Quantum Algorithm for Triangle Finding in Sparse Graphs

This paper presents a quantum algorithm for triangle finding over sparse graphs that improves over the previous best quantum algorithm for this task by Buhrman et al. [SIAM Journal on Computing, 2005]. Our algorithm is based on the recent $\tilde O(n^{5/4})$-query algorithm given by Le Gall [FOCS 2014] for triangle finding over dense graphs (here $n$ denotes the number of vertices in the graph). We show in particular that triangle finding can be solved with $O(n^{5/4-\epsilon})$ queries for some constant $\epsilon>0$ whenever the graph has at most $O(n^{2-c})$ edges for some constant $c>0$.Comment: 13 page

Diffusion Limited Aggregation on a Cylinder

We consider the DLA process on a cylinder G x N. It is shown that this process "grows arms", provided that the base graph G has small enough mixing time. Specifically, if the mixing time of G is at most (log|G|)^(2-\eps), the time it takes the cluster to reach the m-th layer of the cylinder is at most of order m |G|/loglog|G|. In particular we get examples of infinite Cayley graphs of degree 5, for which the DLA cluster on these graphs has arbitrarily small density. In addition, we provide an upper bound on the rate at which the "arms" grow. This bound is valid for a large class of base graphs G, including discrete tori of dimension at least 3. It is also shown that for any base graph G, the density of the DLA process on a G-cylinder is related to the rate at which the arms of the cluster grow. This implies, that for any vertex transitive G, the density of DLA on a G-cylinder is bounded by 2/3.Comment: 1 figur

A novel uncultured marine cyanophage lineage with lysogenic potential linked to a putative marine Synechococcus 'relic' prophage

Marine cyanobacteria are important contributors to primary production in the ocean and their viruses (cyanophages) affect the ocean microbial communities. Despite reports of lysogeny in marine cyanobacteria, a genome sequence of such temperate cyanophages remains unknown although genomic analysis indicate potential for lysogeny in certain marine cyanophages. Using assemblies from Red Sea and Tara Oceans metagenomes, we recovered genomes of a novel uncultured marine cyanophage lineage, which contain, in addition to common cyanophage genes, a phycobilisome degradation protein NblA, an integrase and a split DNA polymerase. The DNA polymerase forms a monophyletic clade with a DNA polymerase from a genomic island in Synechococcus WH8016. The island contains a relic prophage that does not resemble any previously reported cyanophage but shares several genes with the newly identified cyanophages reported here. Metagenomic recruitment indicates that the novel cyanophages are widespread, albeit at low abundance. Here, we describe a novel potentially lysogenic cyanophage family, their abundance and distribution in the marine environment

Universal Word Segmentation: Implementation and Interpretation

Word segmentation is a low-level NLP taskt hat is non-trivial for a considerable number of languages. In this paper, we present asequence tagging framework and apply it to word segmentation for a wide range of languages with different writing systems and typological characteristics. Additionally, we investigate the correlations between various typological factors and word segmentation accuracy. The experimental results indicate that segmentation accuracy is positively related to word boundary markers and negatively to the number of unique non-segmental terms. Based on the analysis, we design a small set of language-specific settings and extensively evaluate the segmentation system on the Universal Dependencies datasets. Our model obtains state-of-the-art accuracies on all the UD languages. It performs substantially better on languages that are non-trivial to segment, such as Chinese, Japanese, Arabic and Hebrew, when compared to previous work

Enumerating Cyclic Orientations of a Graph

Acyclic and cyclic orientations of an undirected graph have been widely studied for their importance: an orientation is acyclic if it assigns a direction to each edge so as to obtain a directed acyclic graph (DAG) with the same vertex set; it is cyclic otherwise. As far as we know, only the enumeration of acyclic orientations has been addressed in the literature. In this paper, we pose the problem of efficiently enumerating all the \emph{cyclic} orientations of an undirected connected graph with $n$ vertices and $m$ edges, observing that it cannot be solved using algorithmic techniques previously employed for enumerating acyclic orientations.We show that the problem is of independent interest from both combinatorial and algorithmic points of view, and that each cyclic orientation can be listed with $\tilde{O}(m)$ delay time. Space usage is $O(m)$ with an additional setup cost of $O(n^2)$ time before the enumeration begins, or $O(mn)$ with a setup cost of $\tilde{O}(m)$ time

Extended h-Index Parameterized Data Structures for Computing Dynamic Subgraph Statistics

We present techniques for maintaining subgraph frequencies in a dynamic graph, using data structures that are parameterized in terms of h, the h-index of the graph. Our methods extend previous results of Eppstein and Spiro for maintaining statistics for undirected subgraphs of size three to directed subgraphs and to subgraphs of size four. For the directed case, we provide a data structure to maintain counts for all 3-vertex induced subgraphs in O(h) amortized time per update. For the undirected case, we maintain the counts of size-four subgraphs in O(h^2) amortized time per update. These extensions enable a number of new applications in Bioinformatics and Social Networking research

Exact Weight Subgraphs and the k-Sum Conjecture

We consider the Exact-Weight-H problem of finding a (not necessarily induced) subgraph H of weight 0 in an edge-weighted graph G. We show that for every H, the complexity of this problem is strongly related to that of the infamous k-Sum problem. In particular, we show that under the k-Sum Conjecture, we can achieve tight upper and lower bounds for the Exact-Weight-H problem for various subgraphs H such as matching, star, path, and cycle. One interesting consequence is that improving on the O(n^3) upper bound for Exact-Weight-4-Path or Exact-Weight-5-Path will imply improved algorithms for 3-Sum, 5-Sum, All-Pairs Shortest Paths and other fundamental problems. This is in sharp contrast to the minimum-weight and (unweighted) detection versions, which can be solved easily in time O(n^2). We also show that a faster algorithm for any of the following three problems would yield faster algorithms for the others: 3-Sum, Exact-Weight-3-Matching, and Exact-Weight-3-Star

Efficient enumeration of graph orientations with sources

International audienceAn orientation of an undirected graph is obtained by assigning a direction to each of its edges. It is called cyclic when a directed cycle appears, and acyclic otherwise. We study efficient algorithms for enumerating the orientations of an undirected graph. To get the full picture, we consider both the cases of acyclic and cyclic orientations, under some rules specifying which nodes are the sources (i.e. their incident edges are all directed outwards). Our enumeration algorithms use linear space and provide new bounds for the delay, which is the maximum elapsed time between the output of any two consecutively listed solutions. We obtain a delay of O(m) for acyclic orientations and ˜Oand˜ and˜O(m) for cyclic ones. When just a single source is specified, these delays become O(m · n) and O(m · h + h 3), respectively, where h is the girth of the graph without the given source. When multiple sources are specified, the delays are the same as in the single source case.