Near-Optimal Low-Congestion Shortcuts on Bounded Parameter Graphs

• Published in: Distributed Computing Vol. 9888; pp. 158 - 172
• Main Authors: Haeupler, Bernhard, Izumi, Taisuke, Zuzic, Goran
• Format: Book Chapter
• Language: English
• Published: Germany Springer Berlin / Heidelberg 2016; Springer Berlin Heidelberg
• Series: Lecture Notes in Computer Science
• Subjects: Bounded-genus graph; CONGEST model; Distributed algorithm; Minimum cut; Minimum spanning tree; Pathwidth; Treewidth
• ISBN: 3662534258; 9783662534250
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-662-53426-7_12

We show that many distributed network optimization problems can be solved much more efficiently in structured and topologically simple networks. It is known that solving essentially any global network optimization problem in a general network requires \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega (\sqrt{n})$$\end{document} rounds in the CONGEST model, even if the network diameter is small, e.g., logarithmic. Many networks of interest, however, have more structure which allows for significantly more efficient algorithms. Recently Ghaffari, Haeupler, Izumi and Zuzic [SODA’16,PODC’16] introduced low-congestion shortcuts as a suitable abstraction to capture this phenomenon. In particular, they showed that graphs with diameter D embeddable in a genus-g surface have good shortcuts and that these shortcuts lead to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{O}(g D)$$\end{document}-round algorithms for MST, Min-Cut and other problems. We generalize these results by showing that networks with pathwidth or treewidth k allow for good shortcuts leading to fast \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{O}(k D)$$\end{document} distributed optimization algorithms. We also improve the dependence on genus g from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{O}(gD)$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{O}(\sqrt{g}D)$$\end{document}. Lastly, we prove lower bounds which show that the dependence on k and g in our shortcuts is optimal. Overall, this significantly refines and extends the understanding of how the complexity of distributed optimization problems depends on the network topology.