Reconfiguration of Hamiltonian cycles and paths in grid graphs
| dc.contributor.author | Nishat, Rahnuma Islam | |
| dc.contributor.supervisor | Whitesides, Sue H. | |
| dc.date.accessioned | 2020-05-11T19:23:36Z | |
| dc.date.available | 2020-05-11T19:23:36Z | |
| dc.date.copyright | 2020 | en_US |
| dc.date.issued | 2020-05-11 | |
| dc.degree.department | Department of Computer Science | |
| dc.degree.level | Doctor of Philosophy Ph.D. | en_US |
| dc.description.abstract | A grid graph is a finite embedded subgraph of the infinite integer grid. A solid grid graph is a grid graph without holes, i.e., each bounded face of the graph is a unit square. The reconfiguration problem for Hamiltonian cycle or path in a sold grid graph G asks the following question: given two Hamiltonian cycles (or paths) of G, can we transform one cycle (or path) to the other using some "operation" such that we get a Hamiltonian cycle (or path) of G in the intermediate steps (i.e., after each application of the operation)? In this thesis, we investigate reconfiguration problems for Hamiltonian cycles and paths in the context of two types of solid graphs: rectangular grid graphs, which have a rectangular outer boundary, and L- shaped grid graphs, which have a single reflex corner on the outer boundary, under three operations we define, flip, transpose and switch, that are local in the grid. Reconfiguration of Hamiltonian cycles and paths in embedded grid graphs has potential applications in path planning, robot navigation, minimizing turn costs in milling problems, minimizing angle costs in TSP, additive manufacturing and 3D printing, and in polymer science. In this thesis, we introduce a complexity measure called bend complexity for Hamiltonian paths and cycles in grid graphs, and using those measures we measure complexity of a grid graph G and give upper and lower bounds on the maximum bend complexity of an mxn grid graph. We define three local operations, flip, transpose and switch, where local means that the operations are applied on vertices that are close in the grid graph but may not be close on the path or cycle. We show that any Hamiltonian cycle or path can be reconfigured to any other Hamiltonian cycle or path in an mxn rectangular grid graph, where m <= 4, using O(|G|) flips and transposes, regardless of the bend complexities of the two cycles. We give algorithms to reconfigure 1-complex Hamiltonian cycles in a rectangular or L-shaped grid graph G using O(|G|) flips and transposes, where the intermediate steps are also 1-complex Hamiltonian cycles. Finally, we establish the structure of 1-complex Hamiltonian paths between diagonally opposite corners s and t of a rectangular grid graph, and then provide a strategy, based on work in progress, for designing an algorithm to reconfigure between any two 1-complex s, t Hamiltonian paths using switch operations. | en_US |
| dc.description.scholarlevel | Graduate | en_US |
| dc.identifier.uri | http://hdl.handle.net/1828/11746 | |
| dc.language | English | eng |
| dc.language.iso | en | en_US |
| dc.rights | Available to the World Wide Web | en_US |
| dc.subject | Hamiltonian cycle | en_US |
| dc.subject | Hamiltonian path | en_US |
| dc.subject | Grid graph | en_US |
| dc.subject | Reconfiguration | en_US |
| dc.subject | Local operation | en_US |
| dc.subject | Algorithm | en_US |
| dc.title | Reconfiguration of Hamiltonian cycles and paths in grid graphs | en_US |
| dc.type | Thesis | en_US |