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

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

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

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

PET radioligand injection for pig neuroimaging

Pigs are useful models in neuroimaging studies with positron emission tomography (PET). Radiolabeled ligands are injected intravenously at the start of the scan and in pigs the most easily accessible route of administration is the ear vein. However, in brain studies the short distance between the brain and ear vein can be problematic as both are localized inside the field of view and, as a consequence, tracer residues in the catheter may influence the outcome of the scan. Here, we discuss options to avoid this problem. The femoral vein can be used in studies where repeated arterial blood sampling is needed because surgical incision has to be performed to allow access to the artery. When a non-invasive technique is preferred, the ear vein is a good alternative although it is recommended to dilute the tracer sufficiently in saline (20-50 mL) prior to injection. In addition, the tracer can be injected through an extension tube (filled with saline before injection), which is removed together with the syringe immediately after tracer injection. This avoids placing the syringe with tracer inside the PET gantry while injecting. By applying these simple techniques, it is our experience that it is possible to obtain high-quality images without exposing pigs to invasive procedures

Tree Compression with Top Trees Revisited

We revisit tree compression with top trees (Bille et al, ICALP'13) and present several improvements to the compressor and its analysis. By significantly reducing the amount of information stored and guiding the compression step using a RePair-inspired heuristic, we obtain a fast compressor achieving good compression ratios, addressing an open problem posed by Bille et al. We show how, with relatively small overhead, the compressed file can be converted into an in-memory representation that supports basic navigation operations in worst-case logarithmic time without decompression. We also show a much improved worst-case bound on the size of the output of top-tree compression (answering an open question posed in a talk on this algorithm by Weimann in 2012).Comment: SEA 201

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

Compressed Membership for NFA (DFA) with Compressed Labels is in NP (P)

In this paper, a compressed membership problem for finite automata, both deterministic and non-deterministic, with compressed transition labels is studied. The compression is represented by straight-line programs (SLPs), i.e. context-free grammars generating exactly one string. A novel technique of dealing with SLPs is introduced: the SLPs are recompressed, so that substrings of the input text are encoded in SLPs labelling the transitions of the NFA (DFA) in the same way, as in the SLP representing the input text. To this end, the SLPs are locally decompressed and then recompressed in a uniform way. Furthermore, such recompression induces only small changes in the automaton, in particular, the size of the automaton remains polynomial. Using this technique it is shown that the compressed membership for NFA with compressed labels is in NP, thus confirming the conjecture of Plandowski and Rytter and extending the partial result of Lohrey and Mathissen; as it is already known, that this problem is NP-hard, we settle its exact computational complexity. Moreover, the same technique applied to the compressed membership for DFA with compressed labels yields that this problem is in P; for this problem, only trivial upper-bound PSPACE was known

Free radical scavenging and formation by multi-walled carbon nanotubes in cell free conditions and in human bronchial epithelial cells

Background: Certain multi-walled carbon nanotubes (MWCNTs) have been shown to elicit asbestos-like toxicological effects. To reduce needs for risk assessment it has been suggested that the physicochemical characteristics or reactivity of nanomaterials could be used to predict their hazard. Fibre-shape and ability to generate reactive oxygen species (ROS) are important indicators of high hazard materials. Asbestos is a known ROS generator, while MWCNTs may either produce or scavenge ROS. However, certain biomolecules, such as albumin – used as dispersants in nanomaterial preparation for toxicological testing in vivo and in vitro - may reduce the surface reactivity of nanomaterials. Methods: Here, we investigated the effect of bovine serum albumin (BSA) and cell culture medium with and without BEAS 2B cells on radical formation/scavenging by five MWCNTs, Printex 90 carbon black, crocidolite asbestos, and glass wool, using electron spin resonance (ESR) spectroscopy and linked this to cytotoxic effects measured by trypan blue exclusion assay. In addition, the materials were characterized in the exposure medium (e.g. for hydrodynamic size-distribution and sedimentation rate). Results: The test materials induced highly variable cytotoxic effects which could generally be related to the abundance and characteristics of agglomerates/aggregates and to the rate of sedimentation. All carbon nanomaterials were found to scavenge hydroxyl radicals (•OH) in at least one of the solutions tested. The effect of BSA was different among the materials. Two types of long, needle-like MWCNTs (average diameter >74 and 64.2 nm, average length 5.7 and 4.0 µm, respectively) induced, in addition to a scavenging effect, a dose-dependent formation of a unique, yet unidentified radical in both absence and presence of cells, which also coincided with cytotoxicity. Conclusions: Culture medium and BSA affects scavenging/production of •OH by MWCNTs, Printex 90 carbon black, asbestos and glass-wool. An unidentified radical is generated by two long, needle-like MWCNTs and these two CNTs were more cytotoxic than the other CNTs tested, suggesting that this radical could be related to the adverse effects of MWCNTs