Transfer Theorems and Asymptotic Distributional Results for m-ary Search Trees

We derive asymptotics of moments and identify limiting distributions, under the random permutation model on m-ary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the so-called shape functional fall under this framework. The approach is based on establishing transfer theorems that link the order of growth of the input into a particular (deterministic) recurrence to the order of growth of the output. The transfer theorems are used in conjunction with the method of moments to establish limit laws. It is shown that (i) for small toll sequences $(t_n)$ [roughly, $t_n =O(n^{1 / 2})$] we have asymptotic normality if $m \leq 26$ and typically periodic behavior if $m \geq 27$; (ii) for moderate toll sequences [roughly, $t_n = \omega(n^{1 / 2})$ but $t_n = o(n)$] we have convergence to non-normal distributions if $m \leq m_0$ (where $m_0 \geq 26$) and typically periodic behavior if $m \geq m_0 + 1$; and (iii) for large toll sequences [roughly, $t_n = \omega(n)$] we have convergence to non-normal distributions for all values of m.Comment: 35 pages, 1 figure. Version 2 consists of expansion and rearragement of the introductory material to aid exposition and the shortening of Appendices A and B.

Distributional convergence for the number of symbol comparisons used by QuickSort

Most previous studies of the sorting algorithm QuickSort have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the n independent and identically distributed (i.i.d.) keys are each represented as a sequence of symbols from a probabilistic source and that QuickSort operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild "tameness" condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first centering by the mean and then dividing by n. Additionally, under a condition that grows more restrictive as p increases, we have convergence of moments of orders p and smaller. In particular, we have convergence in distribution and convergence of moments of every order whenever the source is memoryless, that is, whenever each key is generated as an infinite string of i.i.d. symbols. This is somewhat surprising; even for the classical model that each key is an i.i.d. string of unbiased ("fair") bits, the mean exhibits periodic fluctuations of order n.Comment: Published in at http://dx.doi.org/10.1214/12-AAP866 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

A learning design toolkit to create pedagogically effective learning activities

Despite the plethora of Information and Communication Technologies (ICT) tools and resources available, practitioners are still not making effective use of e-learning to enrich the student experience. This article describes a learning design toolkit which guides practitioners through the process of creating pedagogically informed learning activities which make effective use of appropriate tools and resources. This work is part of a digital libraries project in which teaching staff at two universities in the UK and two in the USA are collaborating to share e-learning resources in the subject domains of Physical, Environmental and Human Geography. Finding, or creating, suitable e-learning resources and embedding them in well designed learning activities can be both challenging and time consuming. Sharing and adapting effective designs and solutions is both a stimulant and a time saver. This article describes the background to the specification of a learning activities design toolkit to support teachers as they create or adapt e-learning activities. This uses a model of pedagogical approaches as a basis for developing effective learning design plans and illustrates its use. The authors share their definition of a learning activity and taxonomies for the constituent elements. Real examples are discussed to illustrate their approach

Embedding blended learning in a university’s teaching culture: experiences and reflections

Blended learning, the combination of traditional face-to-face teaching methods with authentic on-line learning activities, has the potential to transform student learning experiences and outcomes. In spite of this advantage, university teachers often find it difficult to adopt new on-line techniques, in part because institutional practices are still geared to support more traditional approaches. This paper describes how a project, funded to support international collaboration to enhance learning and teaching in Geography, has allowed a university to explore models for change. It briefly examines the associated issues of sharing and repurposing resources; it reflects on the impact of the project on local strategy, and the importance of sustaining the collaborations and approaches to learning and teaching after the funding is completed