DISC 2000: Distributed Computing pp 283-296

# Locating Information with Uncertainty in Fully Interconnected Networks

• Lefteris M. Kirousis
• Evangelos Kranakis
• Danny Krizanc
• Yannis C. Stamatiou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1914)

## Abstract

We consider the problem of searching for a piece of informa- tion in a fully interconnected computer network (or clique) by exploiting advice about its location from the network nodes. Each node contains a database that “knows” what kind of documents or information are stored in other nodes (e.g. a node could be a Web server that answers queries about documents stored on the Web). The databases in each node, when queried, provide a pointer that leads to the node that contains the infor- mation. However, this information is up-to-date (or correct) with some bounded probability. While, in principle, one may always locate the infor- mation by simply visiting the network nodes in some prescribed ordering, this requires a time complexity in the order of the number of nodes of the network. In this paper, we provide algorithms for locating an informa- tion node in the complete communication network, that take advantage of advice given from network nodes. The nodes may either give correct advice, by pointing directly to the information node, or give wrong advice by pointing elsewhere. We show that, on the average, if the probability p that a node provides correct advice is asymptotically larger than 1/n, where n is the number of the computer nodes, then the average time com- plexity for locating the information node is, asymptotically, 1/p or 2/p depending on the available memory. The probability p may, in general, be a function of the number of network nodes n. On the lower bounds side, we prove that no fixed memory deterministic algorithm can locate the information node in finite expected number of steps. We also prove a lower bound of - for the expected number of steps of any algorithm that locates the information node in the complete network.

## KeyWords and Phrases

Search Information Retrieval Complete

## References

1. 1.
Y. Afek, E. Gafni, and M. Ricklin, Upper and lower bounds for routing schemes in dynamic networks, in: Proc. 30th Symposium on Foundations of Computer Science, (1989), 370–375.Google Scholar
2. 2.
S. Albers and M. Henzinger, Exploring unknown environments, in Proc. 29th Symposium on Theory of Computing, (1999), 416–425.Google Scholar
3. 3.
B. Awerbuch, B. Patt-Shamir, and G. Varghese, Self-stabilizing end-to-end communication, Journal of High Speed Networks 5 (1996), 365–381.Google Scholar
4. 4.
R.A. Baeza-Yates, J.C. Culberson, and G.J.E. Rawlins, Searching in the plane, Information and Computation 106(2) (1993), 234–252.
5. 5.
D. Bienstock and P. Seymour, Monotonicity in graph searching, Journal of Algorithms 12 (1991), 239–245.
6. 6.
R. Cole, B. Maggs, and R. Sitaraman, Routing on butterfly networks with random faults, in: Proc. 36th Symposium on Foundations of Computer Science, (1995), 558–570.Google Scholar
7. 7.
X. Deng and C. Papadimitriou, Exploring an unknown graph, in Proc: 31st Symposium on Foundations of Computer Science, 1990, 356–361.Google Scholar
8. 8.
S. Dolev, E. Kranakis, D. Krizanc, and D. Peleg, Bubbles: Adaptive routing scheme for high-speed networks, SI AM Journal on Computing, to appear.Google Scholar
9. 9.
W. Evans and N. Pippenger, Lower bounds for noisy boolean decision trees, in Proc. 28th Symposium on Theory of Computing, (1996), 620–628.Google Scholar
10. 10.
U. Fiege, D. Peleg, P. Raghavan, and E. Upfal, Computing with uncertainty, in Proc. 22nd Symposium on Theory of Computing, (1990), 128–137.Google Scholar
11. 11.
L. Kirousis and C. Papadimitriou, Interval graphs and searching, Discrete Mathematics 55 (1985), 181–184.
12. 12.
E. Kranakis and D. Krizanc, Searching with uncertainty, in: Proc. 6th International Colloquium on Structural Information and Communication Complexity (SIROCCO), (1999), C. Gavoille, J.-C, Bermond, and A. Raspaud, eds., pp, 194–203, Carleton Scientific, 1999.Google Scholar
13. 13.
E. Kushilevitz and Y. Mansour, Computation in noisy radio networks, in Proc. 9th Symposium on Discrete Algorithms, 1998, 236–243.Google Scholar
14. 14.
T. Leighton and B. Maggs, Expanders might be practical, in: 30th Proc. Symposium on Foundations of Computer Science, (1989), 384–389.Google Scholar
15. 15.
L. Lovasz, RandomWalks on Graphs: A Survey, Combinatorics 2 (1993), 1–46.Google Scholar
16. 16.
N. Megiddo, S. Hakimi, M. Garey, D. Johnson, and C. Papadimitriou, The complexity of searching a graph, Journal of the ACM 35 (1988), 18–44.
17. 17.
P. Panaite and A. Pelc, Exploring unknown undirected graphs, in: Proc. 9th Symposium on Discrete Algorithms, (1998), 316–322.Google Scholar
18. 18.
C.H. Papadimitriou and M. Yannakakis, Shortest paths without a map, Theoretical Computer Science 84(1) (1991), 127–150.
19. 19.
N. Pippenger, On networks of noisy gates, in: Proc. 26th Symposium on Foundations of Computer Science, (1985), 30–36.Google Scholar
20. 20.
P. Raghavan, Robust algorithms for packet routing in a mesh, in: Proc. 1st Symposium on Parallel Algorithms and Architectures, (1989), 344–350.Google Scholar
21. 21.
T.E.S. Raghavan, T.S. Ferguson, T. Part has apathy, and O. J. Vrieze eds., Stochastic games and related topics. Kluwer Academic Publishers, 1991.Google Scholar
22. 22.
W.H. Ruckle, Geometric games and their applications. Research Notes in Mathematics, Pitman Publishing Inc., 1983.Google Scholar

## Authors and Affiliations

• Lefteris M. Kirousis
• 1
• Evangelos Kranakis
• 2
• Danny Krizanc
• 3
• Yannis C. Stamatiou
• 4
1. 1.Department of Computer Engineering and InformaticsUniversity of PatrasPatrasGreece
2. 2.Carleton University School of Computer ScienceOttawaCanada
3. 3.Department of MathematicsWesleyan University
4. 4.Department of Computer Engineering and InformaticsUniversity of PatrasPatrasPatrasGreeceGreece