On Optimal Algorithms for List Ranking in the Parallel External Memory Model with Applications to Treewidth and other Elementary Graph Problems

The performance of many algorithms on large input instances substantially depends on the number of triggered cache misses instead of the number of executed operations. This behavior is captured by the external memory model in a natural way. It models a computer by a fast cache of bounded size and a conceptually infinite (external) memory. In contrast to the classical RAMmodel, the complexity measure is the number of cache lines transferred between the cache and the memory. Computations on elements in the cache are not counted. Recent trends in processor design and advances in big data computing require massively parallel algorithms. The parallel external memory (PEM) model extends the external memory model so that it also captures parallelism. It consists of multiple processors which each have a private cache and share the (external) memory. This thesis considers three computational problems in the context of (parallel) external memory algorithms. For the fundamental problem of list ranking, previously, an algorithm was known that has sorting complexity for many settings of the PEM model. In the first part of this thesis, this algorithm is complemented by matching lower bounds for most practical settings. Interestingly, a stronger lower bound for parameter ranges which previously have not been considered is shown. By modeling how list ranking algorithms retrieve information on the structure of the list in the memory, we give a lower bound that is quadratic in sorting complexity for certain parameter settings. It is noteworthy that this result implies the first non-trivial lower bounds for list ranking for the bulk synchronous parallel and the MapReduce model. These lower bounds are complemented by a list ranking algorithm which is, in contrast to previous algorithms, analyzed for all parameter settings of the PEM model. In the second part, an efficient algorithm for the PEM model to compute a tree decomposition of bounded width for a graph is presented. The main challenge is to implement a load balancing strategy such that the running

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