# Informative Labeling Schemes for Graphs

Extended Abstract
• David Peleg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1893)

## Abstract

This paper introduces and studies the notion of informative labeling schemes for arbitrary graphs. Let f(W) be a function on subsets of vertices W. An f labeling scheme labels the vertices of a weighted graph G in such a way that f(W) can be inferred efficiently for any vertex subset W of G by merely inspecting the labels of the vertices of W, without having to use any additional information sources.

The paper develops f labeling schemes for some functions f over the class of n-vertex trees, including SepLevel, the separation level of any two vertices in the tree, LCA, the least common ancestor of any two vertices, and Center, the center of any three given vertices in the tree. These schemes use O(log2 n)-bit labels, which is asymptotically optimal.

We then turn to weighted graphs and consider the function Steiner(W), denoting the weight of the Steiner tree spanning the vertices of W in the graph. For n-vertex weighted trees with M-bit edge weights, it is shown that there exists a Steiner labeling scheme using O((M+log n) log n) bit labels, which is asymptotically optimal. In the full paper it is shown that for the class of arbitrary n-vertex graphs with M-bit edge weights, there exists an approximate- Steiner labeling scheme, providing an estimate (up to a logarithmic factor) for Steiner(W) using O((M + logn) log2n) bit labels.

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### References

1. 1.
M.A. Breuer and J. Folkman. An unexpected result on coding the vertices of a graph. J. Mathemat. Analysis and Applic., 20:583–600, 1967.
2. 2.
M.A. Breuer. Coding the vertexes of a graph. IEEE Trans. on Information Theory, IT-12:148–153, 1966.
3. 3.
C. Gavoille, D. Peleg, S. Pérennes and R. Raz. Distance labeling in graphs. Unpublished manuscript, September 1999.Google Scholar
4. 4.
M. Katz, N.A. Katz and D. Peleg. Distance labeling schemes for well-separated graph classes. In Proc. 17th STACS, pages 516–528, 2000.Google Scholar
5. 5.
L. Kou, G. Markowsky and L. Berman. A fast algorithm for Steiner trees. Acta Informatica, 15:141–145, 1981.
6. 6.
S. Kannan, M. Naor and S. Rudich. Implicit representation of graphs. In Proc. 20th STOC, pages 334–343, May 1988.Google Scholar
7. 7.
D. Peleg. Proximity-preserving labeling schemes and their applications. In Proc. 25th WG, pages 30–41, June 1999.Google Scholar
8. 8.
D. Peleg. Informative labeling schemes for graphs. Technical Report MCS00-05, The Weizmann Institute of Science, 2000.Google Scholar
9. 9.
N. Santoro and R. Khatib. Labelling and implicit routing in networks. The Computer Journal, 28:5–8, 1985.