Dynamic connectivity is a well-studied problem, but so far the most compelling progress has been confined to the edge-update model: maintain an understanding of connectivity in an undirected graph, subject to edge insertions and deletions. In this paper, we study two more challenging, yet equally fundamental problems. Subgraph connectivity asks to maintain an understanding of connectivity under vertex updates: updates can turn vertices on and off, and queries refer to the subgraph induced by "on" vertices. (For instance, this is closer to applications in networks of routers, where node faults may occur.) We describe a data structure supporting vertex updates in O (m^ 2 3 ) amortized time, where m denotes the number of edges in the graph. This greatly improves over the previous result [Chan, STOC'02], which required fast matrix multiplication and had an update time of O(m^0.94). The new data structure is also simpler. Geometric connectivity asks to maintain a dynamic set of n geometric objects, and query connectivity in their intersection graph. (For instance, the intersection graph of balls describes connectivity in a network of sensors with bounded transmission radius.) Previously, nontrivial fully dynamic results were known only for special cases like axis-parallel line segments and rectangles. We provide similarly improved update times, O (n^ 2 3 ), for these special cases. Moreover, we show how to obtain sublinear update bounds for virtually all families of geometric objects which allow sublinear-time range queries, such as arbitrary 2D line segments, d-dimensional simplices, and d-dimensional balls.
It was shown @cite_32 that subgraph connectivity can be reduced to dynamic connectivity of axis-parallel line segments in 3 dimensions. Thus, as soon as one gets enough combinatorial richness in the host geometric space, subgraph connectivity becomes the possible way to solve geometric connectivity.
@cite_32
2011182146
2-Edge connectivity is an important fault tolerance property of a network because it maintains network communication despite the deletion of a single arbitrary edge. Planar spanning subgraphs have been shown to play a significant role for achieving local decentralized routing in wireless networks. Existing algorithmic constructions of spanning planar subgraphs of unit disk graphs (UDGs) such as Minimum Spanning Tree, Gabriel Graph, Nearest Neighborhood Graph, etc. do not always ensure connectivity of the resulting graph under single edge deletion. Furthermore, adding edges to the network so as to improve its edge connectivity not only may create edge crossings (at points which are not vertices) but it may also require edges of unbounded length. Thus we are faced with the problem of constructing 2-edge connected geometric planar spanning graphs by adding edges of bounded length without creating edge crossings (at points which are not vertices). To overcome this difficulty, in this paper we address the problem of augmenting the edge set (i.e., adding new edges) of planar geometric graphs with straight line edges of bounded length so that the resulting graph is planar and 2-edge connected. We provide bounds on the number of newly added straight-line edges, prove that such edges can be of length at most 3 times the max length of an edge of the original graph, and also show that the factor 3 is optimal. It is shown to be NP-Complete to augment a geometric planar graph to a 2-edge connected geometric planar graph with the minimum number of new edges of a given bounded length. We also provide a constant time algorithm that works in location-aware settings to augment a planar graph into a 2-edge connected planar graph with straight-line edges of length bounded by 3 times the longest edge of the original graph. It turns out that knowledge of vertex coordinates is crucial to our construction and in fact we prove that this problem cannot be solved locally if the vertices do not know their coordinates. Moreover, we provide a family of k-connected UDGs which does not have 2-edge connected spanning planar subgraphs, for any . 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. A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2-connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified)? Unfortunately, the problem is known to be NP-hard. We consider the problem of finding a better approximation to the smallest 2-connected subgraph, by an efficient algorithm. For 2-edge connectivity, our algorithm guarantees a solution that is no more than 3 2 times the optimal. For 2-vertex connectivity, our algorithm guarantees a solution that is no more than 5 3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP-hard as well. We also consider the case where the graph has edge weights. For this case, we show that an approximation factor of 2 is possible in polynomial time for finding a k -edge connected spanning subgraph. This