Low-Degree Spanning Trees of Small Weight

The degree-d spanning tree problem asks for a minimum-weight spanning tree in which the degree of each vertex is at most d. When d=2 the problem is TSP, and in this case, the well-known Christofides algorithm provides a 1.5-approximation algorithm (assuming the edge weights satisfy the triangle inequality). In 1984, Christos Papadimitriou and Umesh Vazirani posed the challenge of finding an algorithm with performance guarantee less than 2 for Euclidean graphs (points in R^n) and d > 2. This paper gives the first answer to that challenge, presenting an algorithm to compute a degree-3 spanning tree of cost at most 5/3 times the MST. For points in the plane, the ratio improves to 3/2 and the algorithm can also find a degree-4 spanning tree of cost at most 5/4 times the MST.Comment: conference version in Symposium on Theory of Computing (1994

There are Plane Spanners of Maximum Degree 4

Let E be the complete Euclidean graph on a set of points embedded in the plane. Given a constant t >= 1, a spanning subgraph G of E is said to be a t-spanner, or simply a spanner, if for any pair of vertices u,v in E the distance between u and v in G is at most t times their distance in E. A spanner is plane if its edges do not cross. This paper considers the question: "What is the smallest maximum degree that can always be achieved for a plane spanner of E?" Without the planarity constraint, it is known that the answer is 3 which is thus the best known lower bound on the degree of any plane spanner. With the planarity requirement, the best known upper bound on the maximum degree is 6, the last in a long sequence of results improving the upper bound. In this paper we show that the complete Euclidean graph always contains a plane spanner of maximum degree at most 4 and make a big step toward closing the question. Our construction leads to an efficient algorithm for obtaining the spanner from Chew's L1-Delaunay triangulation

Degree Four Plane Spanners: Simpler and Better

Let P be a set of n points embedded in the plane, and let C be the complete Euclidean graph whose point-set is P. Each edge in C between two points p, q is realized as the line segment [pq], and is assigned a weight equal to the Euclidean distance |pq|. In this paper, we show how to construct in O(nlg{n}) time a plane spanner of C of maximum degree at most 4 and of stretch factor at most 20. This improves a long sequence of results on the construction of bounded degree plane spanners of C. Our result matches the smallest known upper bound of 4 by Bonichon et al. on the maximum degree while significantly improving their stretch factor upper bound from 156.82 to 20. The construction of our spanner is based on Delaunay triangulations defined with respect to the equilateral-triangle distance, and uses a different approach than that used by Bonichon et al. Our approach leads to a simple and intuitive construction of a well-structured spanner, and reveals useful structural properties of the Delaunay triangulations defined with respect to the equilateral-triangle distance

An optimal algorithm for computing angle-constrained spanners

Let S be a set of n points in ℝd. A graph G = (S,E) is called a t-spanner for S, if for any two points p and q in S, the shortest-path distance in G between p and q is at most t|pq|, where |pq| denotes the Euclidean distance between p and q. The graph G is called θ-angle-constrained, if any two distinct edges sharing an endpoint make an angle of at least θ. It is shown that, for any θ with 0 < θ < π/3, a θ-angle-constrained t-spanner can be computed in O(n logn) time, where t depends only on θ

An adaptive distributed algorithm for path aggregation.

Zhang, Zhenyi.Thesis (M.Phil.)--Chinese University of Hong Kong, 2008.Includes bibliographical references (leaves 55-[58]).Abstracts in Chinese and English.Chapter 1 --- Introduction --- p.1Chapter 2 --- Problem Formulation --- p.4Chapter 3 --- Examples --- p.7Chapter 3.1 --- Examples of Undirected Graph --- p.7Chapter 3.1.1 --- Example 1: SPF Routing --- p.7Chapter 3.1.2 --- Example 2: rings --- p.7Chapter 3.1.3 --- Example 3: grid --- p.8Chapter 3.1.4 --- Example 4: cube --- p.9Chapter 3.1.5 --- Example 5: random graph X --- p.10Chapter 3.1.6 --- Example 6: random graph Y --- p.10Chapter 3.2 --- An Example for Directive Graph --- p.11Chapter 4 --- The Framework --- p.13Chapter 4.1 --- The distributed algorithm --- p.13Chapter 4.2 --- The modules --- p.14Chapter 4.3 --- Path control --- p.15Chapter 4.4 --- The forwarding module --- p.18Chapter 4.5 --- The routing module --- p.19Chapter 4.5.1 --- Non-weighted Routing (NWR) --- p.19Chapter 4.5.2 --- Weighted Routing (WR) --- p.20Chapter 4.6 --- Packet Aggregation (PKA) --- p.21Chapter 5 --- Experiments of Path Aggregation --- p.23Chapter 5.1 --- System Setup --- p.24Chapter 5.2 --- Experiment Results --- p.25Chapter 6 --- Convergence --- p.28Chapter 6.1 --- Simulation study --- p.34Chapter 6.2 --- Optimality --- p.34Chapter 6.3 --- Speed of Convergence --- p.37Chapter 7 --- The adaptive property --- p.41Chapter 7.1 --- Adapting to new links --- p.42Chapter 7.2 --- Adapting to topology changing --- p.43Chapter 7.3 --- Adapting to interference and congestion --- p.45Chapter 7.4 --- Adapting to traffic flows --- p.45Chapter 7.5 --- Adapting to capacity --- p.46Chapter 8 --- Related works --- p.48Chapter 8.1 --- Spanning Tree --- p.48Chapter 8.2 --- Minimum Equivalent Directed Graph Problem --- p.49Chapter 8.3 --- Topology Control --- p.50Chapter 8.4 --- The Relationship with our problem --- p.53Chapter 9 --- Conclusion --- p.5