i E l e c t r o n J o u r n a l o f

Abstract We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be approximated well.

When algorithms for sorting and searching are applied to keys that are represented as bit strings, we can quantify the performance of the algorithms not only in terms of the number of key comparisons required by the algorithms but also in terms of the number of bit comparisons. Some of the standard sorting and searching algorithms have been analyzed with respect to key comparisons but not with respect to bit comparisons. In this paper, we investigate the expected number of bit comparisons required by Quickselect (also known as Find). We develop exact and asymptotic formulae for the expected number of bit comparisons required to find the smallest or largest key by Quickselect and show that the expectation is asymptotically linear with respect to the number of keys. Similar results are obtained for the average case. For finding keys of arbitrary rank, we derive an exact formula for the expected number of bit comparisons that (using rational arithmetic) requires only finite summation (rather than such operations as numerical integration) and use it to compute the expectation for each target rank.

An algorithm for perfect simulation from the unique solution of the distributional fixed point equation Y =d UY +U(1−U) is constructed, where Y and U are independent and U is uniformly distributed on [0, 1]. This distribution comes up as a limit distribution in the probabilistic analysis of the Quickselect algorithm. Our simulation algorithm is based on coupling from the past with a multigamma coupler. It has four lines of code.

An dieser Stelle möchte ich mich bei all jenen bedanken, die mir diese Diplomarbeit ermöglicht und zu ihrer Erstellung beigetragen haben. Herrn Prof. Dr. R. Neininger danke ich für das interessante Thema und die hervorragende Betreuung. Sein Interesse am Fortgang der Arbeit, seine konstruktiven

Abstract. We use coupling into and from the past to sample perfectly in a simple and provably fast fashion from the Vervaat family of perpetuities. The family includes the Dickman distribution, which arises both in number theory and in the analysis of the Quickselect algorithm, which was the motivation for our work.