Christian Icking

The Polygon Exploration Problem

We present an on-line strategy that enables a mobile robot with vision to explore an unknown simple polygon. We prove that the resulting tour is less than 26.5 times as long as the shortest watchman tour that could be computed off-line. Our analysis is doubly founded on a novel geometric structure called angle hull. Let D be a connected region inside a simple polygon, P. We define the angle hull of D,

THE TWO GUARDS PROBLEM

{\cal AH}(D)

Searching for the kernel of a polygon—a competitive strategy

, to be the set of all points in P that can see two points of D at a right angle. We show that the perimeter of

Competitive searching in a generalized street

{\cal AH}(D)

Voronoi Diagram for services neighboring a highway

cannot exceed in length the perimeter of D by more than a factor of 2. This upper bound is tight.

Smallest Color-Spanning Objects

Given a simple polygon in the plane with two distinguished vertices, s and g, is it possible for two guards to simultaneously walk along the two boundary chains from s to g in such a way that they are always mutually visible? We decide this question in time O (n log n) and in linear space, where n is the number of edges of the polygon. Moreover, we compute a walk of minimum length within time O(n log n+k), where k is the size of the output, and we prove that this is optimal.

The two guards problem

We present a competitive strategy for walking into the kernel of an initially unknown star-shaped polygon. From an arbitrary start point, s, within the polygon, our strategy finds a path to the closest kernel point, k, whose length does not exceed 5:3331...times the distance from s to k. This is complemented by a general lower bound of v2. Our analysis relies on a result about a new and interesting class of curves which are self-approaching in the following sense. For any three consecutive points a, b, c on the curve the point b is closer to c than a to c. We show a tight upper bound of 5:3331... for the length of a self-approaching curve over the distance between its endpoints.

On the Competitive Complexity of Navigation Tasks

We consider the problem of a robot which has to find a path in an unknown simple polygon from one point <italic>s</italic> to another point <italic>t</italic>, based only on what it has seen so far. A <italic>Street</italic> is a polygon for which the two boundary chains from <italic>s</italic> to <italic>t</italic> are mutually weakly visible, and the set of streets was the only class of polygons for which a competitive search algorithm was known. We define a new, strictly larger class of polygons, called <italic>generalized streets</italic> or <inline-equation> <f> <sc>G</sc></f> </inline-equation>-streets which are characterized by the property that every point on the boundary of a <inline-equation> <f> <sc>G</sc></f> </inline-equation>-street is visible from a point on a horizontal line segment connecting the two boundary chains from <italic>s</italic> to <italic>t</italic>. We present an on-line strategy for a robot placed at <italic>s</italic> to find <italic>t</italic> in an unknown rectilinear <inline-equation> <f> <sc>G</sc></f> </inline-equation>-street; the length of the path created is at most 9 times the length of the shortest path in the <italic>L</italic><subscrpt>1</subscrpt> metric. This is optimal since we show that no strategy can achieve a smaller competitive factor for all rectilinear <inline-equation> <f> <sc>G</sc></f> </inline-equation>-streets. Compared to the <italic>L</italic><subscrpt>2</subscrpt>-shortest path, the strategy is 9.06-competitive which leaves only a very small gap to the lower bound of 9.

Self-Approaching Curves

We are given a transportation line where displacements happen at a bigger speed than in the rest of the plane. A shortest time path is a path between two points which takes less than or equal time to any other. We consider the time to follow a shortest time path to be the time distance between the two points. In this paper, we give a simple algorithm for computing the Time Voronoi Diagram, that is, the Voronoi Diagram of a set of points using the time distance.

Prescont: Predicting protein‐protein interfaces utilizing four residue properties

Motivated by questions in location planning, we show for a set of colored point sites in the plane how to compute the smallest-- by perimeter or area--axis-parallel rectangle and the narrowest strip enclosing at least one site of each color.