Abstract
A new characterization of tree medians is presented: we show that a vertex m is a median of a tree T with n vertices iff there exists a partition of the vertex set into ⌊n/2⌋ disjoint pairs (excluding m when n is odd), such that all the paths connecting the two vertices in any of the pairs pass through m. We show that in this case the sum of the distances between these pairs of vertices is the largest possible among all such partitions, and we use this fact to discuss lower bounds on the message complexity of the distributed sorting problem. We show that, given a network of a tree topology, choosing a median and then routing all the information through it is the best possible strategy, in terms of worst-case number of messages sent during any execution of any distributed sorting algorithm. We also discuss the implications for networks of a general topology and for the distributed ranking problem.
The work of this author was supported in part by the ESPRIT II Basic Research Actions Program of the EC under contract no. 3075 (project ALCOM), while visiting the Department of Computer Science, Aarhus University, DK-8000 Aarhus C, Denmark.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, North-Holland, 1976.
R.G. Gallager, P.A. Humblet and P.M. Spira, A distributed algorithm for minimum spanning tree, ACM Trans. on Programming Languages and Systems, 5, 1, 1983, pp. 66–77.
O. Gerstel, Y. Mansour and S. Zaks, Bit complexity of order statistics on a distributed star network, Information Processing Letters, 30, 1989, pp. 127–132.
S.L. Hakimi, Optimum locations of switching centers and the absolute centers and medians of a graph, Operations Res., 12, 1964, pp. 450–459.
C. Jordan, Sur les assembblages des lignes, J. Reine Angew. Math., 70, 1869.
O. Kariv and S.L. Hakimi, An algorithmic approach to network location problems. I: the p-centers, SIAM J. Appl. Math., Vol. 37, No. 3, December 1979, pp. 513–538.
O. Kariv and S.L. Hakimi, An algorithmic approach to network location problems. II: the p-medians, SIAM J. Appl. Math., Vol. 37, No. 3, December 1979, pp. 539–560.
E. Korach, D. Rotem and N. Santoro, Distributed algorithms for finding centers and medians in networks, ACM Trans. on Programming Languages and Systems, Vol. 6, No. 3, July 1984, pp. 380–401.
E. Korach, D. Rotem and N. Santoro, Distributed algorithms for ranking the vertices of a network, Proceedings of the 13th Southeastern Conference on Combinatorics, Graph Theory and Computing, 1982, pp. 235–246.
M.C. Loui, The complexity of sorting on distributed systems, Information and Control, Vol. 60, 1–3, 1984, Vol. 60, 1–3, 1984, pp. 70–85.
K.B. Reid, Centroids to centers in trees, Networks, Vol. 21, 1991, pp. 11–17.
P.J. Slater, Centers to centroids in graphs, J. of Graph Theory, 2, 1978, pp. 209–222.
O. Wolfson and A. Milo, The multicast policy and its relationship to replicated data placement, ACM Trans. on Database Systems, Vol. 16, 1, 1991, pp. 181–2055.
S. Zaks, Optimal distributed algorithms for sorting and ranking, IEEE Trans. on Computers, c-34,4,April 1985, pp. 376–379.
B. Zelinka, Medians and peripherians of trees, Arch. Math. (Brno) 4, 1968, pp. 87–95.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gerstel, O., Zaks, S. (1993). A new characterization of tree medians with applications to distributed algorithms. In: Mayr, E.W. (eds) Graph-Theoretic Concepts in Computer Science. WG 1992. Lecture Notes in Computer Science, vol 657. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56402-0_43
Download citation
DOI: https://doi.org/10.1007/3-540-56402-0_43
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56402-7
Online ISBN: 978-3-540-47554-5
eBook Packages: Springer Book Archive