Investigating the Existence and the Regularity of Logarithmic Harary Graphs

This paper studies the existence and the regularity of Logarithmic Harary Graphs (LHGs). This study is motivated by the fact that these graphs are employed for modeling the communication topology to support efficient flooding in presence of link and node failures when considering an initial arbitrary number of nodes n. Therefore, the capability to identify graph constraints that allow the construction of LHGs for the largest number of pairs (n, k) (where k is the desired degree of connectivity to be tolerant to failures) becomes of primary importance. The paper presents several results in that direction. We introduce a graph constraint, namely K-TREE, that allows the construction of a LHG for every pair (n, k) such that n ges 2k. Secondly we presents another graph constraint for LHG, namely KDIAMOND, which is equivalent to K-TREE in terms of capability to construct LHGs for any pair (n, k). The interest of K-DIAMOND lies in the fact that, for a given k, KDIAMOND allows to construct more regular graphs than K-TREE does. A k-regular graph shows the minimal number of links required by a k-connected graph, leading tominimal flooding cost. The paper formally shows, in particular, that there are an infinite number of pairs (n, k), such that there exists a k-regular LHG for the pair (n, k) that satisfies K-DIAMOND and does not satisfy K-TREE.

[1]  Eli Upfal,et al.  Building low-diameter P2P networks , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[2]  Antony I. T. Rowstron,et al.  Pastry: Scalable, Decentralized Object Location, and Routing for Large-Scale Peer-to-Peer Systems , 2001, Middleware.

[3]  Kirk L. Johnson,et al.  Overcast: reliable multicasting with on overlay network , 2000, OSDI.

[4]  Keith Marzullo,et al.  Gossip versus Deterministic Flooding: Low Message Overhead and High Reliability for Broadcasting on Small Networks , 1999 .

[5]  Alan J. Demers,et al.  Logarithmic Harary graphs , 2001, Proceedings 21st International Conference on Distributed Computing Systems Workshops.

[6]  Kai-Yeung Siu,et al.  Distributed construction of random expander networks , 2003, IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No.03CH37428).

[7]  Anne-Marie Kermarrec,et al.  Lightweight probabilistic broadcast , 2003, TOCS.

[8]  Flavio D'Alessandro.  A combinatorial problem on Trapezoidal words , 2002, Theor. Comput. Sci..

[9]  de Ng Dick Bruijn.  A combinatorial problem , 1946 .

[10]  Miguel Castro,et al.  Scribe: a large-scale and decentralized application-level multicast infrastructure , 2002, IEEE J. Sel. Areas Commun..

[11]  Dmitri Loguinov,et al.  Graph-theoretic analysis of structured peer-to-peer systems: routing distances and fault resilience , 2003, IEEE/ACM Transactions on Networking.

[12]  Xiaozhou Li,et al.  Active and Concurrent Topology Maintenance , 2004, DISC.

[13]  Idit Keidar,et al.  Araneola: a scalable reliable multicast system for dynamic environments , 2004, Third IEEE International Symposium on Network Computing and Applications, 2004. (NCA 2004). Proceedings..

[14]  ZHANGLi-xia,et al.  A reliable multicast framework for light-weight sessions and application level framing , 1995 .

[15]  K. Menger.  Zur allgemeinen Kurventheorie , 1927 .

[16]  James Aspnes,et al.  Skip graphs , 2003, SODA '03.

[17]  David R. Karger,et al.  Koorde: A Simple Degree-Optimal Distributed Hash Table , 2003, IPTPS.

[18]  Steven Skiena,et al.  Implementing discrete mathematics - combinatorics and graph theory with Mathematica , 1990 .

[19]  Katherine Guo,et al.  Scalable Stability Detection Using Logical Hypercube , 2002, IEEE Trans. Parallel Distributed Syst..

[20]  Moni Naor,et al.  Viceroy: a scalable and dynamic emulation of the butterfly , 2002, PODC '02.

[21]  Hartmut Ehrig,et al.  Theory of Constraints and Application Conditions: From Graphs to High-Level Structures , 2004, Fundam. Informaticae.