A PTAS for assigning sporadic tasks on two-type heterogeneous multiprocessors

Consider the problem of determining a task-toprocessor assignment for a given collection of implicit-deadline sporadic tasks upon a multiprocessor platform in which there are two distinct kinds of processors. We propose a polynomialtime approximation scheme (PTAS) for this problem. It offers the following guarantee: for a given task set and a given platform, if there exists a feasible task-to-processor assignment, then given an input parameter, ϵ, our PTAS succeeds, in polynomial time, in finding such a feasible task-to-processor assignment on a platform in which each processor is 1+3ϵ times faster. In the simulations, our PTAS outperforms the state-of-the-art PTAS [1] and also for the vast majority of task sets, it requires significantly smaller processor speedup than (its upper bound of) 1+3ϵ for successfully determining a feasible task-to-processor assignment

Verbesserte Approximationsalgorithmen für Packungs- und Ablaufplanungsprobleme

This thesis presents approximation algorithms for geometric packing and scheduling problems. First, improved AFPTAS for the Bin Packing Problem (BP) and its generalization, the Variable-sized Bin Packing Problem (VBP), are explained. Our algorithms have to solve the unbounded variant of the Knapsack Problem (KP) and of the Knapsack Problem with Inversely Proportional Profits (KPIP) as subproblems. In the normal 0-1 variant of KP, an item can be chosen only once. In the bounded variant, an individual bounded number of copies can be taken of every item. The unbounded variant (UKP) allows for an infinite number of copies of every item. KPIP is a generalization of KP in which we have not only one, but several knapsack sizes. The profit of an item is inversely proportional to the size of the knapsack into which it has been packed. This makes it non-trivial to choose the knapsack size that maximizes the profit over all knapsack sizes. Similar to KP, there are the 0-1, the bounded, and the unbounded variant of KPIP. We first present FPTAS for every of the three variants of KPIP. They are faster than the natural approach to separately solve for every knapsack size the corresponding Knapsack Problem. Second, we present an FPTAS for UKP that is faster and needs less storage space than previously known algorithms. Finally, we combine the approaches of the KPIP and of the UKP FPTAS to get an FPTAS for the Unbounded KPIP that has again a better time and space complexity. All these results improve the running time for our BP and VBP algorithms. As a corollary, we also improve the running time for a Strip Packing AFPTAS. Finally, we consider Scheduling on Unrelated Machines of which we study the special case with a constant number K of machine types: one job has the same processing time on every machine of the same type. We present a PTAS for this special case. The algorithm has a better running time than the previously known algorithm for general (but constant) K.Diese Dissertation stellt Approximationsalgorithmen für geometrische Packungs- und Ablaufplanungsprobleme (Packing and Scheduling Problems) vor. Zuerst werden verbesserte AFPTAS für das Behälterproblem (Bin Packing, BP) und seine Verallgemeinerung, das Behälterproblem mit verschiedenen Behältergrößen (Variable-sized Bin Packing, VBP), erklärt. Unsere Algorithmen müssen die unbeschränkte (unbounded) Variante des Rucksackproblems (Knapsack Problem, KP) und des Rucksackproblems mit invers proportionalen Profiten (Knapsack Problem with Inversely Proportional Profits, KPIP) als Unterprobleme lösen. Bei KP gibt es die Varianten 0-1, beschränkt und unbeschränkt. KPIP ist eine Verallgemeinerung des Rucksackproblems mit mehreren Rucksackgrößen, die in dieser Dissertation eingeführt wird. Wie bei KP gibt es bei KPIP ebenfalls die Varianten 0-1, beschränkt und unbeschränkt. Wir stellen zuerst FPTAS für alle drei Varianten von KPIP vor. Sie sind schneller als der natürliche Ansatz, für jede Rucksackgröße das entsprechende Rucksackproblem einzeln zu lösen. Danach stellen wir ein FPTAS für UKP vor, das schneller ist und weniger Speicherplatz benötigt als zuvor bekannte Algorithmen. Schließlich kombinieren wir den Ansatz für KPIP und für das unbeschränkte KPIP, um ein FPTAS für die unbeschränkte Variante von KPIP zu erhalten, das wiederum eine kleinere Zeit- und Speicherkomplexität besitzt. All diese Resultate verbessern die Laufzeit unserer BP- und VBP-Algorithmen. Als Korollar verbessern wir außerdem die Laufzeit eines AFPTAS für das geometrische Zuschnittproblem (Strip Packing). Schließlich betrachten wir das Ablaufplanungsproblem auf heterogenen Maschinen (Scheduling on Unrelated Machines), bei dem wir den Spezialfall mit einer konstanten Anzahl K an Maschinentypen untersuchen: Eine Aufgabe (Job) hat auf jeder Maschine desselben Typs die gleiche Ausführungszeit. Wir stellen für diesen Spezialfall ein PTAS vor. Der Algorithmus ist schneller als das zuvor bekannte Verfahren für allgemeines (aber konstantes) K

Real-Time Scheduling on Heterogeneous Multiprocessors

Embedded computing is one of the most important areas in computer science today, witnessed by the fact that 98 % of all computers are embedded. Given that many embedded systems have to interact “promptly” with their physical environment, the scientific community has invested signifi-cant efforts in developing algorithms for scheduling the workload, which is generally implemented as a set of tasks, at the right time and in proving before run-time that all the timing requirements will be satisfied at run-time. This field of study is referred to as the real-time scheduling theory. The scheduling theory for a unicore processor is well-developed; the scientific results are taught at all major universities world-wide and the results are widely-used in industry. Scheduling theory for multicores is emerging but the focus so far has been for multicores with identical pro-cessing units. This is unfortunate because the computer industry is moving towards heterogeneous multicores with a constant number of distinct processor types — AMD Fusion, Intel Atom and NVIDIA Tegra are some of the examples of such multicores. This work deals with the problem of scheduling a set of tasks to meet their deadlines on het-erogeneous multiprocessors with a constant number of distinct processor types. On heterogeneou