Kombinatorické algoritmy se zameřením na on line problémy: semi -on line rozvrhování na strojích s různými rychlostmi

Mgr. Tomáš Ebenlendr Combinatorial algorithms for online problems: Semi-online scheduling on related machines Abstract of doctoral thesis We construct a framework that gives optimal algorithms for a whole class of scheduling problems. This class covers the most studied semi-online variants of preemptive online scheduling on uniformly related machines with the objective to minimize makespan. The algorithms from our framework are deterministic, yet they are optimal even among all randomized algorithms. In addition, they are optimal for any fixed combination of speeds of the machines, and thus our results subsume all the previous work on various special cases. We provide new lower bound of 2.112 for the original online problem. The (deterministic) upper bound is e ≈ 2.718 as there was known e-competitive randomized algorithm before. Our framework applies to all semi-online variants which are based on some knowledge about the input sequence. I.e., they are restrictions of the set of valid inputs. We use our framework to study restrictions that were studied before, and we derive some new bounds. Namely we study known sum of processing times, known maximal processing time, sorted (decreasing) jobs, tightly grouped processing times, approximately known optimal makespan and few combinations. Based on the analysis...Mgr. Tomáš Ebenlendr Kombinatorické algoritmy se zaměřením na online problémy: Semi-online rozvrhování na strojích s různými rychlostmi Abstrakt disertační práce Hlavním výsledkem této práce je konstrukce optimálních algoritmů pro celou třídu rozvrhovacích problémů. Tato třída zahrnuje většinu zkoumaných semi-online vari- ant preemptivního rozvrhování na strojích s různými rychlostmi s cílem minimali- zovat délku rozvrhu. Takto zkonstruované algoritmy jsou deterministické, nicméně dosahují optimální kompetitivní poměr i mezi pravděpodobnostními algoritmy. Navíc jsou optimální i pro libovolnou pevnou kombinaci rychlostí strojů, proto lze naši konstrukci uplatnit i na veškeré dříve studované speciální případy. Ukážeme nový dolní odhad 2.112 pro obecné online rozvrhování. Deterministický horní odhad e ≈ 2.718 pak plyne z dřívější existence e-kompetitivního pravděpodob- nostního algoritmu. Zmíněnou konstrukci lze aplikovat ve všech semi-online variantách, které jsou založeny na znalosti o vstupní sekvenci. Ty lze chápat jako omezení množiny plat- ných vstupů. Tuto konstrukci pak použijeme ke studiu dříve zkoumaných omezení, čímž získáme nové odhady kompetitivního poměru. Jmenovitě zkoumáme známou sumu...Katedra aplikované matematikyDepartment of Applied MathematicsFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

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