Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

\(D^2\)-Tree: A New Overlay with Deterministic Bounds

  • 155 Accesses

  • 3 Citations


We present a new overlay, called the Deterministic Decentralized tree (\(D^2\)-tree). The \(D^2\)-tree compares favorably to other overlays for the following reasons: (a) it provides matching and better complexities, which are deterministic for the supported operations; (b) the management of nodes (peers) and elements are completely decoupled from each other; and (c) an efficient deterministic load-balancing mechanism is presented for the uniform distribution of elements into nodes, while at the same time probabilistic optimal bounds are provided for the congestion of operations at the nodes. The load-balancing scheme of elements into nodes is deterministic and general enough to be applied to other hierarchical tree-based overlays. This load-balancing mechanism is based on an innovative lazy weight-balancing mechanism, which is interesting in its own right.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2


  1. 1.

    We have chosen this \(\epsilon _{j}\) for simplicity. In fact for any \(\eta >0\), choosing \(\epsilon _{j}=\frac{1}{j^{1+\eta }}\) is sufficient.

  2. 2.

    The alternative of following a height-based approach, resulting in a height (instead of weight) balanced overlay, would render update operations inefficient.


  1. 1.

    Abraham, I., Awerbuch, B., Azar, Y., Bartal, Y., Malkhi, D., Pavlov, E.: A generic scheme for building overlay networks in adversarial scenarios. In: Proceedings of the 17th IPDPS, 40 (2003)

  2. 2.

    Arge, L., Eppstein, D., Goodrich, M.T.: Skip-Webs: efficient distributed data structures for multidimensional data sets. In: Proeedings of the 24th PODC, 69–76 (2005)

  3. 3.

    Arge, L., Vitter, J.: Optimal external memory interval management. SIAM J. Comput. 32(6), 1488–1508 (2003)

  4. 4.

    Aspnes, J., Kirsch, J., Krishnamurthy, A.: Load-balancing and locality in range-queriable data structures. In: Proceeding of the 23rd PODC, 115–124 (2004)

  5. 5.

    Aspnes, J., Shah, G.: Skip graphs. In: Proceedings of the 14th SODA, 384–393 (2003)

  6. 6.

    Awerbuch, B., Scheideler, C.: Consistent and compact data management in distributed storage systems. In: Proceedings of the 16th SPAA, 44–53 (2004)

  7. 7.

    Brodal, G., Sioutas, S., Tsichlas, K., Zaroliagis, C.: \(D^2\)-Tree: A New Overlay with Deterministic Bounds. In: Algorithms and Computation—ISAAC 2010, Lecture Notes in Computer Science, vol. 6507, Part II, Springer, pp. 1–12 (2010)

  8. 8.

    Li, Dongsheng, Cao, Jiannong: Efficient range query processing in peer-to-peer systems. IEEE Tran. Knowl. Data Eng. 21(1), 78–91 (2009)

  9. 9.

    Gasenan, P., Bawa, M., Garcia-Molina, H.: Online balancing of range-partitioned data with applications to peer-to-peer systems. In: Proceedings of the 13th VLDB, 444–455 (2004)

  10. 10.

    Goodrich, M.T., Nelson, M.J., Sun, J.Z.: The rainbow skip graph: a fault-tolerant constant-degree distributed data structure. In: Proceedings of the 17th SODA, 384–393 (2006)

  11. 11.

    Gupta, A., Agrawal, D., El Abbadi, A.: Approximate range selection queries in peer-to-peer systems. In: Proceedings of the 1st CIDR (2003)

  12. 12.

    Harvey, N., Munro, J.I.: Deterministic SkipNet. In: Proceedings of the 22nd PODC, 152–153 (2003)

  13. 13.

    Jagadish, H.V., Ooi, B.C., Vu, Q. H.: BATON: a balanced tree structure for peer-to-peer networks, In: Proceedings of the 31st VLDB, 661–672 (2005)

  14. 14.

    Jagadish, H.V., Ooi, B.C., Tan, K.L., Vu, Q.H., Zhang, R.: Speeding up search in P2P networks with a multi-way tree structure. In: Proceedings ACM International Conference on Management of Data (SIGMOD), pp. 1–12, Chicago, IL (2006)

  15. 15.

    Karger, D., Kaashoek, F., Stoica, I., Morris, R., Balakrishnan, H.: Chord: A scalable peer-to-peer lookup service for internet applications. In: Proceedings of the SIGCOMM, 149–160 (2001)

  16. 16.

    Manku, G.S., Bawa, M., Raghavan, P.: Symphony: distributed hashing in a small world. In: 4th USENIX Symposium on Internet Technologies and Systems (2003)

  17. 17.

    Manku, G.S., Naor, M., Wieder, U.: Know thy neighbor’s neighbor: the power of lookahead in randomized P2P networks. In: Proceedings of the 36th STOC, 54–63 (2004)

  18. 18.

    Pugh, W.: Skip lists: a probabilistic alternative to balanced trees. Commun. ACM 33(6), 668–676 (1990)

  19. 19.

    Rowstron, A., Druschel, P.: Pastry: a scalable. Decentralized Object Location, and Routing for Large-Scale Peer-to-Peer Systems, In: Middleware, LNCS 2218, 329–350 (2001)

  20. 20.

    Ratnasamy, S., Francis, P., Handley, M., Karp, R., Shenker, S.: A scalable content addressable network. In: Proceedings ACM Conference on Applications, Technologies, Architectures, and Protocols for Computer Communication (SIGCOMM), pp. 161–172, San Diego, CA (2001)

  21. 21.

    Sahin, O.D., Gupta, A., Agrawal, D., El Abbadi, A.: A peer-to-peer framework for caching range queries. In: Proceedings of the 20th ICDE, 165 (2004)

  22. 22.

    Scheideler, C., Schmid, S.: A distributed and oblivious Heap. In: Proceedings of the 36th ICALP, 571–582 (2009)

  23. 23.

    Zatloukal, K.C., Harvey, N.J.A.: Family trees: an ordered dictionary with optimal congestion, locality, degree and search time. In: Proceedings of the 15th SODA, 301–310 (2004)

  24. 24.

    Zhang, Y., Liu, L., Li, D., Liu, F., Lu, X.: DHT-based range query processing for web service discovery. In: Proceedings of the 2009 IEEE ICWS, 477–484 (2009)

  25. 25.

    Zhao, B.Y., Huang, L., Stribling, J., Rhea, S.C., Joseph, A.D., Kubiatowicz, J.D.: Tapestry: a resilient global-scale overlay for service deployment. IEEE J. Sel. Areas Commun. 22(1), 41–53 (2004)

Download references


We would like to thank the referees for their valuable comments that improved the presentation of this paper.

Author information

Correspondence to Christos Zaroliagis.

Additional information

Part of this work was done while the last author was visiting the Karlsruhe Institute of Technology.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Brodal, G.S., Sioutas, S., Tsichlas, K. et al. \(D^2\)-Tree: A New Overlay with Deterministic Bounds. Algorithmica 72, 860–883 (2015). https://doi.org/10.1007/s00453-014-9878-4

Download citation


  • Overlay
  • Indexing scheme
  • Decentralized system
  • Distributed data structure
  • Load-balancing