improves an approximation factor of 3 for k = 2, due to Frederickson and Ja´Ja´ [1981], and extends it for any k (with an increased running time though). In this paper we study connectivity augmentation problems. Given a connected graph G with some desirable property, we want to make G 2-vertex connected (or 2-edge connected) by adding edges such that the resulting graph keeps the property. The aim is to add as few edges as possible. The property that we consider is planarity, both in an abstract graph-theoretic and in a geometric setting, where vertices correspond to points in the plane and edges to straight-line segments. We show that it is NP-hard to nd a minimum-cardinality augmentation that makes a planar graph 2-edge connected. For making a planar graph 2-vertex connected this was known. We further show that both problems are hard in the geometric setting, even when restricted to trees. The problems remain hard for higher degrees of connectivity. On the other hand we give polynomial-time algorithms for the special case of convex geometric graphs. We also study the following related problem. Given a planar (plane geometric) graph G, two vertices s and t of G, and an integer c, how many edges have to be added to G such that G is still planar (plane geometric) and contains c edge- (or vertex-) disjoint s t paths? For the planar case we give a linear-time algorithm for c = 2. For the plane geometric case we give optimal worst-case bounds for c = 2; for c = 3 we characterize the cases that have a solution. An efficient heuristic is presented for the problem of finding a minimum-size k-connected spanning subgraph of an (undirected or directed) simple graph G=(V,E). There are four versions of the problem, and the approximation guarantees are as follows: minimum-size k-node connected spanning subgraph of an undirected graph 1 + [1 k], minimum-size k-node connected spanning subgraph of a directed graph 1 + [1 k], minimum-size k-edge connected spanning subgraph of an undirected graph 1+[2 (k+1)], minimum-size k-edge connected spanning subgraph of a directed graph 1 + [4 k ]. The heuristic is based on a subroutine for the degree-constrained subgraph (b-matching) problem. It is simple and deterministic and runs in time O(k|E|2). The following result on simple undirected graphs is used in the analysis: The number of edges required for augmenting a graph of minimum degree k to be k-edge connected is at most k,|V| (k+1). For undirected graphs and k=2, a (deterministic) parallel NC version of the heuristic finds a 2-node connected (or 2-edge connected) spanning subgraph whose size is within a factor of ( @math ) of minimum, where @math is a constant. Random geometric graphs result from taking n uniformly distributed points in the unit cube, [0,1]d, and connecting two points if their Euclidean distance is at most r, for some prescribed r. We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching on two sets of @math points distributed uniformly in [0,1]d. We present upper bounds on the threshold width, and show that our bound is sharp for d = 1 and at most a sublogarithmic factor away for d ≥ 2. Interestingly, the threshold width is much sharper for random geometric graphs than for Bernoulli random graphs. Further, a random geometric graph is shown to be a subgraph, with high probability, of another independently drawn random geometric graph with a slightly larger radius; this property is shown to have no analogue for Bernoulli random graphs. (MATH) Inspired by dynamic connectivity applications in computational geometry, we consider a problem we call dynamic subgraph connectivity: design a data structure for an undirected graph @math and a subset of vertices @math , to support insertions and deletions in @math and connectivity queries (are two vertices connected @?) in the subgraph induced by @math . We develop the first sublinear, fully dynamic method for this problem for general sparse graphs, using an elegant combination of several simple ideas. Our method requires linear space, @math amortized update time, and @math query time, where @math is the matrix multiplication exponent and @math hides polylogarithmic factors. We consider supervised learning problems where the features are embedded in a graph, such as gene expressions in a gene network. In this context, it is of much interest to automatically select a subgraph with few connected components; by exploiting prior knowledge, one can indeed improve the prediction performance or obtain results that are easier to interpret. Regularization or penalty functions for selecting features in graphs have recently been proposed, but they raise new algorithmic challenges. For example, they typically require solving a combinatorially hard selection problem among all connected subgraphs. In this paper, we propose computationally feasible strategies to select a sparse and well-connected