The average number of linear extensions of a partial order
Journal of Combinatorial Theory Series A
Tree structure for distributive lattices and its applications
Theoretical Computer Science
An Efficient Data Structure for Lattice Operations
SIAM Journal on Computing
Introduction to algorithms
Succinct indexable dictionaries with applications to encoding k-ary trees and multisets
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Succinct Representation of Balanced Parentheses and Static Trees
SIAM Journal on Computing
Compact oracles for reachability and approximate distances in planar digraphs
Journal of the ACM (JACM)
Journal of the ACM (JACM)
Space-efficient static trees and graphs
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
Succinct Representations of Arbitrary Graphs
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Sorting and selection in posets
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Succinct Representation of Labeled Graphs
Algorithmica
Compact representation of posets
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
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We describe an algorithm for compressing a partially ordered set, or poset, so that it occupies space matching the information theory lower bound (to within lower order terms), in the worst case. Using this algorithm, we design a succinct data structure for representing a poset that, given two elements, can report whether one precedes the other in constant time. This is equivalent to succinctly representing the transitive closure graph of the poset, and we note that the same method can also be used to succinctly represent the transitive reduction graph. For an n element poset, the data structure occupies n2/4+o(n2) bits, in the worst case, which is roughly half the space occupied by an upper triangular matrix. Furthermore, a slight extension to this data structure yields a succinct oracle for reachability in arbitrary directed graphs. Thus, using roughly a quarter of the space required to represent an arbitrary directed graph, reachability queries can be supported in constant time.