The on-line nearest-neighbour graph on a sequence of uniform random points in (0, 1)d (d ∈ N) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge length of this graph, with weight exponent α ∈ (0, d/2), we prove a central limit theorem (in the large-sample limit), including an expression for the limiting variance. In contrast, we...

Given a set of n points (nodes) on a line and a set of m weighted intervals defined on the nodes, we consider a particular dynamic programming (DP) problem on these intervals. If the weight function of the DP has convex or concave property, we can solve this DP problem efficiently by using matrix searching in Monge matrices, together with a new query data structure, which we call the consecutiv...

We present an image filter that transforms an input line drawing into an image of a fractured surface, where the cracks echo the input drawing. The basis of our algorithm is the Voronoi diagram of a weighted graph, where the distance between nodes is path cost in the graph. Modifying the edge costs gives us control over the placement of region boundaries; we interpret region boundaries as crack...

Clustering techniques have proven to be useful in reducing the complexity of large networks. We present a method for drawing a clustered graph in 2.5D, given the clusters of the graph are connected in a planar way. The main contribution is to develop a weighted version of the well known 2D straight line drawing algorithm of de Fraysseix, Pach and Pollack [5] for planar graphs. This version allo...

A graph is called a weighted graph when each edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d(v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H . In this paper, we give a new sufficient condition for a weighted graph to...

We construct a quantum searching model of signed edge driven by walk. The time evolution operator this walk provides weighted adjacency matrix induced the assignment sign to each edge. This can be regarded as so-called coloring. Then an application, under arbitrary coloring which gives matching on complete graph $$n+1$$ vertices we consider search colored from set graph. show that finds within...

The diameter of a connected graph $G$, denoted by $diam(G)$, is the maximum distance between any pair of vertices of $G$. Let $L(G)$ be the line graph of $G$. We establish necessary and sufficient conditions under which for a given integer $k geq 2$, $diam(L(G)) leq k$.

this paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``advanced school and conference on homological and geometrical methods in representation theory'' at the international centre for theoretical physics (ictp), miramare - trieste, italy, 11-29 january 2010. we show how to associate to a triple of positive integer...