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Computing path queries such as the shortest path in public transport networks is challenging because the path costs between nodes change over time. A reachability query from a node at a given start time on such a network retrieves all points of interest (POIs) that are reachable within a given cost budget. Reachability queries are essential building blocks in many applications, for example, group recommendations, ranking spatial queries, or geomarketing. We propose an efficient solution for reachability queries in public transport networks. Currently, there are two options to solve reachability queries. (1) Execute a modified version of Dijkstra's algorithm that supports time-dependent edge traversal costs; this solution is slow since it must expand edge by edge and does not use an index. (2) Issue a separate path query for each single POI, i.e., a single reachability query requires answering many path queries. None of these solutions scales to large networks with many POIs. We propose a novel and lightweight reachability index. The key idea is to partition the network into cells. Then, in contrast to other approaches, we expand the network cell by cell. Empirical evaluations on synthetic and real-world networks confirm the efficiency and the effectiveness of our index-based reachability query solution. [ABSTRACT FROM AUTHOR]

Abstract Fast reachability detection is one of the key problems in graph applications. Most of the existing works focus on creating an index and answering reachability based on that index. For these approaches, the index construction time and index size can become a concern for large graphs. More recently query-preserving graph compression has been proposed, and searching reachability over the compressed graph has been shown to be able to significantly improve query performance as well as reducing the index size. In this paper, we introduce a multilevel compression scheme for DAGs, which builds on existing compression schemes, but can further reduce the graph size for many real-world graphs. We propose an algorithm to answer reachability queries using the compressed graph. Extensive experiments with four existing state-of-the-art reachability algorithms and 12 real-world datasets demonstrate that our approach outperforms the existing methods. Experiments with synthetic datasets ensure the scalability of this approach. We also provide a discussion on possible compression for k-reachability.

Fast reachability detection is one of the key problems in graph applications. Most of the existing works focus on creating an index and answering reachability based on that index. For these approaches, the index construction time and index size can become a concern for large graphs. More recently query-preserving graph compression has been proposed, and searching reachability over the compressed graph has been shown to be able to significantly improve query performance as well as reducing the index size. In this paper, we introduce a multilevel compression scheme for DAGs, which builds on existing compression schemes, but can further reduce the graph size for many real-world graphs. We propose an algorithm to answer reachability queries using the compressed graph. Extensive experiments with four existing state-of-the-art reachability algorithms and 12 real-world datasets demonstrate that our approach outperforms the existing methods. Experiments with synthetic datasets ensure the scalability of this approach. We also provide a discussion on possible compression for k-reachability. [ABSTRACT FROM AUTHOR]

Many schema labelling approaches have been designed to facilitate querying of XML documents. The proposed algorithms are based on the fact that ancestor–descendant relationships among nodes can be quickly determined. Schema labelling is a family of technologies widely used in indexing tree, graph, or structured XML graph, in which a unique identifier is assigned to each node in the tree/graph. The generated identifier is then used in indexing as a reference to the actual node so that structural relationship among the nodes can be quickly captured. In this paper, we extend the prime number schema labelling algorithm for labelling DAG XML graph. Our main contribution is scaling down the original XML graph size substantially based on the Strongly Connected Component (SCC) principles. Labelling each node in DAG with an integer that is the arithmetical multiplication of the prime number associating with the node and its parent label. The schema does not depend on spanning tree. Thus, subsumption hierarchies represented in a DAG can be efficiently explored by checking the divisibility among the labels. Also, it inherits dynamic update ability and compact size features from its predecessors. Our theoretical analysis and the experimental results showed that the generated labelled schema is an efficient and a scalable one for processing reachability queries on large XML graphs.

For many large scale graph reachability query problems in reality, this paper proposed a new algorithm based on RDD which decomposed a graph into a series of spanning trees and a residue graph. In this way, it transformed any reachability query into these two queries to reduce the cost. Compared with the traditional algorithms, like interval labeling, chain de composition, 2-hop labeling, and path-trees etc, this algorithm could lead to smaller space overhead, and lesser time complexity. The simulation illustrates that the algorithm has a smaller storage and more effectively to deal with the large scale groph reachability query problem. [ABSTRACT FROM AUTHOR]

