Improving the dilation of a metric graph by adding edges

Most of the literature on spanners focuses on building the graph from scratch. This paper instead focuses on adding edges to improve an existing graph. A major open problem in this field is: given a graph embedded in a metric space, and a budget of k edges, which k edges do we add to produce a minimum-dilation graph? The special case where k=1 has been studied in the past, but no major breakthroughs have been made for k > 1. We provide the first positive result, an O(k)-approximation algorithm that runs in O(n^3 \log n) time

Feed-links for network extensions

Road network data is often incomplete, making it hard to perform network analysis. This paper discusses the problem of extending partial road networks with reasonable links, using the concept of dilation (also known as crow flight conversion coefficient). To this end, we study how to connect a point (relevant location) inside a polygon (face of the known part of the road network) to the boundary so that the dilation from that point to any point on the boundary is not too large. We provide algorithms and heuristics, and give a computational and experimental analysis

Dilation-optimal edge deletion in polygonal cycles

Consider a geometric network G in the plane. The dilation between any two vertices x and y in G is the ratio of the shortest path distance between x and y in G to the Euclidean distance between them. The maximum dilation over all pairs of vertices in G is called the dilation of G. In t

Dilation-optimal edge deletion in polygonal cycles

Consider a geometric network G in the plane. The dilation between any two vertices x and y in G is the ratio of the shortest path distance between x and y in G to the Euclidean distance between them. The maximum dilation over all pair of vertices in G is called the dilation of G. In this paper, a randomized algorithm is presented which, when given a polygonal cycle C on n vertices in the plane, computes in O(n log 3 n) expected time, the edge of C whose removal results in a polygonal path of smallest possible dilation. It is also shown that the edge whose removal gives a polygonal path of largest possible dilation can be computed in O(n log n) time. If C is a convex polygon, the running time for the latter problem becomes O(n). Finally, it is shown that a (1 − ǫ)-approximation to the dilation of all the path C \ {e}, for all edge e of C, can be computed in O(n log n) total time.

DILATION-OPTIMAL EDGE DELETION IN POLYGONAL CYCLES

Consider a geometric network G in the plane. The dilation between any two vertices x and y in G is the ratio of the shortest path distance between x and y in G to the Euclidean distance between them. The maximum dilation over all pairs of vertices in G is called the dilation of G. In this paper, a randomized algorithm is presented which, when given a polygonal cycle C on n vertices in the plane, computes in O(n log(3) n) expected time, the edge of C whose removal results in a polygonal path of smallest possible dilation. It is also shown that the edge whose removal gives a polygonal path of largest possible dilation can be computed in O(n log n) time. If C is a convex polygon, the running time for the latter problem becomes O(n). Finally, it is shown that a(1 - epsilon)-approximation to the dilation of every path C\{e}, for all edges e of C, can be computed in O(n log n) total time.X111sciescopu

Analysing trajectory similarity and improving graph dilation

In this thesis, we focus on two topics in computational geometry. The first topic is analysing trajectory similarity. A trajectory tracks the movement of an object over time. A common way to analyse trajectories is by finding similarities. The Fr\'echet distance is a similarity measure that has gained popularity in the theory community, since it takes the continuity of the curves into account. One way to analyse trajectories using the Fr\'echet distance is to cluster trajectories into groups of similar trajectories. For vehicle trajectories, another way to analyse trajectories is to compute the path on the underlying road network that best represents the trajectory. The second topic is improving graph dilation. Dilation measures the quality of a network in applications such as transportation and communication networks. Spanners are low dilation graphs with not too many edges. Most of the literature on spanners focuses on building the graph from scratch. We instead focus on adding edges to improve the dilation of an existing graph