Succinct Ordinal Trees Based on Tree Covering
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Various methods have been used to represent a tree of n nodes in essentially the information-theoretic minimum space while supporting various navigational operations in constant time, but different representations usually support different operations. Our main contribution is a succinct representation of ordinal trees, based on that of Geary et al. (7), that supports all the navigational operations supported by various succinct tree representations while requiring only 2n + o(n) bits. It also supports efficient level-order traversal, a useful ordering previously supported only with a very limited set of operations (8).
Our second contribution expands on the notion of a single succinct representation supporting more than one traversal ordering, by showing that our method supports two other encoding schemes as abstract data types. In particular, it supports extracting a word (\(O(\lg n)\) bits) of the balanced parenthesis sequence (11) or depth first unary degree sequence (3;4) in O(f(n)) time, using at most n/f(n) + o(n) additional bits, for any f(n) in \(O(\lg n)\) and Ω(1).
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