subset of features sitting on a directed acyclic graph (DAG). We introduce structured sparsity penalties over paths on a DAG called "path coding" penalties. Unlike existing regularization functions that model long-range interactions between features in a graph, path coding penalties are tractable. The penalties and their proximal operators involve path selection problems, which we efficiently solve by leveraging network flow optimization. We experimentally show on synthetic, image, and genomic data that our approach is scalable and leads to more connected subgraphs than other regularization functions for graphs. An efficient heuristic is presented for the problem of finding a minimum-size k-connected spanning subgraph of a given (undirected or directed) graph G=(V,E). There are four versions of the problem, depending on whether G is undirected or directed, and whether the spanning subgraph is required to be k-node connected (k-NCSS) or k-edge connected (k-ECSS). The approximation guarantees are as follows: min-size k-NCSS of an undirected graph 1+[1 k], min-size k-NCSS of a directed graph 1+[1 k], min-size k-ECSS of an undirected graph 1+[7 k], & min-size k-ECSS of a directed graph 1+[4 spl radic k]. The heuristic is based on a subroutine for the degree-constrained subgraph (b-matching) problem. It is simple, deterministic, and runs in time O(k|E| sup 2 ). For undirected graphs and k=2, a (deterministic) parallel NC version of the heuristic finds a 2-node connected (or a-edge connected) spanning subgraph whose size is within a factor of (1.5+ spl epsiv ) of minimum, where spl epsiv >0 is a constant. The connectivity graph of wireless networks, under many models as well as in practice, may contain unidirectional links. The simplifying assumption that such links are useless is often made, mainly because most wireless protocols use per-hop acknowledgments. However, two-way communication between a pair of nodes can be established as soon as there exists paths in both directions between them. Therefore, instead of discarding unidirectional links, one might be interested in studying the strongly connected components of the connectivity graph. In this paper, we look at the percolation phenomenon in some directed random geometric graphs that can be used to model wireless networks. We show that among the nodes that can be reached from the origin, a non-zero fraction can also reach the origin. In other words, the percolation threshold for strong connectivity is equal to the threshold for one-way connectivity. A 5 4-approximation algorithm is presented for the minimum cardinality 2-edge-connected spanning subgraph problem in undirected graphs. This improves the previous best approximation ratio of 4 3. It is shown that our ratio is tight with respect to current lower bounds, and any further improvement is possible only if new lower bounds are discovered. In this paper, we presented a fully distributed algorithm to compute a planar subgraph of the underlying wireless connectivity graph. We considered the idealized unit disk graph model in which nodes are assumed to be connected if and only if nodes are within their transmission range. The main contribution of this work is a fully distributed algorithm to extract the connected, planar graph for routing in the wireless networks. The communication cost of the proposed algorithm is O(d log d) bits, where d is the degree of anode. In addition, this paper also presented a geometric routing algorithm. The algorithm is fully distributed and nodes know only the position of other nodes and can communicate with neighboring nodes in their transmission range We present factor 4 3 approximation algorithms for the problems of finding theminimum 2-edge connected and theminimum 2-vertex connected spanning subgraph of a given undirected graph. It is shown that every connected planar straight line graph with n>=3 vertices has an embedding preserving augmentation to a 2-edge connected planar straight line graph with at most @?(2n-2) 3@? new edges. It is also shown that every planar straight line tree with n>=3 vertices has an embedding preserving augmentation to a 2-edge connected planar topological graph with at most @?n 2@? new edges. These bounds are the best possible. However, for every n>=3, there are planar straight line trees with n vertices that do not have an embedding preserving augmentation to a 2-edge connected planar straight line graph with fewer than 1733n-O(1) new edges. We consider the Directed Steiner Network (DSN) problem, also called the Point-to-Point Connection problem, where given a directed graph G and p pairs (s sub 1 ,t sub 1 ), ..., (s sub p ,t sub p ) of nodes in the graph, one has to find the smallest subgraph H of G that contains paths from s sub i to t sub i for all i. The problem is NP-hard for general p, since the Directed Steiner Tree problem is a special case. Until