Smaller Cuts, Higher Lower Bounds
Autor: | Seri Khoury, Amir Abboud, Ami Paz, Keren Censor-Hillel |
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Rok vydání: | 2021 |
Předmět: |
FOS: Computer and information sciences
Discrete mathematics Dense graph Logarithm 010102 general mathematics Vertex cover Context (language use) 0102 computer and information sciences 01 natural sciences Upper and lower bounds Mathematics (miscellaneous) Quadratic equation Computer Science - Distributed Parallel and Cluster Computing 010201 computation theory & mathematics Distributed algorithm Independent set Computer Science - Data Structures and Algorithms Data Structures and Algorithms (cs.DS) Distributed Parallel and Cluster Computing (cs.DC) 0101 mathematics Mathematics |
Zdroj: | ACM Transactions on Algorithms. 17:1-40 |
ISSN: | 1549-6333 1549-6325 |
Popis: | This paper proves strong lower bounds for distributed computing in the CONGEST model, by presenting the bit-gadget: a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs. Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortest-paths problem is discussed. Finally, it is shown that graph constructions for CONGEST lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature. Comment: This is work is a merger of arXiv:1605.05109 and arXiv:1705.05646 |
Databáze: | OpenAIRE |
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