19,725 research outputs found
Non-deterministic graph searching in trees
International audienceNon-deterministic graph searching was introduced by Fomin et al. to provide a unified approach for pathwidth, treewidth, and their interpretations in terms of graph searching games. Given q ≥ 0, the q-limited search number, s q (G), of a graph G is the smallest number of searchers required to capture an invisible fugitive in G, when the searchers are allowed to know the position of the fugitive at most q times. The search parameter s 0 (G) corresponds to the pathwidth of a graph G, and s ∞ (G) to its treewidth. Determining s q (G) is NP-complete for any fixed q ≥ 0 in general graphs and s 0 (T) can be computed in linear time in trees, however the complexity of the problem on trees has been unknown for any q > 0. We introduce a new variant of graph searching called restricted non-deterministic. The corresponding parameter is denoted by rs q and is shown to be equal to the non-deterministic graph searching parameter s q for q = 0, 1, and at most twice s q for any q ≥ 2 (for any graph G). Our main result is a polynomial time algorithm that computes rs q (T) for any tree T and any q ≥ 0. This provides a 2-approximation of s q (T) for any tree T , and shows that the decision problem associated to s 1 is polynomial in the class of trees. Our proofs are based on a new decomposition technique for trees which might be of independent interest
IBBT: Informed Batch Belief Trees for Motion Planning Under Uncertainty
In this work, we propose the Informed Batch Belief Trees (IBBT) algorithm for
motion planning under motion and sensing uncertainties. The original stochastic
motion planning problem is divided into a deterministic motion planning problem
and a graph search problem. We solve the deterministic planning problem using
sampling-based methods such as PRM or RRG to construct a graph of nominal
trajectories. Then, an informed cost-to-go heuristic for the original problem
is computed based on the nominal trajectory graph. Finally, we grow a belief
tree by searching over the graph using the proposed heuristic. IBBT interleaves
between batch state sampling, nominal trajectory graph construction, heuristic
computing, and search over the graph to find belief space motion plans. IBBT is
an anytime, incremental algorithm. With an increasing number of batches of
samples added to the graph, the algorithm finds motion plans that converge to
the optimal one. IBBT is efficient by reusing results between sequential
iterations. The belief tree searching is an ordered search guided by an
informed heuristic. We test IBBT in different planning environments. Our
numerical investigation confirms that IBBT finds non-trivial motion plans and
is faster compared with previous similar methods.Comment: arXiv admin note: substantial text overlap with arXiv:2110.0017
Latent Tree Language Model
In this paper we introduce Latent Tree Language Model (LTLM), a novel
approach to language modeling that encodes syntax and semantics of a given
sentence as a tree of word roles.
The learning phase iteratively updates the trees by moving nodes according to
Gibbs sampling. We introduce two algorithms to infer a tree for a given
sentence. The first one is based on Gibbs sampling. It is fast, but does not
guarantee to find the most probable tree. The second one is based on dynamic
programming. It is slower, but guarantees to find the most probable tree. We
provide comparison of both algorithms.
We combine LTLM with 4-gram Modified Kneser-Ney language model via linear
interpolation. Our experiments with English and Czech corpora show significant
perplexity reductions (up to 46% for English and 49% for Czech) compared with
standalone 4-gram Modified Kneser-Ney language model.Comment: Accepted to EMNLP 201
Faster Fully-Dynamic Minimum Spanning Forest
We give a new data structure for the fully-dynamic minimum spanning forest
problem in simple graphs. Edge updates are supported in
amortized time per operation, improving the amortized bound of
Holm et al. (STOC'98, JACM'01). We assume the Word-RAM model with standard
instructions.Comment: 13 pages, 2 figure
Search for an Immobile Hider on a Stochastic Network
Harry hides on an edge of a graph and does not move from there. Sally,
starting from a known origin, tries to find him as soon as she can. Harry's
goal is to be found as late as possible. At any given time, each edge of the
graph is either active or inactive, independently of the other edges, with a
known probability of being active. This situation can be modeled as a zero-sum
two-person stochastic game. We show that the game has a value and we provide
upper and lower bounds for this value. Finally, by generalizing optimal
strategies of the deterministic case, we provide more refined results for trees
and Eulerian graphs.Comment: 28 pages, 9 figure
Parallel Batch-Dynamic Graph Connectivity
In this paper, we study batch parallel algorithms for the dynamic
connectivity problem, a fundamental problem that has received considerable
attention in the sequential setting. The most well known sequential algorithm
for dynamic connectivity is the elegant level-set algorithm of Holm, de
Lichtenberg and Thorup (HDT), which achieves amortized time per
edge insertion or deletion, and time per query. We
design a parallel batch-dynamic connectivity algorithm that is work-efficient
with respect to the HDT algorithm for small batch sizes, and is asymptotically
faster when the average batch size is sufficiently large. Given a sequence of
batched updates, where is the average batch size of all deletions, our
algorithm achieves expected amortized work per
edge insertion and deletion and depth w.h.p. Our algorithm
answers a batch of connectivity queries in expected
work and depth w.h.p. To the best of our knowledge, our algorithm
is the first parallel batch-dynamic algorithm for connectivity.Comment: This is the full version of the paper appearing in the ACM Symposium
on Parallelism in Algorithms and Architectures (SPAA), 201
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
- …