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

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 $(e+1)\sqrt{n} < \Delta \leq n/5$

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

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

Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus, and Minor-Free Graphs

A (1 + eps)-approximate distance oracle for a graph is a data structure that supports approximate point-to-point shortest-path-distance queries. The most relevant measures for a distance-oracle construction are: space, query time, and preprocessing time. There are strong distance-oracle constructions known for planar graphs (Thorup, JACM'04) and, subsequently, minor-excluded graphs (Abraham and Gavoille, PODC'06). However, these require Omega(eps^{-1} n lg n) space for n-node graphs. We argue that a very low space requirement is essential. Since modern computer architectures involve hierarchical memory (caches, primary memory, secondary memory), a high memory requirement in effect may greatly increase the actual running time. Moreover, we would like data structures that can be deployed on small mobile devices, such as handhelds, which have relatively small primary memory. In this paper, for planar graphs, bounded-genus graphs, and minor-excluded graphs we give distance-oracle constructions that require only O(n) space. The big O hides only a fixed constant, independent of \epsilon and independent of genus or size of an excluded minor. The preprocessing times for our distance oracle are also faster than those for the previously known constructions. For planar graphs, the preprocessing time is O(n lg^2 n). However, our constructions have slower query times. For planar graphs, the query time is O(eps^{-2} lg^2 n). For our linear-space results, we can in fact ensure, for any delta > 0, that the space required is only 1 + delta times the space required just to represent the graph itself

Utility of (11)C-methionine and (11)C-donepezil for imaging of Staphylococcus aureus induced osteomyelitis in a juvenile porcine model:comparison to autologous (111)In-labelled leukocytes, (99m) Tc-DPD, and (18)F-FDG

The aim of this study was to compare (11)C-methionine and (11)C-donepezil positron emission tomography (PET) with (111)In-labeled leukocyte and (99m)Tc-DPD (Tc-99m 3,3-diphosphono-1,2-propanedicarboxylic acid) single-photon emission computed tomography (SPECT), and (18)F-fluorodeoxyglucose ((18)F-FDG) PET to improve detection of osteomyelitis. The tracers’ diagnostic utility where tested in a juvenile porcine hematogenously induced osteomyelitis model comparable to osteomyelitis in children. Five 8-9 weeks old female domestic pigs were scanned seven days after intra-arterial inoculation in the right femoral artery with a porcine strain of Staphylococcus aureus. The sequential scan protocol included Computed Tomography, (11)C-methionine and (11)C-donepezil PET, (99m)Tc-DPD and (111)In-labelled leukocytes scintigraphy, and (18)F-FDG PET. This was followed by necropsy of the pigs and gross pathology, histopathology, and microbial examination. The pigs developed a total of 24 osteomyelitic lesions, 4 lesions characterized as contiguous abscesses and pulmonary abscesses (in two pigs). By comparing the 24 osteomyelitic lesions, (18)F-FDG accumulated in 100%, (111)In-leukocytes in 79%, (11)C-methionine in 79%, (11)C-donepezil in 58%, and (99m)Tc-DPD in none. Overall, (18)F-FDG PET was superior to (111)In-leukocyte SPECT and (11)C-methionine in marking infectious lesions

Kinetic Modelling of [⁶⁸Ga]Ga-DOTA-Siglec-9 in Porcine Osteomyelitis and Soft Tissue Infections

