TopCom: Index for Shortest Distance Query in Directed Graph

Finding shortest distance between two vertices in a graph is an important problem due to its numerous applications in diverse domains, including geo-spatial databases, social network analysis, and information retrieval. Classical algorithms (such as, Dijkstra) solve this problem in polynomial time, but these algorithms cannot provide real-time response for a large number of bursty queries on a large graph. So, indexing based solutions that pre-process the graph for efficiently answering (exactly or approximately) a large number of distance queries in real-time is becoming increasingly popular. Existing solutions have varying performance in terms of index size, index building time, query time, and accuracy. In this work, we propose T OP C OM , a novel indexing-based solution for exactly answering distance queries. Our experiments with two of the existing state-of-the-art methods (IS-Label and TreeMap) show the superiority of T OP C OM over these two methods considering scalability and query time. Besides, indexing of T OP C OM exploits the DAG (directed acyclic graph) structure in the graph, which makes it significantly faster than the existing methods if the SCCs (strongly connected component) of the input graph are relatively small

Temporal Cliques Admit Sparse Spanners

Let G=(V,E) be an undirected graph on n vertices and lambda:E -> 2^{N} a mapping that assigns to every edge a non-empty set of positive integer labels. These labels can be seen as discrete times when the edge is present. Such a labeled graph {G}=(G,lambda) is said to be temporally connected if a path exists with non-decreasing times from every vertex to every other vertex. In a seminal paper, Kempe, Kleinberg, and Kumar (STOC 2000) asked whether, given such a temporal graph, a sparse subset of edges can always be found whose labels suffice to preserve temporal connectivity - a temporal spanner. Axiotis and Fotakis (ICALP 2016) answered negatively by exhibiting a family of Theta(n^2)-dense temporal graphs which admit no temporal spanner of density o(n^2). The natural question is then whether sparse temporal spanners always exist in some classes of dense graphs. In this paper, we answer this question affirmatively, by showing that if the underlying graph G is a complete graph, then one can always find temporal spanners of density O(n log n). The best known result for complete graphs so far was that spanners of density binom{n}{2}- floor[n/4] = O(n^2) always exist. Our result is the first positive answer as to the existence of o(n^2) sparse spanners in adversarial instances of temporal graphs since the original question by Kempe et al., focusing here on complete graphs. The proofs are constructive and directly adaptable as an algorithm

Scaling distance labeling on small-world networks

© 2019 Association for Computing Machinery. Distance labeling approaches are widely adopted to speed up the online performance of shortest distance queries. The construction of the distance labeling, however, can be exhaustive especially on big graphs. For a major category of large graphs, small-world networks, the state-of-the-art approach is Pruned Landmark Labeling (PLL). PLL prunes distance labels based on a node order and directly constructs the pruned labels by performing breadth-first searches in the node order. The pruning technique, as well as the index construction, has a strong sequential nature which hinders PLL from being parallelized. It becomes an urgent issue on massive small-world networks whose index can hardly be constructed by a single thread within a reasonable time. This paper scales distance labeling on small-world networks by proposing a Parallel Shortest-distance Labeling (PSL) scheme and further reducing the index size by exploiting graph and label properties. PSL insightfully converts the PLL's node-order dependency to a shortest-distance dependence, which leads to a propagation-based parallel labeling in D rounds where D denotes the diameter of the graph. Extensive experimental results verify our efficiency on billion-scale graphs and near-linear speedup in a multi-core environment

Distributed Community Detection in Dynamic Graphs

Inspired by the increasing interest in self-organizing social opportunistic networks, we investigate the problem of distributed detection of unknown communities in dynamic random graphs. As a formal framework, we consider the dynamic version of the well-studied \emph{Planted Bisection Model} \sdG(n,p,q) where the node set $[n]$ of the network is partitioned into two unknown communities and, at every time step, each possible edge $(u,v)$ is active with probability $p$ if both nodes belong to the same community, while it is active with probability $q$ (with $q<<p$) otherwise. We also consider a time-Markovian generalization of this model. We propose a distributed protocol based on the popular \emph{Label Propagation Algorithm} and prove that, when the ratio $p/q$ is larger than $n^{b}$ (for an arbitrarily small constant $b>0$), the protocol finds the right "planted" partition in $O(\log n)$ time even when the snapshots of the dynamic graph are sparse and disconnected (i.e. in the case $p=\Theta(1/n)$).Comment: Version I