k-edge Connected Components in Graphs
| dc.contributor.author | Mousavi, Azadehsadat | |
| dc.contributor.supervisor | Srinivasan, Venkatesh | |
| dc.contributor.supervisor | Thomo, Alex | |
| dc.date.accessioned | 2024-05-13T22:33:44Z | |
| dc.date.available | 2024-05-13T22:33:44Z | |
| dc.date.issued | 2024 | |
| dc.degree.department | Department of Computer Science | |
| dc.degree.level | Master of Science MSc | |
| dc.description.abstract | A k-edge-connected component (KECC) is a group of vertices in a graph G(V, E) where every pair of vertices has at least k edge-disjoint paths between them. This research delves into the detection and analysis of KECCs within graphs, presenting algorithms and thoroughly evaluating their performance. The study introduces algorithms tailored to find KECCs for a given value of k. One such algorithm, known as the fix-k algorithm, exhibits a time complexity of O(V^2.E^2 ). By leveraging various maximum flow algorithms and exploiting properties of k-edge connectivity, we optimize this algorithm to perform E times faster, resulting in a new approach with time complexity of O(V^2.E). Further enhancements are achieved through randomization, leading to an algorithm with an improved time complexity of O(V^3.ln(V)^O(1)). Additionally, we identify another algorithm with a time complexity of O(V.E^{1+o(1)} ). Moreover, algorithms are proposed to find all KECCs in a graph G(V, E) for all values of 1 ≤ k ≤ V . It is demonstrated that for any graph G(V, E), there exists a threshold value h, where for all k ≥ h, KECCs are singletons. An efficient algorithm is devised to determine this h in O(log(V )) time, which significantly impacts the all-k problem, allowing us to focus solely on finding KECCs for 1 ≤ k < h. This approach yields an algorithm with a time complexity of O(h ∗ O(Fix-k algorithm)) to find all KECCs. Additionally, another algorithm with a time complexity of O(V^2.E^2) is presented, which can be further optimized to achieve time complexities of O(V^2.E), O(V^3.ln(V )^O(1)), and O(V.E^{1+o(1)}). The insights gained from meticulous implementation and experimentation shed light on the behavior of these algorithms across various graph structures and scenarios. | |
| dc.description.scholarlevel | Graduate | |
| dc.identifier.uri | https://hdl.handle.net/1828/16512 | |
| dc.language.iso | en | |
| dc.subject | k-edge-connected components | |
| dc.subject | KECC | |
| dc.subject | graph algorithms | |
| dc.subject | edge connectivity | |
| dc.subject | maximum flow | |
| dc.subject | minimum cut | |
| dc.title | k-edge Connected Components in Graphs | |
| dc.type | project |