Moving Walkways, Escalators, and Elevators

We study a simple geometric model of transportation facility that consists of two points between which the travel speed is high. This elementary definition can model shuttle services, tunnels, bridges, teleportation devices, escalators or moving walkways. The travel time between a pair of points is defined as a time distance, in such a way that a customer uses the transportation facility only if it is helpful. We give algorithms for finding the optimal location of such a transportation facility, where optimality is defined with respect to the maximum travel time between two points in a given set.Comment: 16 pages. Presented at XII Encuentros de Geometria Computacional, Valladolid, Spai

Colorful Strips

Given a planar point set and an integer $k$, we wish to color the points with $k$ colors so that any axis-aligned strip containing enough points contains all colors. The goal is to bound the necessary size of such a strip, as a function of $k$. We show that if the strip size is at least $2k{-}1$, such a coloring can always be found. We prove that the size of the strip is also bounded in any fixed number of dimensions. In contrast to the planar case, we show that deciding whether a 3D point set can be 2-colored so that any strip containing at least three points contains both colors is NP-complete. We also consider the problem of coloring a given set of axis-aligned strips, so that any sufficiently covered point in the plane is covered by $k$ colors. We show that in $d$ dimensions the required coverage is at most $d(k{-}1)+1$. Lower bounds are given for the two problems. This complements recent impossibility results on decomposition of strip coverings with arbitrary orientations. Finally, we study a variant where strips are replaced by wedges

Dynamic Trees with Almost-Optimal Access Cost

An optimal binary search tree for an access sequence on elements is a static tree that minimizes the total search cost. Constructing perfectly optimal binary search trees is expensive so the most efficient algorithms construct almost optimal search trees. There exists a long literature of constructing almost optimal search trees dynamically, i.e., when the access pattern is not known in advance. All of these trees, e.g., splay trees and treaps, provide a multiplicative approximation to the optimal search cost. In this paper we show how to maintain an almost optimal weighted binary search tree under access operations and insertions of new elements where the approximation is an additive constant. More technically, we maintain a tree in which the depth of the leaf holding an element e_i does not exceed min(log(W/w_i),log n)+O(1) where w_i is the number of times e_i was accessed and W is the total length of the access sequence. Our techniques can also be used to encode a sequence of m symbols with a dynamic alphabetic code in O(m) time so that the encoding length is bounded by m(H+O(1)), where H is the entropy of the sequence. This is the first efficient algorithm for adaptive alphabetic coding that runs in constant time per symbol

Surface Ozone Variability and Trends over the South African Highveld from 1990 to 2007

Surface ozone is a secondary air pollutant formed from reactions between nitrogen oxides (NOx = NO + NO2) and volatile organic compounds in the presence of sunlight. In this work we examine effects of the climate pattern known as the El Nio-Southern Oscillation (ENSO) and NOx variability on surface ozone from 1990 to 2007 over the South African Highveld, a heavily populated region in South Africa with numerous industrial facilities. Over summer and autumn (December-May) on the Highveld, El Nio, as signified by positive sea surface temperature (SST) anomalies over the central Pacific Ocean, is typically associated with drier and warmer than normal conditions favoring ozone formation. Conversely, La Nia, or negative SST anomalies over the central Pacific Ocean, is typically associated with cloudier and above normal rainfall conditions, hindering ozone production. We use a generalized regression model to identify any linear dependence that the Highveld ozone, measured at five air quality monitoring stations, may have on ENSO and NOx. Our results indicate that four out of the five stations exhibit a statistically significant sensitivity to ENSO at some point over the December-May period where El Nio amplifies ozone formation and La Nia reduces ozone formation. Three out of the five stations reveal statistically significant sensitivity to NOx variability, primarily in winter and spring. Accounting for ENSO and NOx effects throughout the study period of 18 years, two stations exhibit statistically significant negative ozone trends in spring, one station displays a statistically significant positive trend in August, and two stations show no statistically significant change in surface ozone

