Abstract
Broadcasting is the process of dissemination of a message from one vertex (called originator) to all other vertices in the graph. This task is accomplished by placing a sequence of calls between neighboring vertices, where one call requires one unit of time and each call involves exactly two vertices. Each vertex can participate in one call per one unit of time. Determination of the broadcast time of a vertex x in arbitrary graph G is NP-complete. Problem can be solved in polynomial time for trees and some subclasses of cactus graphs. In this paper broadcasting in cactus graphs is studied. An algorithm that determines broadcast time of any originator with time complexity O(n) in k-restricted cactus graph (where k is constant) is given. Furthermore, another algorithm which calculates broadcast time for all vertices in k-restricted cactus graph within the same time complexity is outlined. The algorithm also provides an optimal broadcast scheme for every vertex. As a byproduct, broadcast center of a k-restricted cactus graph is computed.
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References
Aho AV, Hopcroft JE, Ullman JD (1974) The design and analysis of computer algorithms. Addison-Wesley, Reading
Bar-Noy A, Guha S, Naor J, Schieber B (1998) Multicasting in heterogeneous networks. In: Proceedings of the thirtieth annual ACM symposium on theory of computing
Beier R, Sibeyn JF (2000) A powerful heuristic for telephone gossiping. Max-Planck-Inst. für Informatik, Bibliothek & Dokumentation
Ben-Moshe B, Dvir A, Segal M, Tamir A (2012) Centdian computation in cactus graphs. J Graph Algorithms Appl 16(2):199–224
Brandstädt A, Le VB, Spinrad JP (2000) Graph classes. A survey. SIAM, Philadelphia
Elenbogen B, Fink JF (2007) Distance distributions for graphs modeling computer networks. Discret Appl Math 155(18):2612–2624
Elkin M, Kortsarz G (2005) A combinatorial logarithmic approximation algorithm for the directed telephone broadcast problem. SIAM J Comput 35(3):672–689
Elkin M, Kortsarz G (2006) Sublogarithmic approximation for telephone multicast. J Comput Syst Sci 72(4):648–659
Feige U, Peleg D, Raghavan P, Upfal E (1990) Randomized broadcast in networks. Random Struct Algorithm 1(4):447–460
Fraigniaud P, Vial S (1997) Approximation algorithms for broadcasting and gossiping. J Parallel Distrib Comput 43(1):47–55
Fraigniaud P, Vial S (1997) Heuristic algorithms for personalized communication problems in point-to-point networks. In: Proceedings of the 4th colloquium on structural information and communication complexity, SIROCCO, vol 97, pp 240–252
Fraigniaud P, Vial S (1999) Comparison of heuristics for one-to-all and all-to-all communications in partial meshes. Parallel Process Lett 9(1):9–20
Garey M, Johnson D (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, San Francisco
Harutyunyan HA, Shao B (2006) An efficient heuristic for broadcasting in networks. J Parallel Distrib Comput 66(1):68–76
Harutyunyan HA, Laza G, Maraachlian E (2009) Broadcasting in necklace graphs. In: Proceedings of the 2nd Canadian conference on computer science and software engineering, C3S2E09, pp 253–256
Harutyunyan HA, Maraachlian E (2008) On broadcasting in unicyclic graphs. J Comb Optim 16:307–322
Harutyunyan HA, Maraachlian E (2009) Broadcasting in fully connected trees. In: International conference on parallel and distributed systems, ICPADS09, pp 740–745
Harutyunyan HA, Maraachlian E (2009) Linear algorithm for broadcasting in networks with no intersecting cycles. In: International conference on parallel and distributed processing techniques and applications, PDPTA09, pp 296–301
Hedetniemi ST, Laskar R, Pfaff J (1986) A linear algorithm for finding a minimum dominating set in a cactus. Discret Appl Math 13:287–292
Kortsarz G, Peleg D (1995) Approximation algorithms for minimum time broadcast. SIAM J Discret Math 8:401–427
Markov M, Ionut Andreica M, Manev K, Tapus N (2012) A linear time algorithm for computing longest paths in cactus graphs. Serdica J Comput 6(3):287–298
Paten B, Earl D, Nguyen N, Diekhans M, Zerbino D, Haussler D (2011) Cactus: algorithms for genome multiple sequence alignment. Genome Res 21(9):1512–1528
Ravi R (1994) Rapid rumor ramification: approximating the minimum broadcast time. In: 35th Annual symposium on foundations of computer science, pp 202–213
Scheuermann P, Wu G (1984) Heuristic algorithms for broadcasting in point-to-point computer networks. IEEE Trans Comput 100(9):804–811
Slater PJ, Cockayne EJ, Hedetniemi ST (1981) Information dissemination in trees. SIAM J Comput 10:692–701
Zmazek B, Žerovnik J (2004) The obnoxious center problem on weighted cactus graphs. Discret Appl Math 136:377–386
Zmazek B, Žerovnik J (2005) Estimating the traffic on weighted cactus networks in linear time. In: International conference on information visualisation, pp 536–541
Acknowledgments
The authors wish to sincerely thank to the anonymous referees for careful reading of the manuscript and for many constructive remarks. We also wish to thank to Sergio Cabello Justo for constructive discussion. This work was supported in part by Slovenian Research Agency ARRS (Grant number P1-0285-0101).
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Čevnik, M., Žerovnik, J. Broadcasting on cactus graphs. J Comb Optim 33, 292–316 (2017). https://doi.org/10.1007/s10878-015-9957-8
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DOI: https://doi.org/10.1007/s10878-015-9957-8