This chapter provides various definitions of generalized Gray codes and brings out the intrinsic difference between the Gray codes with odd and even base. The widespread availability of distributed memory architectures based on the binary hypercube topology, there is a growing interest in the portability of algorithms developed for architectures based on other topologies such as linear arrays, rings, multidimensional grids, trees, to hypercube based architectures. While these generalized Gray codes have immediate applications to the embedding of graphs in a base-b cube, the analysis of their properties holds independent interest as well. However, there is a close relation between the cyclic Gray codes and a class of Gray codes called reflected Gray codes.

Consider a private network of geographically dispersed computers with fast and high capacity connections, and an Internet application session, such as a massive multiplayer online game, with a server and a set of clients. We refer to the former as a service overlay network (SON), and assume that it could be connected to the Internet. The problem is to decide how to configure and utilize the SON in support of this application, such that the clients׳ speed of communication with the server is within given communication performance requirements. We provide an Integer Programming formulation of this problem, and prove that it is NP-Hard. In an attempt to solve the problem within strict computational time requirements of actual applications, we develop a solution framework based on partitioning and enumerating the solution space into smaller subproblems, one or more of which contains an optimal solution. In this framework, we develop and test an optimal seeking exact, and a fast polynomial time heuristic algorithm with success. The exact algorithm sets optimally solvable sizes of the subject problem, whereas the heuristic algorithm sets the size of solvable instances in a real application.

Multimedia content now contribute to a huge amount of the Internet traffic due to the popularity and availability of anytime anywhere Internet connection. Unlike the circuit-switched telephone network – in which necessary resources are reserved for communication between two parties at the time the connection is established, a packet-switched network, like the Internet, only guarantees the reachability when the connection between two parties is established. In other words, the end-to-end delay and available bandwidth between two hosts depend on the amount of traffic on the network. The communication paths between the participating hosts are also determined by the routing policies and hence are not under control of the participating hosts. Hence how to improve the performance of delivering multimedia content on the Internet has become an interesting research topic.In this dissertation, we consider the problem of delivering multimedia contents using multicast wherein a group of participants are participating in the same communication session. We assume the networks are flexible such that the end hosts can specify the communication paths. A few examples of this type of networks are overlay networks and IPv6 network with source routing support. This problem is addressed from both routing and network traffic perspectives. First, we assume a two-layer approach which includes a well-provisioned service overlay network and the regular Internet. The participants in the multimedia group communication can take the advantage of the service overlay network by connecting to the nodes in the service overlay network through the Internet. We consider two major assignment problems – Server and Client Assignment Problem (SCAP, Client-Server model) and Client Assignment Problem (CAP, Peer-to-Peer model) as well as several variants of these problems. These problems are NP -hard and we have developed polynomial-time heuristic algorithms to assign the participants to appropriate service nodes such that some real-time constraint(s) are satisfied and the number of service nodes involved are minimal. Integer programming (IP) models for solving these problems are also developed for performance evaluation purpose. Empirical results show that the solution quality of the proposed algorithms compares favorably with the optimal ones obtained from the execution of IP models, while keeping the execution times significantly low.We have also considered the Multi-stream Multi-source Multicast Routing Problem ( MMMRP). Given a network and a set of multicast sessions, each with one or more sources and multiple destinations. The goal of MMMRP is to determine multiple multicast tree for these multicast sessions on the given network in such a way that the overall residual bandwidth on the links that are shared among the trees is maximized. We prove that MMMRP is NP -hard and apart from providing an IP formulation, we have also provided a heuristic algorithm MMForests which runs in polynomial-time. We compared and contrasted the performance of MMMRP with known algorithms for the multicast tree packing problem. Our exhaustive empirical evaluations show that our heuristic has a very low execution-time while achieving the optimal residual bandwidth. In addition, our heuristic is very scalable as it is able to produce results for networks with thousands of nodes, unlike the other ones which are based on Steiner tree heuristics.

Multicasting is an efficient way to deliver multimedia content (streaming, for instance) to different locations in the network. While end-to-end real-time constraints are important for interactive applications, sustained availability of bandwidth is more important to the destinations for multimedia streaming. In this research, we address the problem of multi-stream multi-source multicast routing problem (MMMRP) where each data stream could have multiple sources that will serve it and each source can serve multiple data streams in a sustained manner. The goal of MMMRP is to construct a routing forest for each of the data streams and the destinations while maximizing the residual bandwidth. The residual bandwidth is the available bandwidth after all destinations have been served with their desired streams. Our problem is shown to be NP-hard and we provide an Integer Programming formulation together with an efficient heuristic algorithm (MMForests) based on widest-path algorithm. Our empirical evaluations show that our algorithm MMForests can construct the multicast routing trees both quickly and keeping the residual bandwidth close to the optimal.

