Implicit representation of graphs
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Tree Contractions and Evolutionary Trees
SIAM Journal on Computing
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Nearest common ancestors: a survey and a new distributed algorithm
Proceedings of the fourteenth annual ACM symposium on Parallel algorithms and architectures
Labeling schemes for small distances in trees
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Small Induced-Universal Graphs and Compact Implicit Graph Representations
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Proximity-Preserving Labeling Schemes and Their Applications
WG '99 Proceedings of the 25th International Workshop on Graph-Theoretic Concepts in Computer Science
Approximate Distance Labeling Schemes
ESA '01 Proceedings of the 9th Annual European Symposium on Algorithms
The height of a random binary search tree
Journal of the ACM (JACM)
Compact and localized distributed data structures
Distributed Computing - Papers in celebration of the 20th anniversary of PODC
Labeling schemes for weighted dynamic trees
Information and Computation
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We study how to label the vertices of a tree in such a way that we can decide the distance of two vertices in the tree given only their labels. For trees, Gavoille et al. [7] proved that for any such distance labelling scheme, the maximum label length is at least ${1 \over 8} {\rm log}^{2} n - O({\rm log} n)$ bits. They also gave a separator-based labelling scheme that has the optimal label length ${\it \Theta}({\rm log} {n} \cdot {\rm log}(H_{n}(T)))$, where Hn(T) is the height of the tree. In this paper, we present two new distance labelling schemes that not only achieve the optimal label length ${\it \Theta}({\rm log} n \cdot {\rm log} (H_{n}(T)))$, but also have a much smaller expected label length under certain tree distributions. With these new schemes, we also can efficiently find the least common ancestor of any two vertices based on their labels only.