In this paper, a geometric shortest path problem in weighted regions is discussed. An arrangement of lines A, a source s, and a target t are given. The objective is to find a weighted shortest path, πst, from s to t. Existing approximation algorithms for weighted shortest paths work within bounded regions (typically triangulated). To apply these algorithms to unbounded regions, such as arrangem...

Consider an unweighted k-nearest neighbor graph on n points that have been sampled i.i.d. from some unknown density p on R. We prove how one can estimate the density p just from the unweighted adjacency matrix of the graph, without knowing the points themselves or any distance or similarity scores. The key insights are that local differences in link numbers can be used to estimate a local funct...

This paper proposes a model of placing a liaison which forms relations to two members of different levels in a pyramid organization structure such that the communication of information between every member in the organization becomes the most efficient. For the model of adding a node of liaison which gets adjacent to a node with a depth M and its descendant with a depth N in a complete K-ary tr...

We present a novel graph embedding to speed-up distance-range and k-nearest neighbor queries on static and/or dynamic objects located on a (weighted) graph that is applicable also for very large networks. Our method extends an existing embedding called reference node embedding which can be used to compute accurate lower and upper bounding filters for the true shortest path distance. In order to...

We consider the problem of computing all farthest neighbors (and the diameter) of a given set of n points in the presence of highways and obstacles in the plane. When traveling on the plane, travelers may use highways for faster movement and must avoid all obstacles. We present an efficient solution to this problem based on knowledge from earlier research on shortest path computation. Our algor...

This paper proposes a model of placing a liaison which forms relations to two members of different levels in a pyramid organization structure such that the communication of information between every member in the organization becomes the most efficient. For the model of adding a node of liaison which gets adjacent to a node with a depth M and its descendant with a depth N in a complete K-ary tr...

We give algorithmic results for combinatorial problems with cost arrays possessing certain algebraic Monge properties. We extend Monge-array results for two shortest path problems to a general algebraic setting, with values in an ordered commutative semigroup, if the semigroup operator is strictly compatible with the order relation. We show how our algorithms can be modified to solve bottleneck...

We study the following problem k-constrained shortest path problem: given an acyclic directed graph D = (V,E) with arc weights ci,j , (i, j) ∈ E, two nodes s and t and an integer k, nd a shortest st-path containing at most k arcs. An important application of the problem in linear curve approximation is discussed. Vertices and edges of associated polytopes are determined, and integrality of thes...