We study greedy routing over @math nodes placed in a ring, with the between two nodes defined to be the clockwise or the absolute distance between them along the ring. Such graphs arise in the context of modeling social networks and in routing networks for peer-to-peer systems. We construct the first network over @math nodes in which greedy routing takes @math hops in the worst-case, with @math out-going links per node. Our result has the first asymptotically optimal greedy routing complexity. Previous constructions required @math hops.
Deterministic butterflies have been proposed for DHT routing by Xu al @cite_16 , who subsequently developed their ideas into Ulysses @cite_6 . for distance function @math has structural similarities with Ulysses -- both are butterfly-based networks. The key differences are as follows: (a) Ulysses does not use @math as its distance function, (b) Ulysses does not use routing, and (c) Ulysses uses more links than for distance function @math -- additional links have been introduced to ameliorate non-uniform edge congestion caused by Ulysses' routing algorithm. In contrast, the congestion-free routing algorithm developed in obviates the need for any additional links in (see Theorem ).
@cite_16 @cite_6
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A number of distributed hash table (DHT)-based protocols have been proposed to address the issue of scalability in peer-to-peer networks. In this paper, we present Ulysses, a peer-to-peer network based on the butterfly topology that achieves the theoretical lower bound of log n log log n on network diameter when the average routing table size at nodes is no more than log n. Compared to existing DHT-based schemes with similar routing table size, Ulysses reduces the network diameter by a factor of log log n, which is 2–4 for typical configurations. This translates into the same amount of reduction on query latency and average traffic per link node. In addition, Ulysses maintains the same level of robustness in terms of routing in the face of faults and recovering from graceful ungraceful joins and departures, as provided by existing DHT-based schemes. The performance of the protocol has been evaluated using both analysis and simulation. Copyright © 2004 AEI We study a fundamental tradeoff issue in designing a distributed hash table (DHT) in peer-to-peer (P2P) networks: the size of the routing table versus the network diameter. Observing that existing DHT schemes have either 1) a routing table size and network diameter both of O(log sub 2 n), or 2) a routing table of size d and network diameter of O(n sup 1 d ), S. (2001) asked whether this represents the best asymptotic "state-efficiency" tradeoffs. We show that some straightforward routing algorithms achieve better asymptotic tradeoffs. However, such algorithms all cause severe congestion on certain network nodes, which is undesirable in a P2P network. We rigorously define the notion of "congestion" and conjecture that the above tradeoffs are asymptotically optimal for a congestion-free network. The answer to this conjecture is negative in the strict sense. However, it becomes positive if the routing algorithm is required to eliminate congestion in a "natural" way by being uniform. We also prove that the tradeoffs are asymptotically optimal for uniform algorithms. Furthermore, for uniform algorithms, we find that the routing table size of O(log sub 2 n) is a magic threshold point that separates two different "state-efficiency" regions. Our third result is to study the exact (instead of asymptotic) optimal tradeoffs for uniform algorithms. We propose a new routing algorithm that reduces the routing table size and the network diameter of Chord both by 21.4 without introducing any other protocol overhead, based on a novel number-theory technique. Our final result is to present Ulysses, a congestion-free nonuniform algorithm that achieves a better asymptotic "state-efficiency" tradeoff than existing schemes in the probabilistic sense, even under dynamic node joins leaves. The various proposed DHT routing algorithms embody several different underlying routing geometries. These geometries include hypercubes, rings, tree-like structures, and butterfly networks. In this paper we focus on how these basic geometric approaches affect the resilience and proximity properties of DHTs. One factor that distinguishes these geometries is the degree of flexibility they provide in the selection of neighbors and routes. Flexibility is an important factor in achieving good static resilience and effective proximity neighbor and route selection. Our basic finding is that, despite our initial preference for more complex geometries, the ring geometry allows the greatest flexibility, and hence achieves the best resilience and proximity performance. Dipsea is a modular architecture for building a Distributed Hash Table (DHT). A DHT is a large hash table that is cooperatively maintained by a large number of machines communicating over the Internet. Decentralization and automatic re-configuration are two key design goals for a DHT. The architecture of Dipsea consists of three layers: ID Management, Overlay Routing and Data Management. The Overlay Routing layer consists of three modules: Emulation Engine, Ring Management and Choice of Long-Distance Links. Efficient algorithms for ID Management are designed—these algorithms require few messages, require few re-assignments of existing IDs and ensure that the hash table is divided among the participating machines as evenly as possible. Ring Management ensures that participating machines establish connections among themselves, as a function of their IDs, to form a fault-tolerant ring. The Emulation Engine is responsible for “emulation” of arbitrary families of routing networks. It handles issues arising out of dynamism (arrival and departure of participating machines), scale (variation in the average number of participating machines) and physical network proximity. Choice of Long-Distance Links allows a DHT to choose any family of routing networks (deterministic or randomized) for emulation. Several deterministic and randomized routing networks are designed and analyzed. Among these are Symphony (one of the first randomized DHT routing networks), Papillon (a deterministic routing network that guarantees asymptotically optimal route lengths with greedy routing for a fixed out-degree of nodes), and Mariposa (a randomized routing network that also guarantees optimal route lengths for a given out-degree of nodes).