Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph

Data-sensitive metrics adapt distances locally based the density of data points with the goal of aligning distances and some notion of similarity. In this paper, we give the first exact algorithm for computing a data-sensitive metric called the nearest neighbor metric. In fact, we prove the surprising result that a previously published $3$-approximation is an exact algorithm. The nearest neighbor metric can be viewed as a special case of a density-based distance used in machine learning, or it can be seen as an example of a manifold metric. Previous computational research on such metrics despaired of computing exact distances on account of the apparent difficulty of minimizing over all continuous paths between a pair of points. We leverage the exact computation of the nearest neighbor metric to compute sparse spanners and persistent homology. We also explore the behavior of the metric built from point sets drawn from an underlying distribution and consider the more general case of inputs that are finite collections of path-connected compact sets. The main results connect several classical theories such as the conformal change of Riemannian metrics, the theory of positive definite functions of Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop novel proof techniques based on the combination of screw functions and Lipschitz extensions that may be of independent interest.Comment: 15 page

Constant memory routing in quasi-planar and quasi-polyhedral graphs

AbstractWe address the problem of online route discovery for a class of graphs that can be embedded either in two or in three-dimensional space. In two dimensions we propose the class of quasi-planar graphs and in three dimensions the class of quasi-polyhedral graphs. In the former case such graphs are geometrically embedded in R2 and have an underlying backbone that is planar with convex faces; however within each face arbitrary edges (with arbitrary crossings) are allowed. In the latter case, these graphs are geometrically embedded in R3 and consist of a backbone of convex polyhedra and arbitrary edges within each polyhedron. In both cases we provide a routing algorithm that guarantees delivery. Our algorithms need only “remember” the source and destination nodes and one (respectively, two) reference nodes used to store information about the underlying face (respectively, polyhedron) currently being traversed. The existence of the backbone is used only in proofs of correctness of the routing algorithm; the particular choice is irrelevant and does not affect the behaviour of the algorithm

Distributed optimization algorithms for multihop wireless networks

Recent technological advances in low-cost computing and communication hardware design have led to the feasibility of large-scale deployments of wireless ad hoc and sensor networks. Due to their wireless and decentralized nature, multihop wireless networks are attractive for a variety of applications. However, these properties also pose significant challenges to their developers and therefore require new types of algorithms. In cases where traditional wired networks usually rely on some kind of centralized entity, in multihop wireless networks nodes have to cooperate in a distributed and self-organizing manner. Additional side constraints, such as energy consumption, have to be taken into account as well. This thesis addresses practical problems from the domain of multihop wireless networks and investigates the application of mathematically justified distributed algorithms for solving them. Algorithms that are based on a mathematical model of an underlying optimization problem support a clear understanding of the assumptions and restrictions that are necessary in order to apply the algorithm to the problem at hand. Yet, the algorithms proposed in this thesis are simple enough to be formulated as a set of rules for each node to cooperate with other nodes in the network in computing optimal or approximate solutions. Nodes communicate with their neighbors by sending messages via wireless transmissions. Neither the size nor the number of messages grows rapidly with the size of the network. The thesis represents a step towards a unified understanding of the application of distributed optimization algorithms to problems from the domain of multihop wireless networks. The problems considered serve as examples for related problems and demonstrate the design methodology of obtaining distributed algorithms from mathematical optimization methods

Distributed services for mobile ad hoc networks

A mobile ad hoc network consists of certain nodes that communicate only through wireless medium and can move arbitrarily. The key feature of a mobile ad hoc network is the mobility of the nodes. Because of the mobility, communication links form and disappear as nodes come into and go out of each other's communica- tion range. Mobile ad hoc networks are particularly useful in situations like disaster recovery and search, military operations, etc. Research on mobile ad hoc networks has drawn a huge amount of attention recently. The main challenges for mobile ad hoc networks are the sparse resources and frequent mobility. Most of the research work has been focused on the MAC and routing layer. In this work, we focus on distributed services for mobile ad hoc networks. These services will provide some fundamental functions in developing various applications for mobile ad hoc networks. In particular, we focus on the clock synchronization, connected dominating set, and k-mutual exclusion problems in mobile ad hoc networks

ALGORITHMS FOR FAULT TOLERANCE IN DISTRIBUTED SYSTEMS AND ROUTING IN AD HOC NETWORKS

Checkpointing and rollback recovery are well-known techniques for coping with failures in distributed systems. Future generation Supercomputers will be message passing distributed systems consisting of millions of processors. As the number of processors grow, failure rate also grows. Thus, designing efficient checkpointing and recovery algorithms for coping with failures in such large systems is important for these systems to be fully utilized. We presented a novel communication-induced checkpointing algorithm which helps in reducing contention for accessing stable storage to store checkpoints. Under our algorithm, a process involved in a distributed computation can independently initiate consistent global checkpointing by saving its current state, called a tentative checkpoint. Other processes involved in the computation come to know about the consistent global checkpoint initiation through information piggy-backed with the application messages or limited control messages if necessary. When a process comes to know about a new consistent global checkpoint initiation, it takes a tentative checkpoint after processing the message. The tentative checkpoints taken can be flushed to stable storage when there is no contention for accessing stable storage. The tentative checkpoints together with the message logs stored in the stable storage form a consistent global checkpoint. Ad hoc networks consist of a set of nodes that can form a network for communication with each other without the aid of any infrastructure or human intervention. Nodes are energy-constrained and hence routing algorithm designed for these networks should take this into consideration. We proposed two routing protocols for mobile ad hoc networks which prevent nodes from broadcasting route requests unnecessarily during the route discovery phase and hence conserve energy and prevent contention in the network. One is called Triangle Based Routing (TBR) protocol. The other routing protocol we designed is called Routing Protocol with Selective Forwarding (RPSF). Both of the routing protocols greatly reduce the number of control packets which are needed to establish routes between pairs of source nodes and destination nodes. As a result, they reduce the energy consumed for route discovery. Moreover, these protocols reduce congestion and collision of packets due to limited number of nodes retransmitting the route requests

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum