Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling

A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node $v \in G$ stores its distance to the so-called hubs $S_v \subseteq V$, chosen so that for any $u,v \in V$ there is $w \in S_u \cap S_v$ belonging to some shortest $uv$ path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme at least as a key building block. Our interest lies in hub labelings of sparse graphs, i.e., those with $|E(G)| = O(n)$, for which we show a lowerbound of $\frac{n}{2^{O(\sqrt{\log n})}}$ for the average size of the hubsets. Additionally, we show a hub-labeling construction for sparse graphs of average size $O(\frac{n}{RS(n)^{c}})$ for some $0 < c < 1$, where $RS(n)$ is the so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced matchings in dense graphs. This implies that further improving the lower bound on hub labeling size to $\frac{n}{2^{(\log n)^{o(1)}}}$ would require a breakthrough in the study of lower bounds on $RS(n)$, which have resisted substantial improvement in the last 70 years. For general distance labeling of sparse graphs, we show a lowerbound of $\frac{1}{2^{O(\sqrt{\log n})}} SumIndex(n)$, where $SumIndex(n)$ is the communication complexity of the Sum-Index problem over $Z_n$. Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be $\Theta(\frac{n}{2^{(\log n)^c}})$ for some $0<c < 1$

The Tree Inclusion Problem: In Linear Space and Faster

Given two rooted, ordered, and labeled trees $P$ and $T$ the tree inclusion problem is to determine if $P$ can be obtained from $T$ by deleting nodes in $T$. This problem has recently been recognized as an important query primitive in XML databases. Kilpel\"ainen and Mannila [\emph{SIAM J. Comput. 1995}] presented the first polynomial time algorithm using quadratic time and space. Since then several improved results have been obtained for special cases when $P$ and $T$ have a small number of leaves or small depth. However, in the worst case these algorithms still use quadratic time and space. Let $n_S$, $l_S$, and $d_S$ denote the number of nodes, the number of leaves, and the %maximum depth of a tree $S \in \{P, T\}$. In this paper we show that the tree inclusion problem can be solved in space $O(n_T)$ and time: O(\min(l_Pn_T, l_Pl_T\log \log n_T + n_T, \frac{n_Pn_T}{\log n_T} + n_{T}\log n_{T})). This improves or matches the best known time complexities while using only linear space instead of quadratic. This is particularly important in practical applications, such as XML databases, where the space is likely to be a bottleneck.Comment: Minor updates from last tim

A simple and optimal ancestry labeling scheme for trees

We present a $\lg n + 2 \lg \lg n+3$ ancestry labeling scheme for trees. The problem was first presented by Kannan et al. [STOC 88'] along with a simple $2 \lg n$ solution. Motivated by applications to XML files, the label size was improved incrementally over the course of more than 20 years by a series of papers. The last, due to Fraigniaud and Korman [STOC 10'], presented an asymptotically optimal $\lg n + 4 \lg \lg n+O(1)$ labeling scheme using non-trivial tree-decomposition techniques. By providing a framework generalizing interval based labeling schemes, we obtain a simple, yet asymptotically optimal solution to the problem. Furthermore, our labeling scheme is attained by a small modification of the original $2 \lg n$ solution.Comment: 12 pages, 1 figure. To appear at ICALP'1

Compressed Subsequence Matching and Packed Tree Coloring

We present a new algorithm for subsequence matching in grammar compressed strings. Given a grammar of size $n$ compressing a string of size $N$ and a pattern string of size $m$ over an alphabet of size $\sigma$, our algorithm uses $O(n+\frac{n\sigma}{w})$ space and $O(n+\frac{n\sigma}{w}+m\log N\log w\cdot occ)$ or $O(n+\frac{n\sigma}{w}\log w+m\log N\cdot occ)$ time. Here $w$ is the word size and $occ$ is the number of occurrences of the pattern. Our algorithm uses less space than previous algorithms and is also faster for $occ=o(\frac{n}{\log N})$ occurrences. The algorithm uses a new data structure that allows us to efficiently find the next occurrence of a given character after a given position in a compressed string. This data structure in turn is based on a new data structure for the tree color problem, where the node colors are packed in bit strings.Comment: To appear at CPM '1

Cokebildung und Entcoking während der Methanbildung und des Methanzerfalls auf Ni-Cu-Trägerkatalysatoren

The effect of the composition of silica supported Ni-Cu alloy catalysts on the process of coking and decoking during methane decomposition and during methanation was considered. The kinetics of methanation was studied and compared to those of carbon deposition and of strong adsorption of hydrogen. Initiation of the formation of filamentous carbon formation on mono-metallic surfaces may take place if the ratio of the partial pressures, pCO/pH2, is larger than 2 (T 673 K). Once the process starts, the chemical potential of the gas phase may be reduced to lower values without interruption of filament growth. Besides, it was concluded that the methanation reaction includes two steps: the dissociative adsorption of CO and the hydrogenation of the adsorbed species. It was possible to establish the mechanism through which Cu affects the activity of Ni. The effect of the composition of the alloy catalysts on the methane formation and on the simultaneous carbon deposition indicates that those reactions belong to group I and to group II, respectively, following Ponec's classification. It was possible to find the optimal Cu concentration that maximises methanation and minimises carbon deposition. The kinetics of methane decomposition was also considered and is well described by adapting a model developed by other authors for Fe catalysts

Dynamic and Multi-functional Labeling Schemes

We investigate labeling schemes supporting adjacency, ancestry, sibling, and connectivity queries in forests. In the course of more than 20 years, the existence of $\log n + O(\log \log)$ labeling schemes supporting each of these functions was proven, with the most recent being ancestry [Fraigniaud and Korman, STOC '10]. Several multi-functional labeling schemes also enjoy lower or upper bounds of $\log n + \Omega(\log \log n)$ or $\log n + O(\log \log n)$ respectively. Notably an upper bound of $\log n + 5\log \log n$ for adjacency+siblings and a lower bound of $\log n + \log \log n$ for each of the functions siblings, ancestry, and connectivity [Alstrup et al., SODA '03]. We improve the constants hidden in the $O$-notation. In particular we show a $\log n + 2\log \log n$ lower bound for connectivity+ancestry and connectivity+siblings, as well as an upper bound of $\log n + 3\log \log n + O(\log \log \log n)$ for connectivity+adjacency+siblings by altering existing methods. In the context of dynamic labeling schemes it is known that ancestry requires $\Omega(n)$ bits [Cohen, et al. PODS '02]. In contrast, we show upper and lower bounds on the label size for adjacency, siblings, and connectivity of $2\log n$ bits, and $3 \log n$ to support all three functions. There exist efficient adjacency labeling schemes for planar, bounded treewidth, bounded arboricity and interval graphs. In a dynamic setting, we show a lower bound of $\Omega(n)$ for each of those families.Comment: 17 pages, 5 figure

Labeling Schemes for Bounded Degree Graphs

We investigate adjacency labeling schemes for graphs of bounded degree $\Delta = O(1)$. In particular, we present an optimal (up to an additive constant) $\log n + O(1)$ adjacency labeling scheme for bounded degree trees. The latter scheme is derived from a labeling scheme for bounded degree outerplanar graphs. Our results complement a similar bound recently obtained for bounded depth trees [Fraigniaud and Korman, SODA 10], and may provide new insights for closing the long standing gap for adjacency in trees [Alstrup and Rauhe, FOCS 02]. We also provide improved labeling schemes for bounded degree planar graphs. Finally, we use combinatorial number systems and present an improved adjacency labeling schemes for graphs of bounded degree $\Delta$ with $(e+1)\sqrt{n} < \Delta \leq n/5$

Facing danger: exploring personality and reactions of European hedgehogs (Erinaceus europaeus) towards robotic lawn mowers

The populations of European hedgehog (Erinaceus europaeus) are in decline, and it is essential that research identifies and mitigates the factors causing this. Hedgehogs are increasingly sharing habitats with humans, being exposed to a range of dangers in our backyards. Previous research has documented that some models of robotic lawn mowers can cause harm to hedgehogs. This study explored the personality and behaviour of 50 live hedgehogs when facing an approaching, disarmed robotic lawn mower. By combining a novel arena and novel object test, we found that 27 hedgehogs could be categorised as “shy” and 23 as “bold”, independently of sex and age. The encounter tests with a robotic lawn mower showed that the hedgehogs positioned themselves in seven different ways. Personality did not affect their reactions. Adult hedgehogs tended to react in a shyer manner, and the hedgehogs, generally, acted less boldly during their second encounter with the robotic lawn mower. Additionally, our results show that bold individuals reacted in a more unpredictable way, being more behaviourally unstable compared to the shy individuals. This knowledge will be applied in the design of a standardised hedgehog safety test, eventually serving to produce and approve hedgehog-friendly robotic lawn mowers

An untapped potential for imaging of peripheral osteomyelitis in paediatrics using [ ¹⁸ F]FDG PET/CT —the inference from a juvenile porcine model

Abstract Purpose To examine parameters affecting the detection of osteomyelitis (OM) by [18F]FDG PET/CT and to reduce tracer activity in a pig model. Background [18F]FDG PET/CT is recommended for the diagnosis of OM in the axial skeleton of adults. In children, OM has a tendency to become chronic or recurrent, especially in low-income countries. Early diagnosis and initiation of therapy are therefore essential. We have previously demonstrated that [18F]FDG PET/CT is promising in juvenile Staphylococcus aureus (S. aureus) OM of peripheral bones in a pig model, not failing even small lesions. When using imaging in children, radiation exposure should be balanced against fast diagnostics in the individual case. Methods Twenty juvenile pigs were inoculated with S. aureus. One week after inoculation, the pigs were [18F]FDG PET/CT scanned. PET list-mode acquired data of a subgroup were retrospectively processed in order to simulate and examine the image quality obtainable with an injected activity of 132 MBq, 44 MBq, 13.2 MBq, and 4.4 MBq, respectively. Results All lesions were detected by [18F]FDG PET and CT. Some lesions were very small (0.01 cm3), and others were larger (4.18 cm3). SUVmax was higher when sequesters (p = 0.023) and fistulas were formed (p < 0.0001). The simulated data demonstrated that it was possible to reduce the activity to 4.4 MBq without compromising image quality in pigs. Conclusions [18F]FDG PET/CT localized even small OM lesions in peripheral bones. It was possible to reduce the injected activity considerably without compromising image quality, impacting the applicability of PET/CT in peripheral OM in children