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Reachability query is a fundamental graph operation which answers whether a vertex can reach another vertex over a large directed graph G with n vertices and m edges, and has been extensively studied. In the literature, all the approaches compute a label for every vertex in a graph G by index construction offline. The query time for answering reachability queries online is affected by the quality of the labels computed in index construction. The three main costs are the index construction time, the index size, and the query time. Some of the up-to-date approaches can answer reachability queries efficiently, but spend non-linear time to construct an index. Some of the up-to-date approaches construct an index in linear time and space, but may need to depth-first search G at run-time in O(n + m). In this paper, as the first, we propose a new randomized labeling approach to answer reachability queries, and the randomness is by independent permutation. We conduct extensive experimental studies to compare with the up-to-date approaches using 19 large real datasets used in the existing work and synthetic datasets. We confirm the efficiency of our approach.

Abstract:Carbon dots (CDs), as a new member of carbon nanomaterial family, have aroused great interest since their discovery in 2004. Because of their outstanding water solubility, high sensitivity and selectivity to target analytes, low toxicity, favorable biocompatibility, and excellent photostability, researchers from diverse disciplines have come together to further develop the fundamental properties of CDs. Many methods for the production of CDs have been reported, therein, hydrothermal and solvothermal technology needs simple equipments, and microwave synthesis needs less reaction time, hence these methods become current common synthesis methods, in which many precursors have been applied to produce CDs. Due to their excellent fluorescence, CDs have made impressive strides in sensitivity and selectivity to a diverse array of salt ions, organic/biological molecules and target gases. The development of CDs as nanoprobes is still in its infancy, but continued progress may lead to their integration into environmental and biological applications. Hydrothermal, solvothermal, and microwave synthesis of fluorescent carbon dots and their detection applications as nanoprobes in salt ions, organic/biological molecules, and target gases will be reviewed.

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