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

The Plasma and Suprathermal Ion Composition (PLASTIC) investigation on the STEREO observatories

The Plasma and Suprathermal Ion Composition (PLASTIC) investigation provides the in situ solar wind and low energy heliospheric ion measurements for the NASA Solar Terrestrial Relations Observatory Mission, which consists of two spacecraft (STEREO-A, STEREO-B). PLASTIC-A and PLASTIC-B are identical. Each PLASTIC is a time-of-flight/energy mass spectrometer designed to determine the elemental composition, ionic charge states, and bulk flow parameters of major solar wind ions in the mass range from hydrogen to iron. PLASTIC has nearly complete angular coverage in the ecliptic plane and an energy range from ∼0.3 to 80 keV/e, from which the distribution functions of suprathermal ions, including those ions created in pick-up and local shock acceleration processes, are also provided

Abstracts from the NIHR INVOLVE Conference 2017

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Node-Weighted Steiner Tree and Group Steiner Tree in Planar Graphs

We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Θ(log n), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prizecollecting Steiner tree, and network-formation games. The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O(log3 n), or O(log2 n) when the host graph is a tree. We obtain an O(log npolyloglog n) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group. © 2014 ACM

Polynomial Kernels and Faster Algorithms for the Dominating Set Problem on Graphs with an Excluded Minor

Abstract. The domination number of a graph G = (V, E) is the minimum size of a dominating set U ⊆ V, which satisfies that every vertex in V \ U is adjacent to at least one vertex in U. The notion of a problem kernel refers to a polynomial time algorithm that achieves some provable reduction of the input size. Given a graph G whose domination number is k, the objective is to design a polynomial time algorithm that produces a graph G ′ whose size depends only on k, and also has domination number equal to k. Note that the graph G ′ is constructed without knowing the value of k. Problem kernels can be used to obtain efficient approximation and exact algorithms for the domination number, and are also useful in practical settings. In this paper, we present the first nontrivial result for the general case of graphs with an excluded minor, as follows. For every fixed h, given a graph G with n vertices that does not contain Kh as a topological minor, our O(n 3.5 + k O(1) ) time algorithm constructs a subgraph G ′ of G, such that if the domination number of G is k, then the domination number of G ′ is also k and G ′ has at most k c vertices, where c is a constant that depends only on h. This result is improved for graphs that do not contain K3,h as a topological minor, using a simpler algorithm that constructs a subgraph with at most ck vertices, where c is a constant that depends only on h. Our results imply that there is a problem kernel of polynomial size for graphs with an excluded minor and a linear kernel for graphs that are K3,h-minor-free. The only previous kernel results known for the dominating set problem are the existence of a linear kernel for the planar case as well as for graphs of bounded genus. Using the polynomial kernel construction, we give an O(n 3.5 + 2 O( √ k)) time algorithm for finding a dominating set of size at most k in an H-minor-free graph with n vertices. This improves the running time of the previously best known algorithm. Key words: H-minor-free graphs, degenerated graphs, dominating set problem, fixed-parameter tractable algorithms, problem kernel