On the search path length of random binary skip graphs

We present Corona, a deterministic self-stabilizing algorithm for skip list construction in structured overlay networks. Corona operates in the low-atomicity message-passing asynchronous system model. Corona requires constant process memory space for its operation and, therefore, scales well. We prove the general necessary conditions limiting the initial states from which a self-stabilizing structured overlay network in a messagepassing system can be constructed. The conditions require that initial state information has to form a weakly connected graph and it should only contain identifiers that are present in the system. We formally describe Corona and rigorously prove that it stabilizes from an arbitrary initial state subject to the necessary conditions. We extend Corona to construct a skip graph. construct and to verify. Furthermore, a large initial state space may lead to excessive process memory demands during stabilization, especially during initial linearization: topological sorting of the processes [21].

IEEE Open Journal of the Communications Society, 2020

Skip graph is a distributed data structure that provides a scalable structured overlay network by routing in logarithmic time for resource location and dynamic node addition/deletion. However, most of the routing paths are quite longer than the shortest paths because each node in the network knows only its neighbors, rather than the global topology. In general, long routing paths lead to the high latency and the low fault tolerance. Herein, we propose Detouring Skip Graph, which performs more efficient routing through the use of detour routes. It does not require construction of extra links or modification of its topology; thereby, it shortens the paths without additional costs while maintaining the advantages of Skip Graph. Our evaluation experiments show that the proposed method tends to shorten the paths considerably, and in particular, that the average path length is approximately 20%-30% shorter than that of Skip Graph.

1996

We present a new type of search trees isomorphic to Skip-lists, i.e., there is a one-to-one mapping that commutes with the update algorithms. Moreover, as Skip-trees inherit all the race properties from Skip-lists, they can be selected as well as Skip-lists. We design a concurrent algorithm on the fly approach to update skip-trees. Among other advantages, this algorithm is more compressive than the one designed by Pugh for Skip-lists. It is based on the closest relationship between Skip-trees and the family of B-trees that allows the translation of local rules between them. From a practical point of view, although the Skip-list should be in main memory, the Skip-trees can be registered into secondary external storage, therefore it can be likely to analyse its ability to manage Databases.

Workshop on Distributed Data and Structures, 2000

We propose DSL, a new Scalable Distributed Data Structure for the dic- tionary problem, based on a version of Skip Lists, as an alternative to both random trees and deterministic height balanced trees. Our scheme exhibits, with high probability, logarithmic search time, constant reconstruction time, and linear space overhead. Additionally, at the expense of two additional pointers per internal node,

Journal of Algorithms, 1998

Distributed paging BFR92, ABF93b, AK95] deals with the dynamic allocation of copies of les in a distributed network as to minimize the total communication cost over a sequence of read and write requests.

Computer Communications, 2008

The support for complex queries, such as range, prefix and aggregation queries, over structured peer-to-peer systems is currently an active and significant topic of research. This paper demonstrates how Skip Tree Graph, as a novel structure, presents an efficient solution to that problem area through provision of a distributed search tree functionality on decentralised and dynamic environments. Since Skip Tree Graph is based on skip trees, a concurrent approach to skip lists, it constitutes an augmentation of skip graphs that extends its functionality and allows for important performance improvements. This work presents a thorough comparison between these two related peer-to-peer overlay networks, their construction, search algorithms and properties. Being based on tree structures, skip tree graphs supports aggregation queries and multicast/broadcast operations, which cannot be directly implemented in its predecessor. The repair mechanism for healing the structure in case of failures is more efficient and harnesses the parallelism inherent in P2P networks. Particular consideration is given to the performance of different range-query schemes over the two related structures. Theoretical and experimental results conclude that Skip Tree Graphs outperform skip graphs on both exact-match and range searches.

Proceedings of the 28th ACM symposium on Principles of distributed computing - PODC '09, 2009

Peer-to-peer systems rely on scalable overlay networks that enable efficient routing between its members. Hypercubic topologies facilitate such operations while each node only needs to connect to a small number of other nodes. In contrast to static communication networks, peer-to-peer networks allow nodes to adapt their neighbor set over time in order to react to join and leave events and failures. This paper shows how to maintain such networks in a robust manner. Concretely, we present a distributed and self-stabilizing algorithm that constructs a (variant of the) skip graph in polylogarithmic time from any initial state in which the overlay network is still weakly connected. This is an exponential improvement compared to previously known self-stabilizing algorithms for overlay networks. In addition, individual joins and leaves are handled locally and require little work.

arXiv (Cornell University), 2011

Random walks are useful in many applications on complex networks, like, for instance, searching a network for a desired resource. In that case, their simplicity has the drawback that searches may need many hops to find the resource. Modifications of the regular random walk have been suggested to improve search performance. In this work, we propose a network search mechanism based on building random walks connecting together partial walks that have been precomputed at each network node. In an initial stage, each node computes a number of partial walks, associating each of them with a data structure containing the set of resources found in the nodes of that partial walk. We assume these data structures to be Bloom filters, which are very compact but can make false positive mistakes. When a node starts a search, it chooses one of the partial walks of the node and queries its Bloom filter for the desired resource. If the result is negative, the search jumps to the last node of that partial walk, repeating the process. If the result is positive, the search traverses that partial walk checking each node until the resource is found. With this strategy, many hops of the regular random walk are saved because of the jumps over partial walks where we know the resource is not located. However, additional unnecessary hops come from false positives at the Bloom filters. An analytical model has been developed to predict the expected search length of this mechanism. The model also provides expressions for the optimal size of the partial walks and the corresponding optimal (shortest) expected search length. We have found that the optimal search length is proportional to the square root of the expected search length of regular random walks, achieving a significant improvement. We have performed simulation experiments finding that the model predictions are very close to the experimental data. The experimental study is also used to assess the impact of the number of partial walks precomputed by each network node in the statistical behavior of the lengths of the searches. We have found that with just two partial walks per node the results are similar to those obtained for larger values, which is a significant result regarding the practical implementation of the search mechanism.

2008

Abstract We study the problem of optimal skip placement in an inverted list. Assuming the query distribution to be known in advance, we formally prove that an optimal skip placement can be computed quite efficiently. Our best algorithm runs in time O (n log n), n being the length of the list.

Lecture Notes in Computer Science, 2011

We introduce a generalization of the distributed sorting problem on chain network. Our problem consists of sorting values in processes that are separated from each other by any number of intermediate processes which can relay values but do not have their own values. We solve this problem in a chain network by proposing a silent self-stabilizing distributed algorithm.