Implicit representation of graphs
SIAM Journal on Discrete Mathematics
WWW '99 Proceedings of the eighth international conference on World Wide Web
Data on the Web: from relations to semistructured data and XML
Data on the Web: from relations to semistructured data and XML
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Compact labeling schemes for ancestor queries
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Proceedings of the thirteenth annual ACM symposium on Parallel algorithms and architectures
Improved labeling scheme for ancestor queries
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
A comparison of labeling schemes for ancestor queries
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
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
Labeling Schemes for Flow and Connectivity
SIAM Journal on Computing
Compact oracles for reachability and approximate distances in planar digraphs
Journal of the ACM (JACM)
Universal partial order represented by means of oriented trees and other simple graphs
European Journal of Combinatorics
Compact Labeling Scheme for Ancestor Queries
SIAM Journal on Computing
Informative labeling schemes for graphs
Theoretical Computer Science - Mathematical foundations of computer science 2000
Labeling schemes for vertex connectivity
ACM Transactions on Algorithms (TALG)
Compact ancestry labeling schemes for XML trees
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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In this paper, we solve the ancestry problem, which was introduced more than twenty years ago by Kannan et al. [STOC '88], and is among the most well-studied problems in the field of informative labeling schemes. Specifically, we construct an ancestry labeling scheme for n-node trees with label size log2 n + O(log log n) bits, thus matching the log2 n + Ω(log log n) bits lower bound given by Alstrup et al. [SODA '03]. Besides its optimal label size, our scheme assigns the labels in linear time, and guarantees that any ancestry query can be answered in constant time. In addition to its potential impact in terms of improving the performances of XML search engines, our ancestry scheme is also useful in the context of partially ordered sets. Specifically, for any fixed integer k, our scheme enables the construction of a universal poset of size O(nk log4k n) for the family of n-element posets with tree-dimension at most k. This bound is almost tight thanks to a lower bound of nk-o(1) due to Alon and Scheinerman [Order '88].