Grafa diametra un rādiusa aproksimēšana
Date
2006
Authors
Dudelis, Mārtiņš
Journal Title
Journal ISSN
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Publisher
Latvijas Universitāte
Abstract
Mūsdienās pieaugot apstrādājamo datu apjomam, pat mazas pakāpes polinomiāli algoritmi var izrādīties nelietojami lielu datu apstrādei. Šādās situācijas pat nebūtu vajadzīgs optimālais risinājums, bet pietiktu ar pietiekami tuvu tam. Tāda problēma ir arī grafa diametra un rādiusa atrašana, kurus izmanto grafu klasterizēšai, epidēmiju izplatības izpētē, sociālajos tīklos. Lai arī grafa diametra un rādiusa atrašanas algoritma sarežģītība ir O(nm), tas lieliem grafiem izrādās nelietojams. Dažus gadus atpakaļ tika radīts orientēta grafa diametra aproksimācijas algoritms, bet tikai pusgadu atpakaļ tika radīts vēl viens aproksimācijas algoritms grafa diametra un rādiusa novērtēšanai, kas strādā uz neorientētiem grafiem. Abiem šiem algoritmiem ir arī teorētiski pierādīts sliktākā gadījuma novērtējums. Darba mērķis ir šī grafa diametra un rādiusa aproksimācijas algoritma pielāgošana orientētiem grafiem. Pielāgotais algoritms tiks pārbaudīts uz reāliem grafiem un iegūtie rezultāti tiks salīdzināti ar izvirzīto sliktākā gadījuma novērtējumu. Algoritma veiktspēja tika salīdzināta ar otru eksistējošo grafa diametra aproksimācijas algoritmu, kam ir pierādīts sliktākā gadījuma novērtējums. Testēšanai tika izveidoti vairāk kā 26 tūkstoši grafi, kuri sadalīti 131 testu kopās.
Nowadays, with rapid increase in data amount, even polynomial algorithms with small power may turn out practically unusable. When the running time is more important than the optimal result, good approximation plays important role. Finding a diameter and radius of a graph is such problem. Graph diameter and radius have useful applications in graph clustering, epidemic spread research and social networks. Although optimal algorithm for finding graph diameter and radius has complexity O(m*n), it is often too slow for large graphs. Recently an algorithm for approximation of the diameter of oriented graphs was developed. Soon another algorithm was developed, which approximates radius and diameter for unoriented graphs. Both algorithms have theoretically proved worst case performance. The aim of this work was to adjust the latter of these algorithms for oriented graphs. Adjusted algorithm was tested on generated graphs and results compared to theoretical worst case estimate. Performance of the algorithm was compared against another diameter approximation algorithms. As a part of this work, more than 26 thousands of graphs were generated and grouped into 131 test sets.
Nowadays, with rapid increase in data amount, even polynomial algorithms with small power may turn out practically unusable. When the running time is more important than the optimal result, good approximation plays important role. Finding a diameter and radius of a graph is such problem. Graph diameter and radius have useful applications in graph clustering, epidemic spread research and social networks. Although optimal algorithm for finding graph diameter and radius has complexity O(m*n), it is often too slow for large graphs. Recently an algorithm for approximation of the diameter of oriented graphs was developed. Soon another algorithm was developed, which approximates radius and diameter for unoriented graphs. Both algorithms have theoretically proved worst case performance. The aim of this work was to adjust the latter of these algorithms for oriented graphs. Adjusted algorithm was tested on generated graphs and results compared to theoretical worst case estimate. Performance of the algorithm was compared against another diameter approximation algorithms. As a part of this work, more than 26 thousands of graphs were generated and grouped into 131 test sets.
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