Computing the Detour of Polygons
نویسندگان
چکیده
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).
منابع مشابه
A SIMPLE ALGORITHM FOR COMPUTING DETOUR INDEX OF NANOCLUSTERS
Let G be the chemical graph of a molecule. The matrix D = [dij ] is called the detour matrix of G, if dij is the length of longest path between atoms i and j. The sum of all entries above the main diagonal of D is called the detour index of G. In this paper, a new algorithm for computing the detour index of molecular graphs is presented. We apply our algorithm on copper and silver nanoclusters ...
متن کاملOn edge detour index polynomials
The edge detour index polynomials were recently introduced for computing the edge detour indices. In this paper we find relations among edge detour polynomials for the 2-dimensional graph of $TUC_4C_8(S)$ in a Euclidean plane and $TUC4C8(S)$ nanotorus.
متن کاملDetour Monophonic Graphoidal Covering Number of Corona Product Graph of Some Standard Graphs with the Wheel
A chord of a path $P$ is an edge joining two non-adjacent vertices of $P$. A path $P$ is called a monophonic path if it is a chordless path. A longest $x-y$ monophonic path is called an $x-y$ detour monophonic path. A detour monophonic graphoidal cover of a graph $G$ is a collection $psi_{dm}$ of detour monophonic paths in $G$ such that every vertex of $G$ is an internal vertex of at most on...
متن کاملThe explicit relation among the edge versions of detour index
The vertex version of detour index was defined during the works on connected graph in chemistry. The edge versions of detour index have been introduced ecently. In this paper, the explicit relations among edge versions of detour index have been declared and due to these relations, we compute the edge detour indices for some well-known graphs.
متن کاملComputing the Maximum Detour and Spanning Ratio of Planar Paths, Trees, and Cycles
The maximum detour and spanning ratio of an embedded graph G are values that measure how well G approximates Euclidean space and the complete Euclidean graph, respectively. In this paper we describe O(n logn) time algorithms for computing the maximum detour and spanning ratio of a planar polygonal path. These algorithms solve open problems posed in at least two previous works [5,10]. We also ge...
متن کامل