Recent advances in multimedia software and hardware technologies and the availability of high-speed Internet service have been instrumental for growth in the online gaming industry. Multiple servers distributed across the network are commonly used to provide the desired quality-of-service (QoS) for the network game in order to achieve a higher quality-of-experience (QoE) to the players (clients). Each player in this distributed multi-player gaming environment connects to a particular server and it distributes each of the actions to all other players through the servers they are connected to. We imagine the server network to be an overlay network, wherein the latency on a link between two servers is the latency of the Internet path connecting them. We assume that we are given an overlay network of servers with link latencies and a set of players each with a different latency to each of the servers. Now our goal is to develop algorithms that perform the following actions in such a way that delay related QoS constraints are satisfied: (a) choose a subnetwork of the server network (server network selection) and (b) assign each player to a server in the subnetwork (client-assignment). More specifically, the QoS constraints that we address in this paper are a bound on the maximum delay in propagating a player's move to all other players (delay bound) and a bound on the maximum difference in the arrival times of a player's move at all other players (delay-variation bound). We have provided polynomial-time heuristics to determine a minimal cardinality server network and the corresponding client-assignment that satisfy both delay bound and that minimize delay-variation, if such a solution exists. We have considered cases in which the server network follows two communication models: client-server (CS) and peer-to-peer (P2P). Our extensive empirical studies indicate that our heuristic uses significantly less run-time in achieving the tightest delay variation for a given end-- - to-end delay bound while choosing a minimal number of servers.

This paper answers the following question: How do the risk characteristics of serial channels cumulate when an entity traverses through them? The findings can be applied to assess the risk a commercial container would face as it traverses through a number of transportation channels, including, for example a rogue state. The risk associated with each channel is characterized by a log-normal probability density function. The cumulative distribution function is utilized to represent the probability of a potential loss bounded to a pre-specified limit on an end-to-end basis. The results obtained can be used to guide the level of insurance a container should carry in order to cover the potential loss by providing an upper bound to the cumulative loss with an acceptable probability. The loss is largely attributed to natural or man-made disasters, e.g., acts of terrorism.

Improving latency is the key to a successful online game-playing experience. With the use of multiple servers along with a well-provisioned network it is possible to reduce the latency. Given a network of servers, game clients, and a desired delay bound, we have designed algorithms to determine the subnetwork of servers whose cardinality is minimal. We have considered the cases wherein the subnetwork architecture is a client-server and a peer-to-peer. We have also provided exhaustive empirical evaluations of our algorithms and compared their performance with the optimum. Experimental results show that our polynomial-time algorithms could find good solutions quickly.

In this paper we analyze a class of n-person super-modular games that arise in the context of interdependent security analysis. More specifically, we quantify the number and the distribution of Nash equilibria in pure strategies and their impact on the tipping set.

In Chapters 22–25 we have discussed the solution to the off-line, 4DVAR problem of assimilating a given set of observations in deterministic/dynamic models using the classical least squares (Part II) method. In this framework, the adjoint method facilitates the computation of the gradient of the least squares objective function, which when used in conjunction with the minimization methods described in Part III, leads to the optimal initial conditions for the dynamic model. Even in the ideal case of a perfect dynamic model (error free model), the computed values of the optimal initial condition are noisy in response to erroneous observations. The deterministic approach in Chapter 22–25 are predicated on the assumption that the statistical properties of the noise corrupting the observations are not known a priori. The question is: if we are given additional information, say the second-order properties (mean and covariance) of the noisy observations, how can we use this information to derive the second-order properties of the optimal initial conditions? This can only be achieved by reliance on the statistical least squares method described in Chapter 14.

In Chapters 5 and 7 the least squares problem – minimization of the residual norm, f ( x ) = ∥ r ( x )∥ where r ( x ) = ( z − Hx ) in (5.1.11) and (7.1.3) with respect to the state variable x is formulated. There are essentially two mathematically equivalent approaches to this minimization. In the first, compute the gradient ∇ f ( x ) and obtain (the minimizer) x by solving ∇ f ( x ) = 0 . We then check if the Hessian ∇ 2 f ( x ) is positive definite to guarantee that x is indeed a local minimum. In the linear least squares problem in Chapter 5, f ( x ) is a quadratic function x and hence ∇ f ( x ) = 0 leads to the solution of a linear system of the type Ax = b with A a symmetric and positive definite matrix (refer to (5.1.17)) which can be solved by the methods described in Chapter 9. In the nonlinear least squares problem, f ( x ) may be highly nonlinear (far beyond the quadratic nonlinearity). In this case, we can compute x by solving a nonlinear algebraic system given by ∇ f ( x ) = 0 , and then checking for the positive definiteness of the Hessian ∇ 2 f ( x ). Alternatively, we can approximate f ( x ) locally around a current operating point, say, x c by a quadratic form Q ( y ) (using either the first-order or the second-order method described in Chapter 7) where y = ( x – x c ).

In the opening chapter of Part VI we considered a very special dynamical model for pedagogical reasons. Having gained some working knowledge of the methodology for solving the inverse problem using the Lagrangian framework, we now consider the general linear dynamical system. Once we understand the underpinnings of this methodology in the context of a general linear dynamical system, its applicability to a wide variety of linear models is possible.