Background: [68Ga]Ga-DOTA-Siglec-9 is a positron emission tomography (PET) radioligand for vascular adhesion protein 1 (VAP-1), a protein involved in leukocyte trafficking. The tracer facilitates the imaging of inflammation and infection. Here, we studied the pharmacokinetic modelling of [68Ga]Ga-DOTA-Siglec-9 in osteomyelitis and soft tissue infections in pigs. Methods: Eight pigs with osteomyelitis and soft tissue infections in the right hind limb were dynamically PET scanned for 60 min along with arterial blood sampling. The fraction of radioactivity in the blood accounted for by the parent tracer was evaluated with radio-high-performance liquid chromatography. One- and two-tissue compartment models were used for pharmacokinetic evaluation. Post-mortem soft tissue samples from one pig were analysed with anti-VAP-1 immunofluorescence. In each analysis, the animal&rsquo;s non-infected left hind limb was used as a control. Results: Tracer uptake was elevated in soft tissue infections but remained low in osteomyelitis. The kinetics of [68Ga]Ga-DOTA-Siglec-9 followed a reversible 2-tissue compartment model. The tracer metabolized quickly; however, taking this into account, produced more ambiguous results. Infected soft tissue samples showed endothelial cell surface expression of the Siglec-9 receptor VAP-1. Conclusion: The kinetics of [68Ga]Ga-DOTA-Siglec-9 uptake in porcine soft tissue infections are best described by the 2-tissue compartment model

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

Kinetic Modelling of [Ga-68]Ga-DOTA-Siglec-9 in Porcine Osteomyelitis and Soft Tissue Infections

Background: [Ga-68]Ga-DOTA-Siglec-9 is a positron emission tomography (PET) radioligand for vascular adhesion protein 1 (VAP-1), a protein involved in leukocyte trafficking. The tracer facilitates the imaging of inflammation and infection. Here, we studied the pharmacokinetic modelling of [Ga-68]Ga-DOTA-Siglec-9 in osteomyelitis and soft tissue infections in pigs. Methods: Eight pigs with osteomyelitis and soft tissue infections in the right hind limb were dynamically PET scanned for 60 min along with arterial blood sampling. The fraction of radioactivity in the blood accounted for by the parent tracer was evaluated with radio-high-performance liquid chromatography. One- and two-tissue compartment models were used for pharmacokinetic evaluation. Post-mortem soft tissue samples from one pig were analysed with anti-VAP-1 immunofluorescence. In each analysis, the animal's non-infected left hind limb was used as a control. Results: Tracer uptake was elevated in soft tissue infections but remained low in osteomyelitis. The kinetics of [Ga-68]Ga-DOTA-Siglec-9 followed a reversible 2-tissue compartment model. The tracer metabolized quickly; however, taking this into account, produced more ambiguous results. Infected soft tissue samples showed endothelial cell surface expression of the Siglec-9 receptor VAP-1. Conclusion: The kinetics of [Ga-68]Ga-DOTA-Siglec-9 uptake in porcine soft tissue infections are best described by the 2-tissue compartment model.</div

Kinetic Modelling of Infection Tracers [ 18

Introduction. Positron emission tomography (PET) is increasingly applied for infection imaging using [18F]FDG as tracer, but uptake is unspecific. The present study compares the kinetics of [18F]FDG and three other PET tracers with relevance for infection imaging. Methods. A juvenile porcine osteomyelitis model was used. Eleven pigs underwent PET/CT with 60-minute dynamic PET imaging of [18F]FDG, [68Ga]Ga-citrate, [11C]methionine, and/or [11C]donepezil, along with blood sampling. For infectious lesions, kinetic modelling with one- and two-tissue-compartment models was conducted for each tracer. Results. Irreversible uptake was found for [18F]FDG and [68Ga]Ga-citrate; reversible uptake was found for [11C]methionine (two-tissue model) and [11C]donepezil (one-tissue model). The uptake rate for [68Ga]Ga-citrate was slow and diffusion-limited. For the other tracers, the uptake rate was primarily determined by perfusion (flow-limited uptake). Net uptake rate for [18F]FDG and distribution volume for [11C]methionine were significantly higher for infectious lesions than for correspondingly noninfected tissue. For [11C]donepezil in pigs, labelled metabolite products appeared to be important for the analysis. Conclusions. The kinetics of the four studied tracers in infection was characterized. For clinical applications, [18F]FDG remains the first-choice PET tracer. [11C]methionine may have a potential for detecting soft tissue infections. [68Ga]Ga-citrate and [11C]donepezil were not found useful for imaging of osteomyelitis