Hollow Heaps

We introduce the hollow heap, a very simple data structure with the same amortized efficiency as the classical Fibonacci heap. All heap operations except delete and delete-min take $O(1)$ time, worst case as well as amortized; delete and delete-min take $O(\log n)$ amortized time on a heap of $n$ items. Hollow heaps are by far the simplest structure to achieve this. Hollow heaps combine two novel ideas: the use of lazy deletion and re-insertion to do decrease-key operations, and the use of a dag (directed acyclic graph) instead of a tree or set of trees to represent a heap. Lazy deletion produces hollow nodes (nodes without items), giving the data structure its name.Comment: 27 pages, 7 figures, preliminary version appeared in ICALP 201

Faster Shortest Paths in Dense Distance Graphs, with Applications

We show how to combine two techniques for efficiently computing shortest paths in directed planar graphs. The first is the linear-time shortest-path algorithm of Henzinger, Klein, Subramanian, and Rao [STOC'94]. The second is Fakcharoenphol and Rao's algorithm [FOCS'01] for emulating Dijkstra's algorithm on the dense distance graph (DDG). A DDG is defined for a decomposition of a planar graph $G$ into regions of at most $r$ vertices each, for some parameter $r < n$. The vertex set of the DDG is the set of $\Theta(n/\sqrt r)$ vertices of $G$ that belong to more than one region (boundary vertices). The DDG has $\Theta(n)$ arcs, such that distances in the DDG are equal to the distances in $G$. Fakcharoenphol and Rao's implementation of Dijkstra's algorithm on the DDG (nicknamed FR-Dijkstra) runs in $O(n\log(n) r^{-1/2} \log r)$ time, and is a key component in many state-of-the-art planar graph algorithms for shortest paths, minimum cuts, and maximum flows. By combining these two techniques we remove the $\log n$ dependency in the running time of the shortest-path algorithm, making it $O(n r^{-1/2} \log^2r)$. This work is part of a research agenda that aims to develop new techniques that would lead to faster, possibly linear-time, algorithms for problems such as minimum-cut, maximum-flow, and shortest paths with negative arc lengths. As immediate applications, we show how to compute maximum flow in directed weighted planar graphs in $O(n \log p)$ time, where $p$ is the minimum number of edges on any path from the source to the sink. We also show how to compute any part of the DDG that corresponds to a region with $r$ vertices and $k$ boundary vertices in $O(r \log k)$ time, which is faster than has been previously known for small values of $k$

New Algorithms for Position Heaps

We present several results about position heaps, a relatively new alternative to suffix trees and suffix arrays. First, we show that, if we limit the maximum length of patterns to be sought, then we can also limit the height of the heap and reduce the worst-case cost of insertions and deletions. Second, we show how to build a position heap in linear time independent of the size of the alphabet. Third, we show how to augment a position heap such that it supports access to the corresponding suffix array, and vice versa. Fourth, we introduce a variant of a position heap that can be simulated efficiently by a compressed suffix array with a linear number of extra bits

“Wading Through Water” - Parental Experiences Of Their Child’s HE Choice Process

In an increasingly marketised and competitive UK HE environment understanding the student decision-making process has become very important. At the same time, there has been an increase in parental involvement in this choice amongst certain groups of parents. This paper examines parental accounts of their experiences and involvement in their child's HE choice process. It finds that the choice process is experienced as a form of parenting. Participants described their efforts in trying to get their child to talk to them and to achieve a balance in terms of their involvement and that of the child. This idea of relationships impacting on the choice process is one which is almost entirely missing from the choice literature and warrants further investigation. In this paper, parental experiences are examined relating to the literature on choice and student and parental decision-making within HE. The research adopts a qualitative phenomenological approach with parents focusing in detail on their actual experiences and on aspects of importance to them. HEIs should be wary of over-estimating the choice processes which students and their parents engage in and of assuming that parental involvement leads to a more thorough process

Can children withhold consent to treatment

A dilemma exists when a doctor is faced with a child or young person who refuses medically indicated treatment. The Gillick case has been interpreted by many to mean that a child of sufficient age and intelligence could validly consent or refuse consent to treatment. Recent decisions of the Court of Appeal on a child's refusal of medical treatment have clouded the issue and undermined the spirit of the Gillick decision and the Children Act 1989. It is now the case that a child patient whose competence is in doubt will be found rational if he or she accepts the proposal to treat but may be found incompetent if he or she disagrees. Practitioners are alerted to the anomalies now exhibited by the law on the issue of children's consent and refusal. The impact of the decisions from the perspectives of medicine, ethics, and the law are examined. Practitioners should review each case of child care carefully and in cases of doubt seek legal advice