now, the complexity was unknown for constant p spl ges 3. We prove that the problem is polynomially solvable if p is any constant number, even if nodes and edges in G are weighted and the goal is to minimize the total weight of the subgraph H. In addition, we give an efficient algorithm for the Strongly Connected Steiner Subgraph problem for any constant p, where given a directed graph and p nodes in the graph, one has to compute the smallest strongly connected subgraph containing the p nodes. A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The connected domatic number of G is the maximum number of pairwise disjoint, connected dominating sets in V(G). We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3. The High Level Architecture (HLA) is a standard for constructing distributed simulations. The Data Distribution Management services of HLA reduce the amount of data delivered to an HLA federate by allowing communications connections to be based on federates’ expressed data production and requirements. At the core of determining which connections to make is a geometric problem: finding the dynamic intersection of d-dimensional rectilinear hyperrectangles in d-space. Four different algorithms for solving that problem are described, including a new one developed through application of a data structure from computational geometry. Those algorithms are then compared in an experiment designed to reveal how well they perform in the specific context of the data distribution application. Both intersection performance and connectivity efficiency results are reported. In the subgraph-freeness problem, we are given a constant-sized graph H, and wish to de- termine whether the network graph contains H as a subgraph or not. Until now, the only lower bounds on subgraph-freeness known for the CONGEST model were for cycles of length greater than 3; here we extend and generalize the cycle lower bound, and obtain polynomial lower bounds for subgraph-freeness in the CONGEST model for two classes of subgraphs. The first class contains any graph obtained by starting from a 2-connected graph H for which we already know a lower bound, and replacing the vertices of H by arbitrary connected graphs. We show that the lower bound on H carries over to the new graph. The second class is constructed by starting from a cycle Ck of length k ≥ 4, and constructing a graph H from Ck by replacing each edge i, (i + 1) mod k of the cycle with a connected graph Hi, subject to some constraints on the graphs H_ 0 , . . . , H_ k−1 . In this case we obtain a polynomial lower bound for the new graph H , depending on the size of the shortest cycle in H passing through the vertices of the original k-cycle. We give approximation algorithms for the Survivable Network problem. The input consists of a graph G = (V,E) with edge node-costs, a node subset S ⊆ V, and connectivity requirements r(s,t):s,t ∈ T ⊆ V . The goal is to find a minimum cost subgraph H of G that for all s,t ∈ T contains r(s,t) pairwise edge-disjoint st-paths such that no two of them have a node in S s s,t in common. Three extensively studied particular cases are: Edge-Connectivity Survivable Network (S = ∅), Node-Connectivity Survivable Network (S = V), and Element-Connectivity Survivable Network (r(s,t) = 0 whenever s ∈ S or t ∈ S). Let k = maxs,t ∈ T r(s,t). In Rooted Survivable Network, there is s ∈ T such that r(u,t) = 0 for all u ≠ s, and in the Subset k-Connected Subgraph problem r(s,t) = k for all s,t ∈ T. For edge-costs, our ratios are O(k log k) for Rooted Survivable Network and O(k2 log k) for Subset k-Connected Subgraph. This improves the previous ratio O(k2 log n), and for constant values of k settles the approximability of these problems to a constant. For node-costs, our ratios are as follows. —O(k log |T|) for Element-Connectivity Survivable Network, matching the best known ratio for Edge-Connectivity Survivable Network. —O(k2 log |T|) for Rooted Survivable Network and O(k3 log |T|) for Subset k-Connected Subgraph, improving the ratio O(k8 log2 |T|). —O(k4 log2 |T|) for Survivable Network; this is the first nontrivial approximation algorithm for the node-costs version of the problem. We present a simple, efficient, and stable method for computing-with any desired precision-the medial axis of simply connected planar domains. The domain boundaries are assumed to be given as polynomial spline curves. Our approach combines known results from the field of geometric approximation theory with a new algorithm from the field of computational geometry. Challenging steps are (1) the approximation of the boundary spline such that the medial axis is geometrically stable, and (2) the efficient decomposition of the domain into base cases where the medial axis can be computed directly and exactly. We solve these problems via spiral biarc approximation and a randomized divide & conquer algorithm.