Given S1, a finite set of points in the plane, we define a sequence of point sets Si as follows: ... more Given S1, a finite set of points in the plane, we define a sequence of point sets Si as follows: With Si already determined, let Li be the set of all the line segments connecting pairs of points of S i j=1 Sj, and let Si+1 be the set of intersection points of those line segments in Li, which cross but do not overlap. We show that with the exception of some starting configurations the set of all crossing points S ∞ i=1 Si is dense in a particular subset of the plane with nonempty interior. This region is the intersection of all closed half planes which contain all but at most one point from S1.
Indirect reciprocity describes a class of reputation-based mechanisms which may explain the preva... more Indirect reciprocity describes a class of reputation-based mechanisms which may explain the prevalence of cooperation in large groups where partners meet only once. The first model for which this has been demonstrated was the image scoring mechanism. But analytical work on the simplest possible case, the binary scoring model, has shown that even small errors in implementation destabilize any cooperative regime. It has thus been claimed that for indirect reciprocity to stabilize cooperation, assessments of reputation must be based on higher-order information. Is indirect reciprocity relying on first-order information doomed to fail? We use a simple analytical model of image scoring to show that this need not be the case. Indeed, in the general image scoring model the introduction of implementation errors has just the opposite effect as in the binary scoring model: it may stabilize instead of destabilize cooperation.
Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we ca... more Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance.
A pride of lions are prowling among the vertices and edges of an n × n grid. If their paths are k... more A pride of lions are prowling among the vertices and edges of an n × n grid. If their paths are known in advance, is it possible to design a safe path for a man that avoids all lions, assuming that man and lion move at the same speed? In their recent paper [4], Dumitrescu et al. employed probabilistic arguments to show that O ( √ n) lions can always be avoided. They raised the question if it is also possible to avoid O(n) lions. Using a proof technique quite different from theirs, we give a positive answer. Even ⌊ n⌋ lions can be avoided in dimen-2 sion 2. However, there is no escaping from, by order of magnitude, Θ ( nd−1 √ ) lions on the d-dimensional grid.
Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we ca... more Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set, we would like to know the smallest possible dilation of any graph that contains the given points. In this paper we prove that a dilation of 1.678 is always sufficient, and that #/2 1.570 ... is sometimes necessary in order to accommodate a finite set of points.
On the geometric dilation of curves and point sets
Let G be an embedded planar graph whose edges are curves. The detour between two points u and v (... more Let G be an embedded planar graph whose edges are curves. The detour between two points u and v (on edges or vertices) of G is the ratio between the shortest path in G between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebbers-Baumann, Grüne and Klein have recently shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation δ ≥ π/2 ≈ 1.57. We prove a stronger lower bound δ ≥ (1 + 10 −11)π/2 by relating graphs with small dilation to a problem of packing and covering the plane by circular disks. 1
Spanning Ratio and Maximum Detour of Rectilinear Paths in the L1 Plane
The spanning ratio and maximum detour of a graph G embedded in a metric space measure how well G ... more The spanning ratio and maximum detour of a graph G embedded in a metric space measure how well G approximates the minimum complete graph containing G and metric space, respectively. In this paper we show that computing the spanning ratio of a rectilinear path P in L1 space has a lower bound of Ω(n log n) in the algebraic computation tree model and describe a deterministic O(n log² n) time algorithm. On the other hand, we give a deterministic O(n log² n) time algorithm for computing the maximum detour of a rectilinear path P in L1 space and obtain an O(n) time algorithm when P is a monotone rectilinear path.
Let G be an embedded planar graph whose edges may be curves. The detour between two points, p and... more Let G be an embedded planar graph whose edges may be curves. The detour between two points, p and q (on edges or vertices) of G, is the ratio between the shortest path in G between p and q and their Euclidean distance. The supremum over all pairs of points of all these ratios is called the geometric dilation of G. Our research is motivated by the problem of designing graphs of low dilation. We provide a characterization of closed curves of constant halving distance (i.e., curves for which all chords dividing the curve length in half are of constant length) which are useful in this context. We then relate the halving distance of curves to other geometric quantities such as area and width. Among others, this enables us to derive a new upper bound on the geometric dilation of closed curves, as a function of D/w, where D and w are the diameter and width, respectively. We further give lower bounds on the geometric dilation of polygons with n sides as a function of n. Our bounds are tight...
Let S be a set of points in the plane. What is the minimum possible dilation of all plane graphs ... more Let S be a set of points in the plane. What is the minimum possible dilation of all plane graphs that contain S? Even for a set S as simple as five points evenly placed on the circle, this question seems hard to answer; it is not even clear if there exists a lower bound> 1. In this paper we provide the first upper and lower bounds for the embedding problem. 1. Each finite point set can be embedded into the vertex set of a finite triangulation of dilation ≤ 1.1247. 2. Each embedding of a closed convex curve has dilation ≥ 1.00157. 3. Let P be the plane graph that results from intersecting n infinite families of equidistant, parallel lines in general position. Then the vertex set of P has dilation ≥ 2 / √ 3 ≈ 1.1547.
