five decades the problem has been studied by many researchers and all have come to demonstrate that despite its easy definition, the problem of broadcasting does not have trivial properties and symmetries. For general graphs, and even for some very restricted classes of graphs, the question of finding the broadcast time and scheme remains NP-hard. This work uses graph theoretical concepts to explore mathematical bounds on how fast information can be broadcast in a network. The connectivity of a graph is a measure to assess how separable the graph is, or in other words how many machines in a network will have to fail to disrupt communication between all machines in the network. We initiate the study of finding upper bounds on broadcast time b(G) in highly connected graphs. In particular, we give upper bounds on b(G) for k-connected graphs and graphs with a large minimum degree. We explore 2-connected (biconnected) graphs and broadcasting in them. Using Whitney's open ear decomposition in an inductive proof we propose broadcast schemes that achieve an upper bound of ceil(n/2) for classical broadcasting as well as similar bounds for multiple originators. Exploring further, we use a matching-based approach to prove an upper bound of ceil(log(k)) + ceil(n/k) - 1 for all k-connected graphs. For many infinite families of graphs, these bounds are tight. Discussion of broadcasting in highly connected graphs leads to an exploration of dependence between the minimum degree in the graph and the broadcast time of the latter. By using similar techniques and arguments we show that if all vertices of the graph are neighboring linear numbers of vertices, then information dissemination in the graph can be achieved in ceil(log(n)) + C time. To the best of our knowledge, the bounds presented in our work are a novelty. Methods and questions proposed in this thesis open new pathways for research in broadcasting

An Introduction to Temporal Graphs: An Algorithmic Perspective

A \emph{temporal graph} is, informally speaking, a graph that changes with time. When time is discrete and only the relationships between the participating entities may change and not the entities themselves, a temporal graph may be viewed as a sequence $G_1,G_2\ldots,G_l$ of static graphs over the same (static) set of nodes $V$. Though static graphs have been extensively studied, for their temporal generalization we are still far from having a concrete set of structural and algorithmic principles. Recent research shows that many graph properties and problems become radically different and usually substantially more difficult when an extra time dimension in added to them. Moreover, there is already a rich and rapidly growing set of modern systems and applications that can be naturally modeled and studied via temporal graphs. This, further motivates the need for the development of a temporal extension of graph theory. We survey here recent results on temporal graphs and temporal graph problems that have appeared in the Computer Science community

New Heuristic for Message Broadcasting in Arbitrary Networks

Efficient information dissemination in interconnection networks is a key research area because of the major role it plays in the modern interconnected world. A vast number of topics ranging from distributed computing to Internet communication rely on efficient information dissemination. Broadcasting is one of the information dissemination primitives. The minimum broadcast time problem in arbitrary networks has been examined since the 1970s. Finding an optimal broadcasting scheme for any originator in an arbitrary network has been proved to be an NP-Hard problem. In the current thesis, a new heuristic that generates broadcast schemes in arbitrary networks is presented. The heuristic has O(|E|log|V|) time complexity, where V is the set of nodes and E is the set of the links of the network. Computer simulations in some commonly used topologies and network models show that compared to the existing heuristics the new heuristic shows better performance in some network models, and comparable performance in other network models, while having a low complexity similar to the best existing heuristics. Another advantage of the new heuristic is that approximately one half of the vertices receive the message via a shortest path from the broadcast originator, while the rest of the vertices receive the message via a path at most three hops longer

Approximation Algorithms for Broadcasting in Simple Graphs with Intersecting Cycles

Broadcasting is an information dissemination problem in a connected network in which one node, called the originator, must distribute a message to all other nodes by placing a series of calls along the communication lines of the network. Every time the informed nodes aid the originator in distributing the message. Finding the minimum broadcast time of any vertex in an arbitrary graph is NP-Complete. The problem remains NP-Complete even for planar graphs of degree 3 and for a graph whose vertex set can be partitioned into a clique and an independent set. The best theoretical upper bound gives logarithmic approximation. It has been shown that the broadcasting problem is NP-Hard to approximate within a factor of 3-ɛ. The polynomial time solvability is shown only for tree-like graphs; trees, unicyclic graphs, tree of cycles, necklace graphs and some graphs where the underlying graph is a clique; such as fully connected trees and tree of cliques. In this thesis we study the broadcast problem in different classes of graphs where cycles intersect in at least one vertex. First we consider broadcasting in a simple graph where several cycles have common paths and two intersecting vertices, called a k-path graph. We present a constant approximation algorithm to find the broadcast time of an arbitrary k-path graph. We also study the broadcast problem in a simple cactus graph called k-cycle graph where several cycles of arbitrary lengths are connected by a central vertex on one end. We design a constant approximation algorithm to find the broadcast time of an arbitrary k-cycle graph. Next we study the broadcast problem in a hypercube of trees for which we present a 2-approximation algorithm for any originator. We provide a linear algorithm to find the broadcast time in hypercube of trees with one tree. We extend the result for any arbitrary graph whose nodes contain trees and design a linear time constant approximation algorithm where the broadcast scheme in the arbitrary graph is already known. In Chapter 6 we study broadcasting in Harary graph for which we present an additive approximation which gives 2-approximation in the worst case to find the broadcast time in an arbitrary Harary graph. Next for even values of n, we introduce a new graph, called modified-Harary graph and present a 1-additive approximation algorithm to find the broadcast time. We also show that a modified-Harary graph is a broadcast graph when k is logarithmic of n. Finally we consider a diameter broadcast problem where we obtain a lower bound on the broadcast time of the graph which has at least (d+k-1 choose d) + 1 vertices that are at a distance d from the originator, where k >= 1