Abstract
Communication complexity theory is a powerful tool to show time complexity lower bounds of distributed algorithms for global problems such as minimum spanning tree (MST) and shortest path. While it often leads to nearly-tight lower bounds for many problems, polylogarithmic complexity gaps still lie between the currently best upper and lower bounds. In this paper, we propose a new approach for filling the gaps. Using this approach, we achieve tighter deterministic lower bounds for MST and shortest paths. Specifically, for those problems, we show the deterministic \(\Omega(\sqrt{n})\)-round lower bound for graphs of O(n ε) hop-count diameter, and the deterministic \(\Omega(\sqrt{n/\log n})\) lower bound for graphs of O(logn) hop-count diameter. The main idea of our approach is to utilize the two-party communication complexity lower bound for a function we call permutation identity, which is newly introduced in this paper.
This work is supported in part by KAKENHI No. 25106507 and No. 25289114.
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References
Sarma, A.D., Holzer, S., Kor, L., Korman, A., Nanongkai, D., Pandurangan, G., Peleg, D., Wattenhofer, R.: Distributed verification and hardness of distributed approximation. In: Proc. of the 43rd Annual ACM Symposium on Theory of Computing, pp. 363–372 (2011)
Dinitz, Y., Moran, S., Rajsbaum, S.: Bit complexity of breaking and achieving symmetry in chains and rings. Journal of the ACM 55(1) (2008)
Elkin, M.: An unconditional lower bound on the hardness of approximation of distributed minimum spanning tree problem. In: Proc the 30th ACM Symposium on Theory of Computing(STOC), pp. 331–340 (2004)
Garay, J.A., Kutten, S., Peleg, D.: A sublinear time distributed algorithm for minimum-weight spanning trees. SIAM Journal on Computing 27(1), 302–316 (1998)
Ghaffari, M., Kuhn, F.: Distributed minimum cut approximation. In: Afek, Y. (ed.) DISC 2013. LNCS, vol. 8205, pp. 1–15. Springer, Heidelberg (2013)
Holzer, S., Wattenhofer, R.: Optimal distributed all pairs shortest paths and applications. In: Proc. of the 2012 ACM Symposium on Principles of Distributed Computing (PODC), pp. 355–364 (2012)
Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press (1997)
Lenzen, C., Patt-Shamir, B.: Fast routing table construction using small messages: Extended abstract. In: Proc. of the 45th Annual ACM Symposium on Symposium on Theory of Computing (STOC), pp. 381–390 (2013)
Lenzen, C., Peleg, D.: Efficient distributed source detection with limited bandwidth. In: Proc. of the 2013 ACM Symposium on Principles of Distributed Computing (PODC), pp. 375–382 (2013)
Lotker, Z., Patt-Shamir, B., Peleg, D.: Distributed mst for constant diameter graphs. Distributed Computing 18(6), 453–460 (2006)
Nanongkai, D.: Distributed approximation algorithms for weighted shortest paths. In: Proc. of the 46th ACM Symposium on Theory of Computing (STOC), pp. 565–573 (2014)
Nanongkai, D., Das Sarma, A., Pandurangan, G.: A tight unconditional lower bound on distributed random walk computation. In: Proc. of the 30th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC), pp. 257–266 (2011)
Peleg, D., Roditty, L., Tal, E.: Distributed algorithms for network diameter and girth. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 660–672. Springer, Heidelberg (2012)
Peleg, D., Rubinovich, V.: A near-tight lower bound on the time complexity of distributed minimum-weight spanning tree construction. SIAM Journal on Computing 30(5), 1427–1442 (2000)
Yao, A.C.-C.: Some complexity questions related to distributive computing(preliminary report). In: Proc. of the 11th Annual ACM Symposium on Theory of Computing (STOC), pp. 209–213 (1979)
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Ookawa, H., Izumi, T. (2015). Filling Logarithmic Gaps in Distributed Complexity for Global Problems. In: Italiano, G.F., Margaria-Steffen, T., Pokorný, J., Quisquater, JJ., Wattenhofer, R. (eds) SOFSEM 2015: Theory and Practice of Computer Science. SOFSEM 2015. Lecture Notes in Computer Science, vol 8939. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46078-8_31
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DOI: https://doi.org/10.1007/978-3-662-46078-8_31
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