In a geometric graph, G, the stretch factor between two vertices, u and w, is the ratio between the Euclidean length of the shortest path from u to w in G and the Euclidean distance between u and w. The average stretch factor of G is the average stretch factor taken over all pairs of vertices in G. We show that for any set, V , of n points in Rd , there exists a geometric graph with vertex set ...

Suppose n nodes with n0 acquaintances per node are randomly deployed in a two-dimensional Euclidean space with the geographic restriction that each pair of nodes can exchange information between them directly only if the distance between them is at most r, the acquaintanceship between nodes form a random graph, while the physical communication links constitute a random geometric graph. To get a...

Let C be a compact and convex set in the plane that contains the origin in its interior, and let S be a finite set of points in the plane. The Delaunay graph DGC(S) of S is defined to be the dual of the Voronoi diagram of S with respect to the convex distance function defined by C. We prove that DGC(S) is a t-spanner for S, for some constant t that depends only on the shape of the set C. Thus, ...

The Stochastic Eulerian Tour Problem (SETP) seeks the Eulerian tour of minimum expected length on an undirected Eulerian graph, when demand on the arcs that have to be serviced is probabilistic. In an earlier paper, we have shown that the SETP is NP-hard and derived several properties of the optimal tour. In this paper, we develop three constructive heuristics for the SETP. The first two are gr...

The Euclidean Minimum Spanning Tree problem is to decide whether a given graph G = (P,E) on a set of points in the two dimensional plane is a minimum spanning tree with respect to the Euclidean distance. Czumaj, Sohler, and Ziegler [2] gave a 1-sided-error non-adaptive property-tester for this task of query complexity Õ( √ n). We show that every non-adaptive (not necessarily 1-sided-error) prop...

Lyons, Peres and Schramm have shown that minimal spanning forests on randomly weighted lattices exhibit a critical geometry in the sense that adding or deleting only a small number of edges results in a radical change of percolation properties. We show that these results can be extended to a Euclidean setting by considering families of stationary superand subgraphs that approximate the Euclidea...

Two-sample tests for multivariate and especially for non-Euclidean data are not fully explored. This paper presents a novel test based on a similarity graph constructed on the pooled observations from the two samples. It can be applied to multivariate data and non-Euclidean data as long as a dissimilarity measure on the sample space can be defined, which can usually be provided by domain expert...

Point pattern matching is an important topic of computer vision and pattern recognition. In this paper, we propose a point pattern matching algorithm for two planar point sets under Euclidean transform.We view a point set as a complete graph, establish the relation between the point set and the complete graph, and solve the point pattern matching problem by finding congruent complete graphs. Ex...

Each of the twenty amino acids are represented by a graph-theoretic tree. Biomolecular descriptors are defined by graphical invariants and are used to determine similarity based on the corresponding Euclidean distances. The results verify that graph-theoretic measures independent of biochemical and biophysical descriptors are valid measures of similarity of amino acids. These results may prove ...

Two-sample tests for multivariate data and especially for non-Euclidean data are not well explored. This paper presents a novel test statistic based on a similarity graph constructed on the pooled observations from the two samples. It can be applied to multivariate data and non-Euclidean data as long as a dissimilarity measure on the sample space can be defined, which can usually be provided by...