OPODIS 2014: Principles of Distributed Systems pp 60-75

# Time Lower Bounds for Distributed Distance Oracles

• Taisuke Izumi
• Roger Wattenhofer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8878)

## Abstract

Distributed distance oracles consist of a labeling scheme which assigns a label to each node and a local data structure deployed to each node. When a node v wants to know the distance to a node u, it queries its local data structure with the label of u. The data structure returns an estimated distance to u, which must be larger than the actual distance but can be overestimated. The accuracy of the distance oracle is measured by stretch, which is defined as the maximum ratio between actual distances and estimated distances over all pairs (u, v).

In this paper, we focus on the time complexity of constructing distributed distance oracles with a given stretch. We show a number of time lower bounds depending on the stretch:

• Under the assumption that the popular combinatorial girth conjecture is true, any distributed algorithm constructing oracles with stretch 2t requires $$\tilde{\Omega}(n^{1/(t+1)})$$ rounds in unweighted graphs. This bound holds even if we only consider constant diameter graphs.

• For oracles with stretch 2t in weighted graphs, we have a lower bound of $$\Omega(n^{\frac{1}{2} + \frac{1}{5t}})$$ rounds, assuming the girth conjecture. This bound holds even if we only consider O(logn) diameter graphs.

• If we restrict the label size of oracles to o(n ε ) bits, where ε = 1/2t(t + 1) in unweighted graphs and ε = (1/5t 2) in weighted graphs, the same lower bounds are obtained without assuming the girth conjecture.

To the best of our knowledge, this paper is the first that exhibits a non-trivial trade-off between time and stretch for distributed distance oracles.

## Keywords

Short Path Weighted Graph Short Path Tree Unweighted Graph Unweighted Network
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Das Sarma, A., Dinitz, M., Pandurangan, G.: Efficient computation of distance sketches in distributed networks. In: Proc. of the 24th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pp. 318–326 (2012)Google Scholar
2. 2.
Das Sarma, A., 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)Google Scholar
3. 3.
Diestel, R.: Graph Theory, 4th edn., vol. 173. Springer (2012)Google Scholar
4. 4.
Erdös, P.: Graph theory and probability. In: Classic Papers in Combinatorics, pp. 276–280 (1987)Google Scholar
5. 5.
Frischknecht, S., Holzer, S., Wattenhofer, R.: Networks cannot compute their diameter in sublinear time. In: Proc. of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1150–1162 (2012)Google Scholar
6. 6.
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)Google Scholar
7. 7.
Kalyanasundaram, B., Schintger, G.: The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics 5(4), 545–557 (1992)
8. 8.
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)Google Scholar
9. 9.
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)Google Scholar
10. 10.
Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press (2005)Google Scholar
11. 11.
Nanongkai, D.: Distributed approximation algorithms for weighted shortest paths. In: Proc. of the 46th Annual ACM Symposium on Theory of Computing (STOC), pp. 565–573 (2014)Google Scholar
12. 12.
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)
13. 13.
Razborov, A.A.: On the distributional complexity of disjointness. Theoretical Computer Science 106(2), 385–390 (1992)
14. 14.
Thorup, M., Zwick, U.: Approximate distance oracles. Journal of the ACM 52(1), 1–24 (2005)
15. 15.
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)Google Scholar

© Springer International Publishing Switzerland 2014

## Authors and Affiliations

• Taisuke Izumi
• 1
• Roger Wattenhofer
• 2
1. 1.Nagoya Institute of TechnologyNagoyaJapan
2. 2.ETH ZurichZurichSwitzerland