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

Fifteenth century earth science

The earth science content of two late medieval encyclopedias, the Mirrour of the World and Higden's Polychronicon, both printed by William Caxton in the 1480's, is examined in relation to fifteenth century ideas about the physical nature of the earth and the universe. Such topics as the four elements, the earth and the spheres, location of Hell and Paradise, the arrangement of , continents and oceans, the unity of waters, earthquakes and volcanoes, erosion, fossils and mountain building, climatic zones and weather phenomena are summarized and reference made to the Biblical and Classical Greek sources of these ideas

Smooth heaps and a dual view of self-adjusting data structures

We present a new connection between self-adjusting binary search trees (BSTs) and heaps, two fundamental, extensively studied, and practically relevant families of data structures. Roughly speaking, we map an arbitrary heap algorithm within a natural model, to a corresponding BST algorithm with the same cost on a dual sequence of operations (i.e. the same sequence with the roles of time and key-space switched). This is the first general transformation between the two families of data structures. There is a rich theory of dynamic optimality for BSTs (i.e. the theory of competitiveness between BST algorithms). The lack of an analogous theory for heaps has been noted in the literature. Through our connection, we transfer all instance-specific lower bounds known for BSTs to a general model of heaps, initiating a theory of dynamic optimality for heaps. On the algorithmic side, we obtain a new, simple and efficient heap algorithm, which we call the smooth heap. We show the smooth heap to be the heap-counterpart of Greedy, the BST algorithm with the strongest proven and conjectured properties from the literature, widely believed to be instance-optimal. Assuming the optimality of Greedy, the smooth heap is also optimal within our model of heap algorithms. As corollaries of results known for Greedy, we obtain instance-specific upper bounds for the smooth heap, with applications in adaptive sorting. Intriguingly, the smooth heap, although derived from a non-practical BST algorithm, is simple and easy to implement (e.g. it stores no auxiliary data besides the keys and tree pointers). It can be seen as a variation on the popular pairing heap data structure, extending it with a "power-of-two-choices" type of heuristic.Comment: Presented at STOC 2018, light revision, additional figure

Agroeca dentigera and Entelecara omissa (Araneae: Liocranidae, Linyphiidae) found in Sweden

The rare spider species Agroeca dentigera Kulczyński, 1913 (Liocranidae) and Entelecara omissa O. P.-Cambridge, 1902 (Linyphiidae), have been found in a small coastal freshwater fen in Lomma (55°42'N 13°4'E), north of Malmö in Scania in southernmost Sweden. A. dentigera was also found on a salt water meadow south of Malmö. Both species have been found only in a few wet localities in Europe. Entelecara depilata Tullgren, 1955, is a junior synonym of Entelecara omissa O. P.-Cambridge, 1902, new synonymy

Excavations and survey at Coats Hill, near Moffat, 1990-1

This report describes the results of the survey and sample excavations of small cairns, annular structures and other remains on Coats Hill, near Moffat. The difficulties of assessing the dates and functions of certain of the structures are discussed. The project formed part of the archaeological studies for the North Western Ethylene Pipeline (NWEP) Project for Shell Chemicals UK Ltd, which wholly funded the archaeological work and the publication of this report

The Mathematical Relationship between Zipf's Law and the Hierarchical Scaling Law

The empirical studies of city-size distribution show that Zipf's law and the hierarchical scaling law are linked in many ways. The rank-size scaling and hierarchical scaling seem to be two different sides of the same coin, but their relationship has never been revealed by strict mathematical proof. In this paper, the Zipf's distribution of cities is abstracted as a q-sequence. Based on this sequence, a self-similar hierarchy consisting of many levels is defined and the numbers of cities in different levels form a geometric sequence. An exponential distribution of the average size of cities is derived from the hierarchy. Thus we have two exponential functions, from which follows a hierarchical scaling equation. The results can be statistically verified by simple mathematical experiments and observational data of cities. A theoretical foundation is then laid for the conversion from Zipf's law to the hierarchical scaling law, and the latter can show more information about city development than the former. Moreover, the self-similar hierarchy provides a new perspective for studying networks of cities as complex systems. A series of mathematical rules applied to cities such as the allometric growth law, the 2^n principle and Pareto's law can be associated with one another by the hierarchical organization.Comment: 30 pages, 5 figures, 5 tables, Physica A: Statistical Mechanics and its Applications, 201

Gypsum dissolution geohazards at Ripon, North Yorkshire, UK

This guide is for a one-day field excursion to examine gypsum dissolution geohazards at Ripon in North Yorkshire. Gypsum is a highly soluble rock and under suitable groundwater flow conditions it can dissolve forming caves and karstic features including collapse and suffosion dolines. These have the capability of causing subsidence damage of the type that affects much of the Ripon area. The guide details the processes involved, the localities visited and some of the remedial measures undertaken