Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition

We provide efficient constant factor approximation algorithms for the problems of finding a hierarchical clustering of a point set in any metric space, minimizing the sum of minimimum spanning tree lengths within each cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can also be used to provide a pants decomposition, that is, a set of disjoint simple closed curves partitioning the plane minus the input points into subsets with exactly three boundary components, with approximately minimum total length. In the Euclidean case, these curves are squares; in the hyperbolic case, they combine our Euclidean square pants decomposition with our tree clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now Lemma 5.2, as the previous proof was erroneou

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

Adjacency labeling schemes and induced-universal graphs

We describe a way of assigning labels to the vertices of any undirected graph on up to $n$ vertices, each composed of $n/2+O(1)$ bits, such that given the labels of two vertices, and no other information regarding the graph, it is possible to decide whether or not the vertices are adjacent in the graph. This is optimal, up to an additive constant, and constitutes the first improvement in almost 50 years of an $n/2+O(\log n)$ bound of Moon. As a consequence, we obtain an induced-universal graph for $n$-vertex graphs containing only $O(2^{n/2})$ vertices, which is optimal up to a multiplicative constant, solving an open problem of Vizing from 1968. We obtain similar tight results for directed graphs, tournaments and bipartite graphs

Kohlenstoffbildung auf Nickel und Nickel-Kupfer-Legierungskatalysatoren

Equilibrium, kinetic and morphological studies of carbon formation in CH4+H2, CO, and CO+H2 gases on silica supported nickel and nickel-copper catalysts are reviewed. The equilibrium deviates in all cases from graphite equilibrium and more so in CO+CO2 than in CH4+H2. A kinetic model based on information from surface science results with chemisorption of CH4 and possibly also the first dehydrogenation step as rate controlling describes carbon formation on nickel catalyst in CH4+H2 well. The kinetics of carbon formation in CO and CO+H2 gases are in agreement with CO disproportionation as rate determining step. The presence of hydrogen influences strongly the chemisorption of CO. Carbon filaments are formed when hydrogen is present in the gas while encapsulating carbon dominates in pure CO. Small amounts of Cu alloying promotes while larger amounts (Cu : Ni ≥ 0.1) inhibits carbon formation and changes the morphology of the filaments ("octopus" carbon formation). Adsorption induced nickel segregation changes the kinetics of the alloy catalysts at high carbon activities. Modifications suggested in some very recent papers on the basis of new results are also briefly discussed.Center for Surface Reactivity

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

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

Ramified rectilinear polygons: coordinatization by dendrons

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most 3. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group $D_4$), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon.Comment: 27 pages, 6 figure