The diameter of a graph is among its most basic parameters. Since a few years, it moreover became a key issue to compute it for massive graphs in the context of complex network analysis. However, known algorithms, including the ones producing approximate values, have too high a time and/or space complexity to be used in such cases. We propose here a new approach relying on very simple and fast algorithms that compute (upper and lower) bounds for the diameter. We show empirically that, on various real-world cases representative of complex networks studied in the literature, the obtained bounds are very tight (and even equal in some cases). This leads to rigorous and very accurate estimations of the actual diameter in cases which were previously untractable in practice. 1 Context. Throughout the paper, we consider a connected undirected unweighted graph G = (V, E) with n = |V | vertices and m = |E | edges. We denote by d(u, v) the distance between u and v in G, by ecc(v) = maxu d(v, u) the eccentricity of v in G, and by D = maxu,v d(u, v) = maxvecc(v) the diameter of G.

Abstract. We extend the classic notion of well-separated pair decomposition [10] to the unit-disk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We show that for the unit-disk graph metric of n points in the plane and for any constant c ≥ 1, there exists a c-wellseparated pair decomposition with O(n log n) pairs, and the decomposition can be computed in O(n log n) time. We also show that for the unit-ball graph metric in k dimensions where k ≥ 3, there exists a c-wellseparated pair decomposition with O(n 2−2/k) pairs, and the bound is tight in the worst case. We present the application of the well-separated pair decomposition in obtaining efficient algorithms for approximating the diameter, closest pair, nearest neighbor, center, median, and stretch factor, all under the unit-disk graph metric. Keywords Well separated pair decomposition, Unit-disk graph, Approximation algorithm

We propose a novel divide-and-conquer algorithm for the solution of the all-pair shortest-path problem for directed and dense graphs with no negative cycles. We propose R-Kleene, a compact and in-place recursive algorithm inspired by Kleene’s algorithm. R-Kleene delivers a better performance than previous algorithms for randomly generated graphs represented by highly dense adjacency matrices, in which the matrix components can have any integer value. We show that R-Kleene, unchanged and without any machine tuning, yields consistently between 1/7 and 1/2 of the peak performance running on five very different uniprocessor systems.

We present an all-pairs shortest path algorithm for arbitrary graphs that performs O(mn log (m; n)) comparison and addition operations, where m and n are the number of edges and vertices, resp., and is Tarjan's inverse-Ackermann function. Our algorithm eliminates the sorting bottleneck inherent in approaches based on Dijkstra's algorithm, and for graphs with O(n) edges our algorithm is within a tiny O(log (n; n)) factor of optimal. Our algorithm can be implemented to run in polynomial time (granted, a large polynomial). We leave open the problem of providing an efficient implementation.

Computing shortest paths between two given nodes is a fundamental operation over graphs, but known to be nontrivial over large disk-resident instances of graph data. While a numberoftechniquesexistfor answeringreachabilityqueries and approximating node distances efficiently, determining actual shortest paths (i.e. the sequence of nodes involved) is often neglected. However, in applications arising in massive online social networks, biological networks, and knowledge graphs it is often essential to find out many, if not all, shortest paths between two given nodes. In this paper, we address this problem and present a scalable sketch-based index structure that not only supports estimation of node distances, but also computes corresponding shortest paths themselves. Generating the actual path information allows for further improvements to the estimation accuracy of distances (and paths), leading to near-exact shortest-path approximations in real world graphs. We evaluate our techniques – implemented within a fully functional RDF graph database system – over large realworld social and biological networks of sizes ranging from tens of thousand to millions of nodes and edges. Experiments on several datasets show that we can achieve query response times providing several orders of magnitude speedup over traditional path computations while keeping the estimation errors between 0 % and 1 % on average.

We present several new external-memory algorithms for finding all-pairs shortest paths in a V-node, E-edge undirected graph. For all-pairs shortest paths and diameter in unweighted undirected graphs we present cache-oblivious algorithnls with O(V. ~ log. ~ ~) I/Os, where B is the block-size and M is the size of internal memory. For weighted tmdirected graphs we present a cache-aware APSP algorithm that performs O(V. ( V/ ~ + ~ log ~)) I/Os. We also present efficient cacheaware algorithms that find paths between all pairs of vertices in an unweighted graph with lengths within a small additive constant of the shortest path length. All of our results improve earlier results known for these problems. For approximate APSP we provide the first nontrivial results. Our diameter result uses C9(V + E) extra space, and all of our other algorithms use O(V 2) space. 1

We consider the classical geometric problem of determining shortest paths between pairs of points lying on a weighted polyhedral surface P consisting of n triangular faces. We present query algorithms that compute approximate distances and/or approximate (weighted) shortest paths. Our algorithm takes as input an approximation parameter ε ∈ (0, 1) and a query time parameter q and builds a data structure which is then used for answering ǫ-approximate distance queries in O(q) time. This algorithm is source point independent and improves significantly on the best previous solution. For the case where one of the query points is fixed we build a data structure that can answer ǫ-approximate distance queries to any query point in P in O(log 1) time. This is an improve-ε ment upon the previously known solution for the Euclidean fixed source query problem. Our algorithm also generalizes the setting from previously studied unweighted polyhedral to weighted polyhedral surfaces of arbitrary genus. Our solutions are based on a novel graph separator algorithm introduced here which extends and generalizes previously known separator algorithms.

Abstract. We study concurrent processes modelled as workflow Petri nets extended with resource constraints. We define a behavioural correctness criterion called soundness: given a sufficient initial number of resources, all cases in the net are guaranteed to terminate successfully, no matter which schedule is used. We give a necessary and sufficient condition for soundness and an algorithm that checks it.

A disk graph is an intersection graph of a set of disks with arbitrary radii in the plane. In this paper, we consider the problem of efficient construction of sparse spanners of disk (ball) graphs with support for fast distance queries. These problems are motivated by issues arising from topology control and routing in wireless networks. We present the first algorithm for constructing spanners of ball graphs. For a ball graph in R k, we construct a (1+ǫ)-spanner with O(nǫ −k+1) edges in O(n 2ℓ+δ ǫ −k log ℓ S) expected time, using an efficient partitioning of the space into hypercubes and solving intersection problems. Here ℓ = 1 − 1/(⌊k/2 ⌋ + 2), δ is any positive constant, and S is the ratio between the largest and smallest radius. For the special case where all the balls have the same radius, we show that the spanner construction has complexity almost equivalent to the construction of a Euclidean minimum spanning tree. Previously known constructions of spanners of unit ball graphs have time complexity much closer to n 2. Additionally, these spanners have a small vertex separator (hereditary), which is then exploited for fast answering of distance queries. The results on geometric graph separators might be of independent interest. 1

Abstract — Complex networks (internet maps, web graphs, data exchanges, etc) received much attention during these last years. However data on such networks are only available through intricate measurement procedures. Until recently, most studies assumed that these procedures eventually lead to sample large enough to be representative of the whole, at least concerning some key properties. This has crucial impact on network modeling and simulation. We propose here a new way to investigate the relevance of this approach: we put together data on complex network measurements that are representative of data commonly used, but significantly larger. Then we study how the observed properties evolve when the sample grows. The obtained results are in sharp contrast with usual assumptions, with important consequences that we discuss.