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The AREA of a schedule for executing DAGs is the average number of DAG-chores that are eligible for execution at each step of the computation. AREA maximization is a new optimization goal for schedules that execute DAGs within computational environments, such as Internet-based computing, clouds, and volunteer computing projects, that are dynamically heterogeneous, in the sense that the environments' constituent computers can change their effective powers at times and in ways that are not predictable. This paper is motivated by the thesis that, within dynamically heterogeneous environments, DAG-schedules that have larger AREAs execute a computation-DAG with smaller completion time under many circumstances; this thesis is supported by preliminary simulation-based experiments. While every DAG admits an AREA-maximizing schedule, it is likely computationally difficult to find such a schedule for an arbitrary DAG. Earlier work has shown how to craft AREA-maximizing schedules efficiently for a number of families of DAGs whose structures are reminiscent of many scientific computations. The current paper extends this work by showing how to efficiently craft AREA-maximizing schedules for series-parallel DAGs, a family that models a multithreading computing paradigm. The techniques for crafting these schedules promise to apply also to other large families of recursively defined DAGs. Moreover, the ability to derive these schedules efficiently leads to an efficient AREA-oriented heuristic for scheduling arbitrary DAGs.

We consider a Master-Worker distributed system where a master processor assigns, over the Internet, tasks to a collection of n workers, which are untrusted and might act maliciously. In addition, a worker may not reply to the master, or its reply may not reach the master, due to unavailabilities or failures of the worker or the network. Each task returns a value, and the goal is for the master to accept only correct values with high probability. Furthermore, we assume that the service provided by the workers is not free; for each task that a worker is assigned, the master is charged with a work-unit. Therefore, considering a single task assigned to several workers, our objective is to have the master processor to accept the correct value of the task with high probability, with the smallest possible amount of work (number of workers the master assigns the task). We probabilistically bound the number of faulty processors by assuming a known probability p < 1/2 of any processor to be faulty.

Our work demonstrates that it is possible to obtain, with provable analytical guarantees, high probability of correct acceptance with low work. In particular, we first show lower bounds on the minimum amount of (expected) work required, so that any algorithm accepts the correct value with probability of success 1 - ε, where ε ≪ 1 (e.g., 1/n). Then we develop and analyze two algorithms, each using a different decision strategy, and show that both algorithms obtain the same probability of success 1 - ε, and in doing so, they require similar upper bounds on the (expected) work. Furthermore, under certain conditions, these upper bounds are asymptotically optimal with respect to our lower bounds.