Rooted Uniform Monotone Minimum Spanning Trees
نویسندگان
چکیده
We study the construction of the minimum cost spanning geometric graph of a given rooted point set P where each point of P is connected to the root by a path that satisfies a given property. We focus on two properties, namely the monotonicity w.r.t. a single direction (y-monotonicity) and the monotonicity w.r.t. a single pair of orthogonal directions (xy-monotonicity). We propose algorithms that compute the rooted y-monotone (xy-monotone) minimum spanning tree of P in O(|P | log |P |) (resp. O(|P | log |P |)) time when the direction (resp. pair of orthogonal directions) of monotonicity is given, and in O(|P | log |P |) time when the optimum direction (resp. pair of orthogonal directions) has to be determined. We also give simple algorithms which, given a rooted connected geometric graph, decide if the root is connected to every other vertex by paths that are all monotone w.r.t. the same direction (pair of orthogonal directions).
منابع مشابه
Angle-monotone Paths in Non-obtuse Triangulations
We reprove a result of Dehkordi, Frati, and Gudmundsson: every two vertices in a non-obtuse triangulation of a point set are connected by an angle-monotone path— an xy-monotone path in an appropriately rotated coordinate system. We show that this result cannot be extended to angle-monotone spanning trees, but can be extended to boundary-rooted spanning forests. The latter leads to a conjectural...
متن کاملLecture notes on “ Analysis of Algorithms ” : Directed Minimum Spanning Trees ( More complete but still unfinished ) Lecturer : Uri Zwick
We describe an efficient implementation of Edmonds’ algorithm for finding minimum directed spanning trees in directed graphs. 1 Minimum Directed Spanning Trees Let G = (V,E,w) be a weighted directed graph, where w : E → R is a cost (or weight) function defined on its edges. Let r ∈ V . A directed spanning tree (DST) of G rooted at r, is a subgraph T of G such that the undirected version of T is...
متن کاملRIMS-1674 Reduction of Ultrametric Minimum Cost Spanning Tree Games to Cost Allocation Games on Rooted Trees By
A minimum cost spanning tree game is called ultrametric if the cost function on the edges of the underlying network is ultrametric. We show that every ultrametric minimum cost spanning tree game is a cost allocation game on a rooted tree. It follows that there exist optimal algorithms for computing the Shapley value, the nucleolus and the egalitarian allocation of the ultrametric minimum cost s...
متن کاملReduction of Ultrametric Minimum Cost Spanning Tree Games to Cost Allocation Games on Rooted Trees
A minimum cost spanning tree game is called ultrametric if the cost function on the edges of the underlying network is an ultrametric. We show that every ultrametric minimum cost spanning tree game is reduced to a cost allocation game on a rooted tree. It follows that there exist O(n) time algorithms for computing the Shapley value, the nucleolus and the egalitarian allocation of the ultrametri...
متن کاملOptimal Independent Spanning Trees on Hypercubes
Two spanning trees rooted at some vertex r in a graph G are said to be independent if for each vertex v of G, v ≠ r, the paths from r to v in two trees are vertex-disjoint. A set of spanning trees of G is said to be independent if they are pairwise independent. A set of independent spanning trees is optimal if the average path length of the trees is the minimum. Any k-dimensional hypercube has ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2017