An Overview of Reachability Indexes on Graphs

Abstract

Graphs have been the natural choice for modeling entities and the relationships among them. One of the most fundamental graph processing operators is a reachability query, which checks whether a path exists from the source to the target vertex in a plain graph, and additionally whether the path can satisfy a given path constraint based on the edge labels in an edge-labeled graph. Processing reachability queries requires potentially visiting a large portion of the graph due to the inherent transitivity of these queries. This makes it costly to evaluate them on large graphs. Thus, significant effort has been spent to design indexing techniques for reachability queries in the last three decades, building advanced data structures to efficiently compress the transitive closure of the graph so as to accelerate online query processing, aka reachability indexes. In this tutorial, we provide an in-depth technical review of the existing reachability indexes, ranging from those designed for plain graphs to ones for edge-labeled graphs. We conclude the tutorial by summarizing the open challenges for integrating these techniques into GDBMSs.

A Process Algebraic Approach to Predict and Control Uncertainty in Smart IoT Systems for Smart Cities Based on Permissible Probabilistic Equivalence

“…It is one of the actions for A shown in ( 1) and ( 16) of the table: null (empty ( 16)), communication (send (17), receive ( 18)), movement (( 19): ( 23)~( 26)), control (( 20): ( 27)~( 29)).…”

“…(a) Empty ( 16)-It represents a null action, where no action occurs. (b) Communication (send (17), receive ( 18))-It represents a synchronous communication between sender and receiver. In order to perform a communication between two processes, a set of synchronous actions, that is, send and receive, must occur in an overlapping time between them through a communication channel with the same type of message.…”

“…• The execution (EX) model is a system transition model similar to reachability graph [16][17][18] that shows visually all the execution paths of a target ITL model with its included ITS models; its visual representation is as defined on the ADOxx meta-modeling platform for EX models; • The geo-temporal space (GTS) model is a graphical model that shows the processes in a system and their interactions, that is, communication and movement, on a twodimensional geo-temporal space, as defined on the ADOxx meta-modeling platform for GTS models.…”