Broadcasting Spanning Forests on a Multiple-Access Channel

Abstract The problem of finding a spanning forest of a graph in a distributed-processing environment is studied. If an input graph is weighted, then the goal is to find a minimum-weight spanning forest. The processors communicate by broadcasting. The output consists of the edges that make a spanning forest and have been broadcast on the network. Input edges are distributed among the processors, with each edge held by one processor. The underlying broadcast network is implemented as a multiple-access channel. If exactly one processor attempts to perform a broadcast, then the broadcast is successful. A message broadcast successfully is delivered to all the processors in one step. If more than one processors broadcast simultaneously, then the messages interfere with each other and no processor can receive any of them. Optimality of algorithmic solutions is investigated, by way of comparing deterministic with randomized algorithms, and adaptive with oblivious ones. Lower bounds are proved that either justify the optimality of specific algorithms or show that the optimal performance depends on a class of algorithms.

[1]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[2]  Charles U. Martel,et al.  The Complexity of Selection Resolution, Conflict Resolution and Maximum Finding on Multiple Access Channels , 1988, AWOC.

[3]  Andrea E. F. Clementi,et al.  Optimal F-Reliable Protocols for the Do-All Problem on Single-Hop Wireless Networks , 2002, ISAAC.

[4]  Andrzej Pelc,et al.  The Wakeup Problem in Synchronous Broadcast Systems , 2000, SIAM J. Discret. Math..

[5]  Dariusz R. Kowalski,et al.  Finding Spanning Forests by Broadcasting , 2002, SIROCCO.

[6]  Baruch Awerbuch,et al.  Optimal distributed algorithms for minimum weight spanning tree, counting, leader election, and related problems , 1987, STOC.

[7]  Shay Kutten,et al.  A sub-linear time distributed algorithm for minimum-weight spanning trees , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[8]  Joseph Y. Halpern,et al.  Performing work efficiently in the presence of faults , 1992, PODC '92.

[9]  Frank Thomson Leighton,et al.  Analysis of Backoff Protocols for Multiple Access Channels , 1996, SIAM J. Comput..

[10]  Leslie Ann Goldberg,et al.  Contention resolution with guaranteed constant expected delay , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[11]  Tomasz Jurdzinski,et al.  Probabilistic Algorithms for the Wakeup Problem in Single-Hop Radio Networks , 2002, ISAAC.

[12]  Francis Y. L. Chin,et al.  Improving the Time Complexity of Message-Optimal Distributed Algorithms for Minimum-Weight Spanning Trees , 1990, SIAM J. Comput..

[13]  G. Grimmett,et al.  Probability and random processes , 2002 .

[14]  Eli Upfal,et al.  Stochastic contention resolution with short delays , 1995, STOC '95.

[15]  Pierre A. Humblet,et al.  A Distributed Algorithm for Minimum-Weight Spanning Trees , 1983, TOPL.

[16]  Dan E. Willard,et al.  Log-Logarithmic Selection Resolution Protocols in a Multiple Access Channel , 1986, SIAM J. Comput..

[17]  David Peleg,et al.  A Near-Tight Lower Bound on the Time Complexity of Distributed Minimum-Weight Spanning Tree Construction , 2000, SIAM J. Comput..

[18]  Boaz Patt-Shamir,et al.  Distributed MST for constant diameter graphs , 2001, PODC '01.

[19]  R. A. Doney,et al.  4. Probability and Random Processes , 1993 .

[20]  Eyal Kushilevitz,et al.  An Omega(D log (N/D)) Lower Bound for Broadcast in Radio Networks , 1998, SIAM J. Comput..

[21]  Alessandro Panconesi,et al.  Nearly optimal distributed edge coloring in O (log log n ) rounds , 1997 .

[22]  Albert G. Greenberg,et al.  A lower bound on the time needed in the worst case to resolve conflicts deterministically in multiple access channels , 1985, JACM.

[23]  János Komlós,et al.  An asymptotically fast nonadaptive algorithm for conflict resolution in multiple-access channels , 1985, IEEE Trans. Inf. Theory.

[24]  Shay Kutten,et al.  Fast Distributed Construction of Small k-Dominating Sets and Applications , 1998, J. Algorithms.

[25]  Charles U. Martel,et al.  Maximum Finding on a Multiple Access Broadcast Network , 1994, Inf. Process. Lett..

[26]  Gordon Bell,et al.  Ethernet: Distributed Packet Switching for Local Computer Networks , 1976 .

[27]  David Peleg,et al.  A near-tight lower bound on the time complexity of distributed MST construction , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[28]  Michalis Faloutsos,et al.  Optimal distributed algorithm for minimum spanning trees revisited , 1995, PODC '95.

[29]  Nathan Linial,et al.  Locality in Distributed Graph Algorithms , 1992, SIAM J. Comput..

[30]  Dimitri P. Bertsekas,et al.  Data Networks , 1986 .

[31]  Alessandro Panconesi,et al.  A faster distributed algorithm for computing maximal matchings deterministically , 1999, PODC '99.

[32]  Miroslaw Kutylowski,et al.  Efficient algorithms for leader election in radio networks , 2002, PODC '02.

[33]  Norman M. Abramson,et al.  Development of the ALOHANET , 1985, IEEE Trans. Inf. Theory.

[34]  JANOS KOMLGS,et al.  An Asymptotically Nonadaptive Algorithm for Conflict Resolution in Multiple-Access Channels , 1985 .

[35]  Robert G. Gallager,et al.  A perspective on multiaccess channels , 1984, IEEE Trans. Inf. Theory.

[36]  Andrzej Lingas,et al.  Performing work in broadcast networks , 2005, Distributed Computing.

[37]  Eyal Kushilevitz,et al.  An Ω(D log(N/D)) lower bound for broadcast in radio networks , 1993, PODC '93.

[38]  Andrzej Lingas,et al.  The do-all problem in broadcast networks , 2001, PODC '01.