Let P be a simple polygon in R² with n vertices. The detour of P between two points, p, q P , is ... more Let P be a simple polygon in R² with n vertices. The detour of P between two points, p, q P , is the length of a shortest path contained in P and connecting p to q, divided by the distance of these points. The detour of the whole polygon is the maximum detour between any two points in P . We first analyze properties of pairs of points with maximum detour. Next, we
Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (... more Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the ratio between the length of a shortest path connecting p and q in G and their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebbers-Baumann, Grüne and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation δ ≥ π/2 ≈ 1.57. They conjectured that the lower bound is not tight. We use new ideas like the halving pair transformation, a disk packing result and arguments from convex geometry, to prove this conjecture. The lower bound is improved to (1 + 10 −11)π/2. The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H. Additionally, we analyze curves of constant halving distance (h = H),...
Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (... more Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the length of a shortest path connecting p and q in G divided by their Euclidean distance |pq|. The
On the Density of Iterated Line Segment Intersections
Given S1, a finite set of points in the plane, we define a sequence of point sets Si as follows: ... more Given S1, a finite set of points in the plane, we define a sequence of point sets Si as follows: With Si already determined, let Li be the set of all the line segments connecting pairs of points of Si j=1 Sj, and let Si+1 be the set of intersection points of those line segments in Li, which cross but do not overlap. We show that with the exception of some starting configurations the set of all crossing points S∞ i=1 Si is dense in a particular subset of the plane with nonempty interior. This region is the intersection of all closed half planes which contain all but one point from S1.
Let G be a geometric graph in the plane whose edges may be curves. For two arbitrary points on it... more Let G be a geometric graph in the plane whose edges may be curves. For two arbitrary points on its edges, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set S, we would like to know the smallest possible dilation of any graph that contains the given points on its edges. We call this infimum the dilation of S and denote it by δ(S). The main results of this thesis are • a general upper bound to the dilation of any finite point set S, δ(S) < 1.678 • a lower bound for a specific set, δ(P)> (1 + 10 −11)π/2 ≈ 1.571 In order to achieve these results, we first consider closed curves. Their dilation depends on the halving pairs, pairs of points which divide the closed curve in two parts of equal length. In particular the distance between the two points is essential, the halving distance. A transformation technique based on halving pairs, th...
Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (... more Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the length of a shortest path connecting p and q in G divided by their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebbers
Let P be a simple polygon in 2 with n vertices. The detour of P between two points, p, q ∈ P , is... more Let P be a simple polygon in 2 with n vertices. The detour of P between two points, p, q ∈ P , is the length of a shortest path contained in P and connecting p to q, divided by the distance of these points. The detour of the whole polygon is the maximum detour between any two points in P . We first analyze properties of pairs of points with maximum detour. Next, we use these properties to achieve a deterministic O(n)-algorithm for computing the maximum Euclidean detour and a deterministic O(n log n)-algorithm which calculates a (1+ε)approximation. Finally, we consider the special case of monotone rectilinear polygons. Their L-detour can be computed in time O(n).
Let G be an embedded planar graph whose edges are curves. The detour between two points u and v (... more Let G be an embedded planar graph whose edges are curves. The detour between two points u and v (on edges or vertices) of G is the ratio between the shortest path in G between u and v and their Euclidean distance. The maximum detour over all pairs of points is called
Let G be a geometric graph in the plane whose edges may be curves. For two arbitrary points on it... more Let G be a geometric graph in the plane whose edges may be curves. For two arbitrary points on its edges, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set S, we would like to know the smallest possible dilation of any graph that contains the given points on its edges. We call this infimum the dilation of S and denote it by δ(S). The main results of this thesis are • a general upper bound to the dilation of any finite point set S, δ(S) < 1.678 • a lower bound for a specific set, δ(P ) > (1 + 10−11)π/2 ≈ 1.571 In order to achieve these results, we first consider closed curves. Their dilation depends on the halving pairs, pairs of points which divide the closed curve in two parts of equal length. In particular the distance between the two points is essential, the halving distance. A transformation technique based on halving pairs, t...
Indirect reciprocity describes a class of reputation-based mechanisms which may explain the preva... more Indirect reciprocity describes a class of reputation-based mechanisms which may explain the prevalence of cooperation in groups where partners meet only once. The first model for which this has analytically been shown was the binary image scoring mechanism, where one's reputation is only based on one's last action. But this mechanism is known to fail if errors in implementation occur. It has thus been claimed that for indirect reciprocity to stabilize cooperation, reputation assessments must be of higher order, i.e. contingent not only on past actions, but also on the reputations of the targets of these actions. We show here that this need not be the case. A simple image scoring mechanism where more than just one past action is observed provides ample possibilities for stable cooperation to emerge even under substantial rates of implementation errors.
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Papers by Ansgar Grüne