A critical review of health risk assessments of exposure to emissions from coal-fired power stations in South Africa

Emissions from coal-fired power stations increase the incidence of respiratory, cardiovascular and cardiopulmonary diseases and contribute to premature deaths. Wildly varying estimates of the magnitude of these health impacts have been published, however. This paper investigates the reasons for the large discrepancies calculated in five comprehensive health risk assessments of South African coal-fired power station emissions. We review the approaches and input data used by the studies. We also evaluate the exposure-response functions (which relate the pollution concentration to which the population is exposed, to the increase in health risk) used by each study and pay specific attention to whether the exposure-response functions are relevant to the South African context. Health risks are under-estimated in studies that only consider impacts in industrialised areas, or assume high counterfactual concentrations. Health risks are probably over-estimated, however, in other studies which use linear or exponential exposure-response functions that are not applicable in areas where exposure levels are much higher (such as solid fuel-using communities). A more accurate estimate of health effects would be obtained by applying integrated exposure-response functions to quantify health risks at actual exposure levels, and then apportioning the health effects relative to the contribution made by each source to total exposure levels. A fair assessment of the health risk of South Africa’s older coal-fired power stations should also weigh the health costs of the emissions against the health benefits of electricity use in lower income households

Deterministic Sampling and Range Counting in Geometric Data Streams

We present memory-efficient deterministic algorithms for constructing epsilon-nets and epsilon-approximations of streams of geometric data. Unlike probabilistic approaches, these deterministic samples provide guaranteed bounds on their approximation factors. We show how our deterministic samples can be used to answer approximate online iceberg geometric queries on data streams. We use these techniques to approximate several robust statistics of geometric data streams, including Tukey depth, simplicial depth, regression depth, the Thiel-Sen estimator, and the least median of squares. Our algorithms use only a polylogarithmic amount of memory, provided the desired approximation factors are inverse-polylogarithmic. We also include a lower bound for non-iceberg geometric queries.Comment: 12 pages, 1 figur

Cache-oblivious dynamic dictionaries with update/query tradeoffs

Several existing cache-oblivious dynamic dictionaries achieve O(logB N) (or slightly better O(logB N over M )) memory transfers per operation, where N is the number of items stored, M is the memory size, and B is the block size, which matches the classic B-tree data structure. One recent structure achieves the same query bound and a sometimes-better amortized update bound of O (...) memory transfers. This paper presents a new data structure, the xDict, implementing predecessor queries in O(...)worstcase memory transfers and insertions and deletions in O (...) amortized memory transfers, for any constant " with 0 < epsilon < 1. For example, the xDict achieves subconstant amortized update cost when N = ..., whereas the B-tree’s ... is subconstant only when ... is subconstant only when N = .... The xDict attains the optimal tradeoff between insertions and queries, even in the broader external-memory model, for the range where inserts cost between (...) and O(1= lg3 N) memory transfers.Danish National Research Foundation (MADALGO (Center for Massive Data Algorithmics))National Science Foundation (U.S.) (NSF Grants CCF-0541209)National Science Foundation (U.S.) (NSF Grants CCF-0541209)Computing Innovation Fellow

A polynomial bound for untangling geometric planar graphs

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n^\epsilon vertices fixed. We answer this question in the affirmative with \epsilon=1/4. The previous best known bound was \Omega((\log n / \log\log n)^{1/2}). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least (n/3)^{1/2} vertices fixed, while the best upper bound was O(n\log n)^{2/3}. We answer a question of Spillner and Wolff [arXiv:0709.0170 2007] by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than 3(n^{1/2}-1) vertices fixed. Moreover, we improve the lower bound to (n/2)^{1/2}.Comment: 14 pages, 7 figure

Decomposition of Multiple Coverings into More Parts

We prove that for every centrally symmetric convex polygon Q, there exists a constant alpha such that any alpha*k-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and Toth (SoCG'07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery lifetime