Overlay networks with class ∗ G. Chiola DISI, Universit` a di Genova via Dodecaneso 35, Genova, Italy chiolag@acm.org G. Cordasco, L. Gargano, A. Negro, and V. Scarano DIA, Universit` a di Salerno Via S. Allende, Baronissi (Salerno), Italy. {cordasco,lg,alberto,vitsca}@dia.unisa.it Abstract We define a family of Distributed Hash Table sys- tems whose aim is to combine routing efficiency of the randomized networks — i.e. average path length O(log n/ log log n) vs. the O(log n) average path length of the deterministic system — with the pro- grammability and startup efficiency of a uniform sys- tem — that is a system in which the overlay network is transitive, and greedy routing is optimal. It is known that Ω(log n) is a lower bound to the average path length for uniform systems with O(log n) degree. The proposed family is parameterized with a posi- tive integer c which measures the amount of random- ness that is used. Indeed, edges are partitioned into c equivalence classes. Varying the value c, the system goes from the deterministic case (c =1) to an “almost uniform” system. Increasing c to relatively low values allows routing with optimal average path length while retaining most of the advantages of a uniform system, such as easy programmability and quick bootstrap of the nodes entering the system. We also provide a matching lower bound for the av- erage path length of the family of routing schemes for any c. Moreover, we show how to extend the result to other overlay networks. 1 Introduction Peer–to–Peer (P2P) file sharing applications have recently become very popular. Several of the proposed systems are completely distributed and use a scalable Distributed Hash Table (DHT) as a substrate. A DHT is a self–organizing overlay network that allows to add, ∗ This work was partially supported by the Italian FIRB project “WEB–MINDS” (Wide-scalE, Broadband MIddleware for Network Distributed Services), http://web-minds.consorzio-cini.it/. delete, and lookup items in a hash table. Several pro- posals have recently been presented for systems whose hosts configure themselves into a structured network such that lookups require a small number of hops. The greedy routing approach, in which any mes- sage is routed through the neighbor which is nearest to the target, has been used in most of the proposed P2P networks. These include [12, 3, 14, 1, 7]. Several rea- sons make the greedy strategy popular. In particular, greedy routing is very simple to implement and has some “implicit” fault-tolerance capabilities. It was, however, noticed that greedy routing usually produces paths of length larger than what would be required in a network of the given node degree. In fact, randomization [6, 8] and de Bruijn graphs [4] were shown to produce networks with optimal av- erage path length. Unfortunately, these algorithms are not greedy and present some other disadvantages as discussed in [8]. 1.1 NoN routing on randomized networks: Opti- mal average path length. By enlarging the set of nodes where the greedy choice is made, it was recently shown that Θ(log n/ log log n) hops can be obtained (on average) with the Neighbors–of–Neighbors (NoN) routing [8]: the greedy choice is made here from among all the nodes which are at most 2 hops distant from the node itself. Latency can be optimally reduced in several well known topologies [8] (such as, e.g., Chord) provided that randomization is used to establish the neighbors of the nodes, and routing is implemented according to the NoN approach. Hence, the use of randomization inspired by the Small-World idea introduced by Klein- berg [5], together with the NoN routing allows one to keep, to some extent, the advantages of greedy routing