What Graph Properties Are Constant-Time Testable?
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In this paper, we survey what graph properties have been found to be constant-time testable at present. How to handle big data is a very important issue in computer science. Developing efficient algorithms for problems on big data is an urgent task. For this purpose, constant-time algorithms are powerful tools, since they run by reading only a constant-sized part of each input. In other words, the running time is invariant regardless of the size of the input. The idea of constant-time algorithms appeared in the 1990s and spread quickly especially in this century. Research on graph algorithms is one of the best studied areas in theoretical computer science. When we study constant-time algorithms on graphs, we have three models that differ in the way that the graphs are represented: the dense-graph model, the bounded-degree model, and the general model. The first one is used to treat properties on dense graphs, and the other two are used to treat properties on sparse graphs. In this paper, we survey one by one what properties have been found to be constant-time testable in each of the three models.
KeywordsConstant-time algorithms Property testing Graphs Regularity lemma Hyperfinite Universal testers Complex networks
The author thanks Professor Ilan Newman of the University of Haifa, Israel, Associate Professor Yuichi Yoshida of the National Institute of Informatics, Japan, and Associate Professor Suguru Tamaki of University of Hyogo, Japan for their helpful advice.
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