GEODESIC DISKS AND CLUSTERING IN A SIMPLE POLYGON

Let P be a simple polygon of n vertices and let S be a set of N points lying in the interior of P. A geodesic diskGD(p,r) with center p and radius r is the set of points in P that have a geodesic distance ≤ r from p (where the geodesic distance is the length of the shortest polygonal path connection that lies in P). In this paper we present an output sensitive algorithm for finding all N geodesic disks centered at the points of S, for a given value of r. Our algorithm runs in [Formula: see text] time, for some constant c and output size k. It is the basis of a cluster reporting algorithm where geodesic distances are used.

THE CRAB: A CONNECTED FRACTILE OF INFINITE CONNECTIVITY

In the plane IR2, let A0 be the unit interval on the x-axis, and let A(1) be the polygonal path with nodes (0, 0), [Formula: see text], (½, 0), [Formula: see text], (1, 0). Let S be the operator which, applied to a segment B(0) in IR2, replaces it by a polygonal path B(1) = SB(0), a similar copy of A(1), but with the same endpoints as B(0). Denote by S(n) the n-th iterate of S. The limit set (with respect to the Hausdorff metric) A(∞) = lim n → ∞ S(n)A(0) is a space-filling curve which is the closure of its interior and the union of four half-size copies of itself, intersecting only in their boundaries. Although A(∞) is of infinite connectivity, it is a tile tessellating the plane. It is related to the set of Eisenstein fractions and has a boundary of Hausdorff dimension [Formula: see text]

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 - ϵ)-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.