A Self-Stabilizing Memory Efficient Algorithm for the Minimum Diameter Spanning Tree under an Omnipotent Daemon

The diameter of a network is one of the most fundamental network parameters. Being able to compute the diameter is an important problem in the analysis of large networks, and moreover this parameter has many important practical applications in real networks. As a consequence, it is natural to study this problem in a distributed system, and more specifically in a distributed system tolerant to transient faults. More specifically, we are interested in the problem to identify one of the centers of graph. Once done, we construct a minimum diameter spanning tree rooted in this centre. Of course, the challenging problem is to compute one centre of the graph. We present a uniform self-stabilizing algorithm for the minimum diameter spanning tree construction problem in the state model. Our protocol has several attractive features that makes it suitable for practical purposes. It is the first algorithm for this problem that operates under the unfair adversary (also called unfair daemon). In other words, no restriction is made on the distributed behaviour of the system. Consequently, it is the hardest adversary to deal with. Moreover, our algorithm needs only O(log n) bits of memory per process (where n is the number of processes), that improves the previous result by a factor n. These improvements are not achieved to the detriment of the convergence time, that stays reasonable with O(n2) rounds.

[1]  Mikhail J. Atallah,et al.  Algorithms and Theory of Computation Handbook , 2009, Chapman & Hall/CRC Applied Algorithms and Data Structures series.

[2]  Felix C. Gaertner,et al.  A Survey of Self-Stabilizing Spanning-Tree Construction Algorithms , 2003 .

[3]  Tetz C. Huang,et al.  A self-stabilizing algorithm for the shortest path problem assuming read/write atomicity , 2005, J. Comput. Syst. Sci..

[4]  Ajoy Kumar Datta,et al.  An O(n)-time self-stabilizing leader election algorithm , 2011, J. Parallel Distributed Comput..

[5]  Maria Gradinariu Potop-Butucaru,et al.  A Superstabilizing log(n)-Approximation Algorithm for Dynamic Steiner Trees , 2009, SSS.

[6]  Alain Cournier.  A New Polynomial Silent Stabilizing Spanning-Tree Construction Algorithm , 2009, SIROCCO.

[7]  Sébastien Tixeuil,et al.  A Taxonomy of Daemons in Self-stabilization , 2011, ArXiv.

[8]  David Peleg,et al.  Distributed Algorithms for Network Diameter and Girth , 2012, ICALP.

[9]  Boaz Patt-Shamir,et al.  On Proof-Labeling Schemes versus Silent Self-stabilizing Algorithms , 2014, SSS.

[10]  Roger Wattenhofer,et al.  Optimal distributed all pairs shortest paths and applications , 2012, PODC '12.

[11]  Christian Lavault,et al.  A Uniform Self-Stabilizing Minimum Diameter Tree Algorithm (Extended Abstract) , 1995, WDAG.

[12]  Amos Israeli,et al.  Uniform Dynamic Self-Stabilizing Leader Election , 1997, IEEE Trans. Parallel Distributed Syst..

[13]  Franck Petit,et al.  Time and Space Optimality of Distributed Depth-First Token Circulation Algorithms , 1999, WDAS.

[14]  Mehmet Hakan Karaata,et al.  Self-stabilizing algorithms for finding centers and medians of trees , 1994, PODC '94.

[15]  Yehuda Afek,et al.  Self-stabilizing unidirectional network algorithms by power-supply , 1997, SODA '97.

[16]  Sébastien Tixeuil,et al.  Compact Deterministic Self-stabilizing Leader Election - The Exponential Advantage of Being Talkative , 2013, DISC.

[17]  Maria Gradinariu Potop-Butucaru,et al.  Fast Self-Stabilizing Minimum Spanning Tree Construction , 2010, ArXiv.

[18]  Chak-Kuen Wong,et al.  Minimum Diameter Spanning Trees and Related Problems , 1991, SIAM J. Comput..

[19]  Liam Roditty,et al.  Fast approximation algorithms for the diameter and radius of sparse graphs , 2013, STOC '13.

[20]  Nicola Santoro,et al.  Distributed algorithms for finding centers and medians in networks , 1984, TOPL.

[21]  Refael Hassin,et al.  On the Minimum Diameter Spanning Tree Problem , 1995, Inf. Process. Lett..

[22]  Vincent Villain,et al.  The First Fully Polynomial Stabilizing Algorithm for BFS Tree Construction , 2011, OPODIS.

[23]  Shing-Tsaan Huang,et al.  A Self-Stabilizing Algorithm for Constructing Breadth-First Trees , 1992, Inf. Process. Lett..

[24]  Shlomi Dolev,et al.  Self-Stabilizing Depth-First Search , 1994, Inf. Process. Lett..

[25]  Pradip K. Srimani,et al.  A Self-stabilizing Distributed Algorithm to Find the Center of a Tree Graph , 1997, Parallel Algorithms Appl..

[26]  Sandeep K. S. Gupta,et al.  Self-Stabilizing Protocol for Shortest Path Tree for Multi-cast Routing in Mobile Networks (Research Note) , 2000, Euro-Par.

[27]  Adrian Kosowski,et al.  A Self-stabilizing Algorithm for Finding a Spanning Tree in a Polynomial Number of Moves , 2005, PPAM.

[28]  Maria Gradinariu Potop-Butucaru,et al.  Self-stabilizing minimum-degree spanning tree within one from the optimal degree , 2009, 2009 IEEE International Symposium on Parallel & Distributed Processing.

[29]  Ajoy Kumar Datta,et al.  Leader Election and Centers and Medians in Tree Networks , 2013, SSS.

[30]  Sébastien Tixeuil,et al.  Compact deterministic self-stabilizing leader election on a ring: the exponential advantage of being talkative , 2017, Distributed Computing.

[31]  Shlomi Dolev,et al.  Self Stabilization , 2004, J. Aerosp. Comput. Inf. Commun..

[32]  Maria Gradinariu Potop-Butucaru,et al.  Fast Self-stabilizing Minimum Spanning Tree Construction - Using Compact Nearest Common Ancestor Labeling Scheme , 2010, DISC.

[33]  Edsger W. Dijkstra,et al.  Self-stabilizing systems in spite of distributed control , 1974, CACM.

[34]  Amos Israeli,et al.  Resource bounds for self stabilizing message driven protocols , 1991, PODC '91.

[35]  Shay Kutten,et al.  Time Optimal Asynchronous Self-stabilizing Spanning Tree , 2007, DISC.

[36]  Shing-Tsaan Huang,et al.  Self-stabilizing depth-first token circulation on networks , 2005, Distributed Computing.

[37]  Toshimitsu Masuzawa,et al.  Fast and compact self-stabilizing verification, computation, and fault detection of an MST , 2011, PODC '11.