Amortized Rotation Cost in AVL Trees

An AVL tree is the original type of balanced binary search tree. An insertion in an $n$-node AVL tree takes at most two rotations, but a deletion in an $n$-node AVL tree can take $\Theta(\log n)$. A natural question is whether deletions can take many rotations not only in the worst case but in the amortized case as well. A sequence of $n$ successive deletions in an $n$-node tree takes $O(n)$ rotations, but what happens when insertions are intermixed with deletions? Heaupler, Sen, and Tarjan conjectured that alternating insertions and deletions in an $n$-node AVL tree can cause each deletion to do $\Omega(\log n)$ rotations, but they provided no construction to justify their claim. We provide such a construction: we show that, for infinitely many $n$, there is a set $E$ of {\it expensive} $n$-node AVL trees with the property that, given any tree in $E$, deleting a certain leaf and then reinserting it produces a tree in $E$, with the deletion having done $\Theta(\log n)$ rotations. One can do an arbitrary number of such expensive deletion-insertion pairs. The difficulty in obtaining such a construction is that in general the tree produced by an expensive deletion-insertion pair is not the original tree. Indeed, if the trees in $E$ have even height $k$, $2^{k/2}$ deletion-insertion pairs are required to reproduce the original tree

Enhancement of the Kondo effect through Rashba spin-orbit interactions

We analyze the physics of a one-orbital Anderson impurity model in a two-dimensional electron gas in the presence of Rashba spin-orbit (RSO) interactions in the Kondo regime. The spin SU(2) symmetry breaking results in an effective two-band electron gas coupled to the impurity. The Kondo regime is obtained by a Schrieffer-Wolff transformation revealing the existence of a parity breaking term with the form of the Dzyaloshinsky-Moriya (DM) interaction. The DM term vanishes at the particle-hole symmetric point of the system, but it has important effects otherwise. Performing a renormalization group (RG) analysis we find that the model describes a two-channel Kondo system with ferro- and anti-ferromagnetic couplings. Furthermore, the DM term renormalizes the antiferromagnetic Kondo coupling producing an exponential enhancement of the Kondo temperature. We suggest that these effects can be observed in semiconducting systems, as well as in graphene and topological insulators.Comment: 4 pages, 1 figure. Final published versio

Peer-supported learning groups: a collaborative approach to supporting students learning in engineering and technology

peer-reviewedThis paper describes a non-traditional tutoring programme based on collaborative peer-support learning approach, and reflects on two years of its implementation to specific subjects in engineering and information technology based courses at the University of Limerick in Ireland. The programme, known as the Peer-Supported Learning Groups (PSLG), is an academic enrichment scheme which has been developed by adapting the SI model such that it meets the needs of the students in Ireland and, at the same time, fits into the Irish third-level education system. The paper begins by giving a rationale for the introduction of the PSLG to the targeted subjects and the reasons for choosing the SI model. This is followed by description of the operational structure of the programme highlighting the difficulties encountered at the initial stages and the measures taken to alleviate these difficulties. Quantitative measures for evaluating the effect of the PSLG on student’s performance, as well as analysis of feedback collected from the students and the leaders, are presented and discussed. The paper concludes by outlining issues for improving the current programme and associated further developments.PUBLISHEDpeer-reviewe

Effect of Iron/Folic Acid Supplementation on the Outcome of Malaria Episodes Treated with Sulfadoxine-Pyrimethamine

Folic acid supplementation may potentially alter the efficacy of sulfadoxine-pyrimethamine (SP) treatment in children with malaria. However, there is lack of evidence fromrandomized controlled trials and effects of folic acid supplementation on clinical efficacy of SP therapy remainmoderately understood among children. In a doublemasked, placebo-controlled trial among preschool children in Pemba Island (Tanzania), iron and folic acid supplementation (Fe/FA) showed an increased risk of hospitalizations and death. In the present paper, we evaluated if folic acid supplementation reduced the efficacy of malaria treatment and thereby contributed to observed adverse effects. During the study, 1648 children had confirmed malarial episodes and received either sulphadoxinepyrimethamine(SP) treatment and iron folic acid or SP treatment and placebo. These children were evaluated for recovery and incidence of hospitalization during the next 15, 30, and 140 days. Two groups did not differ in malarial episode or hospitalization rate on subsequent 15, 30, and 140 days. Altered efficacy of SP by folic acid was not observed and did not contribute to adverse events in the previous trial.This trial is registered with Controlled-trials.com ISRCTN59549825