Abstract: Many schema labelling approaches have been designed to facilitate querying of XML documents. The proposed algorithms are based on the fact that ancestor–descendant relationships among nodes can be quickly determined. Schema labelling is a family of technologies widely used in indexing tree, graph, or structured XML graph, in which a unique identifier is assigned to each node in the tree/graph. The generated identifier is then used in indexing as a reference to the actual node so that structural relationship among the nodes can be quickly captured. In this paper, we extend the prime number schema labelling algorithm for labelling DAG XML graph. Our main contribution is scaling down the original XML graph size substantially based on the Strongly Connected Component (SCC) principles. Labelling each node in DAG with an integer that is the arithmetical multiplication of the prime number associating with the node and its parent label. The schema does not depend on spanning tree. Thus, subsumption hierarchies represented in a DAG can be efficiently explored by checking the divisibility among the labels. Also, it inherits dynamic update ability and compact size features from its predecessors. Our theoretical analysis and the experimental results showed that the generated labelled schema is an efficient and a scalable one for processing reachability queries on large XML graphs. [Copyright &y& Elsevier]

Given a large directed graph, rapidly answering reachability queries between source and target nodes is an important problem. Existing methods for reachability tradeoff indexing time and space versus query time performance. However, the biggest limitation of existing methods is that they do not scale to very large real-world graphs. We present a simple yet scalable reachability index, called GRAIL, that is based on the idea of randomized interval labeling and that can effectively handle very large graphs. Based on an extensive set of experiments, we show that while more sophisticated methods work better on small graphs, GRAIL is the only index that can scale to millions of nodes and edges. GRAIL has linear indexing time and space, and the query time ranges from constant time to being linear in the graph order and size. Our reference C++ implementations are open source and available for download at . [ABSTRACT FROM AUTHOR]

Reachability query is one of the fundamental queries in graph database. The main idea behind answering reachability queries is to assign vertices with certain labels such that the reachability between any two vertices can be determined by the labeling information. Though several approaches have been proposes for building these reachability labels, it remains open issues on how to handle increasingly large number of vertices in real-world graphs, and how to find the best tradeoff among the labeling size, the query answering time, and the construction time. In this article, we introduce a novel graph structure, referred to as pathtree, to help labeling very large graphs. The path-tree cover is a spanning subgraph of G in a tree shape. We show path-tree can be generalized to chain-tree which theoretically can has smaller labeling cost. On top of path-tree and chain-tree index, we also introduce a new compression scheme which groups vertices with similar labels together to further reduce the labeling size. In addition, we also propose an efficient incremental update algorithm for dynamic index maintenance. Finally, we demonstrate both analytically and empirically the effectiveness and efficiency of our new approaches. [ABSTRACT FROM AUTHOR]

• We propose a strategy to improve FELINE's negative cut index. • We introduce LYNX, a novel indexing approach for reachability queries. • Computational results compare LYNX's performance with state-of-the-art algorithms. Given a directed acyclic graph G = (V , A) and two vertices u , v ∈ V , the reachability problem is to answer if there is a path from u to v in the graph. In the context of very large graphs, with millions of vertices and a series of queries to be answered, it is not practical to search the graph for each query. On the other hand, the storage of the full transitive closure of the graph is also impractical due to its O (| V | 2) size. Scalable approaches aim to create indices used to prune the search during its execution. Negative indices may be able to determine (in constant time) that a query has a negative answer while positive indices may determine (again in constant time) that a query has a positive answer. In this paper we propose a novel scalable approach called LYNX that uses a large number of topological sorts of G as a negative cut index without degrading the query time. A similar strategy is applied regarding a positive cut index. In addition, LYNX proposes a user-defined index size that enables the user to control the ratio between negative and positive cuts depending on the expected query pattern. We show by computational experiments that LYNX consistently outperforms the state-of-the-art approach in terms of query-time using the same index-size for graphs with high reachability ratio. In intelligent computer systems that rely on frequent tests of connectivity in graphs, LYNX can reduce the time delay experience by end users through a reduced query time. This comes at the expense of an increased setup time whenever the underlying graph is updated. [ABSTRACT FROM AUTHOR]

Testing reachability in a graph gains substantial interest as an important operation in network analysis and graph mining. In its simplest form, a reachability query is defined by a pair of nodes (u , v) and a graph G , and detects if there is a path from u to v. This paper addresses a specific case of reachability on multi-labeled distributed graphs, where the query is parameterized by a set of source nodes S , a set of target nodes T and a set of constraints C on the edge labels. We conduct a performance evaluation on both synthetic and real-world datasets, using multiple instances of Neo4j servers (as workers) running simultaneously. The results show that the number of workers, the network density and the number of cross-edges have a significant impact on the overall performance. Moreover, we observe that the proposed approach is scalable and can be used to solve label-constrained distributed set reachability queries in multi-labeled networks. [ABSTRACT FROM AUTHOR]