Randomized Lower Bound for Distributed Spanning-Tree Verification

Author(s):  
Taisuke Izumi
Keyword(s):  
2018 ◽  
Vol 242 ◽  
pp. 82-88
Author(s):  
Zoltán Király
Keyword(s):  

2010 ◽  
Vol 14 (3) ◽  
Author(s):  
Rakesh Kawatra

In this paper we present a new heuristic procedure to solve the degree constrained minimal spanning tree problem. This procedure uses Lagrangian relaxation of the integer programming formulation of the problem to get a lower bound for the optimal objective function value. A subgradient optimization method is used to find multipliers that give good lower bounds. A branch exchange procedure is used after each iteration of the subgradient optimization to generate a feasible solution from an infeasible Lagrangean solution. Computational results are given for problems with up to 300 nodes. The heuristic procedure presented here gives optimal solutions in most instances. For problem sets that were not solved optimally, the gap between the lower bound and the feasible solution was less than 10-2 percent.


1999 ◽  
Vol 10 (02) ◽  
pp. 211-223 ◽  
Author(s):  
SAVIO S. H. TSE ◽  
FRANCIS C. M. LAU

We study the problem of adaptive polling in undirected general networks. Polling, also known as broadcast-confirm, consists a propagation round and a feedback round. In adaptive polling, a spanning tree of unknown topology is built dynamically during the propagation round, and feedback messages are free to choose their paths in order to adapt to traffic and fault situations. We study three adaptive polling algorithms and analyze their worst-case communication bit complexities in the propagation round. Then, we prove a lower bound on the worst-case communication bit complexity of Ω(e+n log n) in the propagation round for all algorithms of the same kind as the three algorithms we study, where n is the number of nodes, and e the number of edges. We conclude that the cost introduced into the network due to the running of an adaptive polling algorithm is mild.


2018 ◽  
Vol 29 (04) ◽  
pp. 505-527
Author(s):  
Maria Paola Bianchi ◽  
Hans-Joachim Böckenhauer ◽  
Tatjana Brülisauer ◽  
Dennis Komm ◽  
Beatrice Palano

In the online minimum spanning tree problem, a graph is revealed vertex by vertex; together with every vertex, all edges to vertices that are already known are given, and an online algorithm must irrevocably choose a subset of them as a part of its solution. The advice complexity of an online problem is a means to quantify the information that needs to be extracted from the input to achieve good results. For a graph of size [Formula: see text], we show an asymptotically tight bound of [Formula: see text] on the number of advice bits to produce an optimal solution for any given graph. For particular graph classes, e.g., with bounded degree or a restricted edge weight function, we prove that the upper bound can be drastically reduced; e.g., [Formula: see text] advice bits allow to compute an optimal result if the weight function equals the Euclidean distance; if the graph is complete and has two different edge weights, even a logarithmic number suffices. Some of these results make use of the optimality of Kruskal’s algorithm for the offline setting. We also study the trade-off between the number of advice bits and the achievable competitive ratio. To this end, we perform a reduction from another online problem to obtain a linear lower bound on the advice complexity for any near-optimal solution. Using our results finally allows us to give a lower bound on the expected competitive ratio of any randomized online algorithm for the problem, even on graphs with three different edge weights.


2016 ◽  
Vol 27 (05) ◽  
pp. 579-594
Author(s):  
Savio S. H. Tse

We study the problem of credit-based adaptive polling in undirected arbitrary point-to-point asynchronous networks. Polling consists of two rounds, namely propagation (broadcast) and feedback (confirmation, response) rounds. By adaptive polling, a spanning tree of unknown topology is built dynamically during the propagation round, and feedback messages are free to choose their paths back to the initiator — a specific node who initiates the polling algorithm. The freedom in the feedback round relies on the use of credits in the propagation round. We re-visit three existing algorithms and analyse their average case communication bit complexities incurred by the credits in the propagation round, and these analyses match with the numerical results. We also give an optimal lower bound on the worst case bit message complexity for the case when the number of nodes in the network is unknown.


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