We study greedy routing over @math nodes placed in a ring, with the between two nodes defined to be the clockwise or the absolute distance between them along the ring. Such graphs arise in the context of modeling social networks and in routing networks for peer-to-peer systems. We construct the first network over @math nodes in which greedy routing takes @math hops in the worst-case, with @math out-going links per node. Our result has the first asymptotically optimal greedy routing complexity. Previous constructions required @math hops.
demonstrates that it is possible to construct a graph in which each node has degree @math and in which 1- has routes of length @math in the worst case, for the metrics @math , @math and @math . Furthermore, for all @math , plain greedy on our network design beats even the results obtained in @cite_9 with @math - lookahead .
@cite_9
2949856235 2079430060 2169273227 2161190897
We introduce a family of directed geometric graphs, denoted @math , that depend on two parameters @math and @math . For @math and @math , the @math graph is a strong @math -spanner, with @math . The out-degree of a node in the @math graph is at most @math . Moreover, we show that routing can be achieved locally on @math . Next, we show that all strong @math -spanners are also @math -spanners of the unit disk graph. Simulations for various values of the parameters @math and @math indicate that for random point sets, the spanning ratio of @math is better than the proven theoretical bounds. We study undirected networks with edge costs that satisfy the triangle inequality. Let @math denote the number of nodes. We present an @math -approximation algorithm for a generalization of the metric-cost subset @math -node-connectivity problem. Our approximation guarantee is proved via lower bounds that apply to the simple edge-connectivity version of the problem, where the requirements are for edge-disjoint paths rather than for openly node-disjoint paths. A corollary is that, for metric costs and for each @math , there exists a @math -node connected graph whose cost is within a factor of @math of the cost of any simple @math -edge connected graph. Based on our @math -approximation algorithm, we present an @math -approximation algorithm for the metric-cost node-connectivity survivable network design problem, where @math denotes the maximum requirement over all pairs of nodes. Our results contrast with the case of edge costs of 0 or 1, where Kortsarz, Krauthgamer, and Lee. [SIAM J. Comput., 33 (2004), pp. 704-720] recently proved, assuming NP @math DTIME( @math ), a hardness-of-approximation lower bound of @math for the subset @math -node-connectivity problem, where @math denotes a small positive number. We conjecture that any planar 3-connected graph can be embedded in the plane in such a way that for any nodes s and t, there is a path from s to t such that the Euclidean distance to t decreases monotonically along the path. A consequence of this conjecture would be that in any ad hoc network containing such a graph as a spanning subgraph, two-dimensional virtual coordinates for the nodes can be found for which the method of purely greedy geographic routing is guaranteed to work. We discuss this conjecture and its equivalent forms show that its hypothesis is as weak as possible, and show a result delimiting the applicability of our approach: any 3-connected K3,3-free graph has a planar 3-connected spanning subgraph. We also present two alternative versions of greedy routing on virtual coordinates that provably work. Using Steinitz's theorem we show that any 3-connected planar graph can be embedded in three dimensions so that greedy routing works, albeit with a modified notion of distance; we present experimental evidence that this scheme can be implemented effectively in practice. We also present a simple but provably robust version of greedy routing that works for any graph with a 3-connected planar spanning subgraph. We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds @math on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a @math -approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex @math is at most @math . This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly @math -Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost @math -edge-connected subgraph with at least @math vertices, which we call the @math -subgraph problem. This generalizes some well-studied classical problems such as the @math -MST and the minimum cost @math -edge-connected subgraph problems. We give a polylogarithmic approximation for the @math -subgraph problem. However, by relating it to the Densest @math -Subgraph problem, we provide evidence that the @math -subgraph problem might be hard to approximate for arbitrary @math .