ORCID ID
0000-0001-6595-4399
Date Awarded
2018
Document Type
Dissertation
Degree Name
Doctor of Philosophy (Ph.D.)
Department
Computer Science
Advisor
Weizhen Mao
Committee Member
Rex K Kincaid
Committee Member
Robert M Lewis
Committee Member
andreas Stathopoulos
Committee Member
Gexin Yu
Abstract
The detection and location of issues in a network is a common problem encompassing a wide variety of research areas. Location-detection problems have been studied for wireless sensor networks and environmental monitoring, microprocessor fault detection, public utility contamination, and finding intruders in buildings. Modeling these systems as a graph, we want to find the smallest subset of nodes that, when sensors are placed at those locations, can detect and locate any anomalies that arise. One type of set that solves this problem is the open locating-dominating set (OLD-set), a set of nodes that forms a unique and nonempty neighborhood with every node in the graph. For this work, we begin with a study of OLD-sets in circulant graphs. Circulant graphs are a group of regular cyclic graphs that are often used in massively parallel systems. We prove the optimal OLD-set size for two circulant graphs using two proof techniques: the discharging method and Hall's Theorem. Next we introduce the mixed-weight open locating-dominating set (mixed-weight OLD-set), an extension of the OLD-set. The mixed-weight OLD-set allows nodes in the graph to have different weights, representing systems that use sensors of varying strengths. This is a novel approach to the study of location-detection problems. We show that the decision problem for the minimum mixed-weight OLD-set, for any weights up to positive integer d, is NP-complete. We find the size of mixed-weight OLD-sets in paths and cycles for weights 1 and 2. We consider mixed-weight OLD-sets in random graphs by providing probabilistic bounds on the size of the mixed-weight OLD-set and use simulation to reinforce the theoretical results. Finally, we build and study an integer linear program to solve for mixed-weight OLD-sets and use greedy algorithms to generate mixed-weight OLD-set estimates in random geometric graphs. We also extend our results for mixed-weight OLD-sets in random graphs to random geometric graphs by estimating the probabilistic upper bound for the size of the set.
DOI
http://dx.doi.org/10.21220/s2-4479-zx78
Rights
© The Author
Recommended Citation
Givens, Robin M., "Mixed-Weight Open Locating-Dominating Sets" (2018). Dissertations, Theses, and Masters Projects. William & Mary. Paper 1550153794.
http://dx.doi.org/10.21220/s2-4479-zx78