Smaller Cuts, Higher Lower Bounds

This article proves strong lower bounds for distributed computing in the congest model, by presenting the bit-gadget : a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs. Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P, which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortest-paths problem is discussed. Finally, it is shown that graph constructions for congest lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.

Parallel Privacy-Preserving Shortest Path Algorithms

In this paper, we propose and present secure multiparty computation (SMC) protocols for single-source shortest distance (SSSD) and all-pairs shortest distance (APSD) in sparse and dense graphs. Our protocols follow the structure of classical algorithms—Bellman–Ford and Dijkstra for SSSD; Johnson, Floyd–Warshall, and transitive closure for APSD. As the computational platforms offered by SMC protocol sets have performance profiles that differ from typical processors, we had to perform extensive changes to the structure (including their control flow and memory accesses) and the details of these algorithms in order to obtain good performance. We implemented our protocols on top of the secret sharing based protocol set offered by the Sharemind SMC platform, using single-instruction-multiple-data (SIMD) operations as much as possible to reduce the round complexity. We benchmarked our protocols under several different parameters for network performance and compared our performance figures against each other and with ones reported previously.

Dynamic graph exploration by interactively linked node-link diagrams and matrix visualizations

The visualization of dynamic graphs is a challenging task owing to the various properties of the underlying relational data and the additional time-varying property. For sparse and small graphs, the most efficient approach to such visualization is node-link diagrams, whereas for dense graphs with attached data, adjacency matrices might be the better choice. Because graphs can contain both properties, being globally sparse and locally dense, a combination of several visual metaphors as well as static and dynamic visualizations is beneficial. In this paper, a visually and algorithmically scalable approach that provides views and perspectives on graphs as interactively linked node-link and adjacency matrix visualizations is described. As the novelty of this technique, insights such as clusters or anomalies from one or several combined views can be used to influence the layout or reordering of the other views. Moreover, the importance of nodes and node groups can be detected, computed, and visualized by considering several layout and reordering properties in combination as well as different edge properties for the same set of nodes. As an additional feature set, an automatic identification of groups, clusters, and outliers is provided over time, and based on the visual outcome of the node-link and matrix visualizations, the repertoire of the supported layout and matrix reordering techniques is extended, and more interaction techniques are provided when considering the dynamics of the graph data. Finally, a small user experiment was conducted to investigate the usability of the proposed approach. The usefulness of the proposed tool is illustrated by applying it to a graph dataset, such as e co-authorships, co-citations, and a Comprehensible Perl Archive Network distribution.

Flows on measurable spaces

The theory of graph limits is only understood to a somewhat satisfactory degree in the cases of dense graphs and of bounded degree graphs. There is, however, a lot of interest in the intermediate cases. It appears that one of the most important constituents of graph limits in the general case will be Markov spaces (Markov chains on measurable spaces with a stationary distribution). This motivates our goal to extend some important theorems from finite graphs to Markov spaces or, more generally, to measurable spaces. In this paper, we show that much of flow theory, one of the most important areas in graph theory, can be extended to measurable spaces. Surprisingly, even the Markov space structure is not fully needed to get these results: all we need a standard Borel space with a measure on its square (generalizing the finite node set and the counting measure on the edge set). Our results may be considered as extensions of flow theory for directed graphs to the measurable case.

Uniformly Most Reliable Three-Terminal Graph of Dense Graphs

A graph G with k specified target vertices in vertex set is a k -terminal graph. The k -terminal reliability is the connection probability of the fixed k target vertices in a k -terminal graph when every edge of this graph survives independently with probability p . For the class of two-terminal graphs with a large number of edges, Betrand, Goff, Graves, and Sun constructed a locally most reliable two-terminal graph for p close to 1 and illustrated by a counterexample that this locally most reliable graph is not the uniformly most reliable two-terminal graph. At the same time, they also determined that there is a uniformly most reliable two-terminal graph in the class obtained by deleting an edge from the complete graph with two target vertices. This article focuses on the uniformly most reliable three-terminal graph of dense graphs with n vertices and m edges. First, we give the locally most reliable three-terminal graphs of n and m in certain ranges for p close to 0 and 1. Then, it is proved that there is no uniformly most reliable three-terminal graph with specific n and m , where n ≥ 7 and n 2 − ⌊ n − 3 / 2 ⌋ ≤ m ≤ n 2 − 2 . Finally, some uniformly most reliable graphs are given for n vertices and m edges, where 4 ≤ n ≤ 6 and m = n 2 − 2 or n ≥ 5 and m = n 2 − 1 .